text
stringlengths 75
3.35M
| length
int64 36
1.04M
| original_index
int64 0
552k
|
|---|---|---|
# 1 chapter 4 curve plotting with matlab matlab provides some very powerful features for plotting and...
<MASK>
## Documents
<MASK>
MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
<MASK>
MATLAB has the capability to generate plots of many types. This includes linear plots, line plots, logarithmic plots on both scales, logarithmic plots on one scale, stem plots, bar graphs, and three-dimensional plots. We will be using these capabilities throughout the text, so the present development is intended as an introduction, with many operations to follow in later chapters.
<MASK>
In the two-dimensional plotting commands, the horizontal axis will be referred to as the x-axis and the vertical axis will be referred to as the y-axis. However, the actual variables can be labeled with any quantities. It is only in the plot commands that x and y are used.
<MASK>
>> x = x1:xstep:x2where x1=beginning point, x2=final point, and xstep=step size. Assuming that the final point coincides with an integer multiple of xstep, the number of points N is
2 1 1step
<MASK>
>> x = linspace(x1, x2, N)where x1=beginning point, x2=final point, and N=number of points. The name linspace represents “linear spacing”. Again, the number of points N is
<MASK>
>> t = 0:0.1:10;
<MASK>
object of Example 4-1 with grid.’)A grid is added.>> grid
<MASK>
21( ) 0.25f t t
<MASK>
time in Example 4-2.’)>> grid
<MASK>
Multiple Plots on Same Graph
<MASK>
24
<MASK>
ky Cx
<MASK>
27
<MASK>
<UNMASK>
# 1 chapter 4 curve plotting with matlab matlab provides some very powerful features for plotting and...
32
1 Chapter 4 Curve Plotting with MATLAB MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
<MASK>
## Documents
<MASK>
Chapter 4Curve Plotting with MATLAB
MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
<MASK>
MATLAB has the capability to generate plots of many types. This includes linear plots, line plots, logarithmic plots on both scales, logarithmic plots on one scale, stem plots, bar graphs, and three-dimensional plots. We will be using these capabilities throughout the text, so the present development is intended as an introduction, with many operations to follow in later chapters.
<MASK>
In the two-dimensional plotting commands, the horizontal axis will be referred to as the x-axis and the vertical axis will be referred to as the y-axis. However, the actual variables can be labeled with any quantities. It is only in the plot commands that x and y are used.
6
<MASK>
7
<MASK>
>> x = x1:xstep:x2where x1=beginning point, x2=final point, and xstep=step size. Assuming that the final point coincides with an integer multiple of xstep, the number of points N is
2 1 1step
x xN
<MASK>
>> x = linspace(x1, x2, N)where x1=beginning point, x2=final point, and N=number of points. The name linspace represents “linear spacing”. Again, the number of points N is
<MASK>
10
<MASK>
Example 4-1. Continuation.
<MASK>
>> t = 0:0.1:10;
Alternately,
<MASK>
object of Example 4-1 with grid.’)A grid is added.>> grid
<MASK>
21( ) 0.25f t t
<MASK>
time in Example 4-2.’)>> grid
<MASK>
Multiple Plots on Same Graph
<MASK>
24
<MASK>
ky Cx
<MASK>
27
<MASK>
Bar and Stem Plots
<MASK>
>> stem (x, y)
<MASK>
>> stem (year, sales)
<MASK>
<UNMASK>
# 1 chapter 4 curve plotting with matlab matlab provides some very powerful features for plotting and...
32
1 Chapter 4 Curve Plotting with MATLAB MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
<MASK>
Category:
## Documents
TRANSCRIPT
<MASK>
Chapter 4Curve Plotting with MATLAB
MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
<MASK>
MATLAB has the capability to generate plots of many types. This includes linear plots, line plots, logarithmic plots on both scales, logarithmic plots on one scale, stem plots, bar graphs, and three-dimensional plots. We will be using these capabilities throughout the text, so the present development is intended as an introduction, with many operations to follow in later chapters.
3
Vector Lengths
<MASK>
4
<MASK>
If the vectors have different lengths, it is possible to use a portion of the longer one as one of the variables. For example, suppose y has 200 values and x has 120 values. One could define y1 by the following command:>> y1 = y(1:120)The variable y1 now has the same number of points as x and the two could be plotted together.
<MASK>
In the two-dimensional plotting commands, the horizontal axis will be referred to as the x-axis and the vertical axis will be referred to as the y-axis. However, the actual variables can be labeled with any quantities. It is only in the plot commands that x and y are used.
6
<MASK>
Whenever a plot is to be created from an equation, and linear plots for both the dependent and independent variables are desired, the most convenient way to achieve the result is to create a linear array or vector for the values of the independent variable. MATLAB offers a number of different commands that can be used for this purpose. For this explanation, assume that the independent variable is x.
7
Command for Linear Array
>> x = x1:xstep:x2where x1=beginning point, x2=final point, and xstep=step size. Assuming that the final point coincides with an integer multiple of xstep, the number of points N is
2 1 1step
x xN
<MASK>
Alternate Command for Linear Array
>> x = linspace(x1, x2, N)where x1=beginning point, x2=final point, and N=number of points. The name linspace represents “linear spacing”. Again, the number of points N is
<MASK>
x
<MASK>
9.8v tUse MATLAB to plot the velocity over a time interval from 0 to 10 s.
10
<MASK>
It should be emphasized that this is a simple linear equation with a vertical intercept of 0 so we actually need only two points to plot the curve. However, our purpose is to learn how to use MATLAB for plotting and we will utilize far more points than necessary as a learning process.
<MASK>
Example 4-1. Continuation.
<MASK>
>> t = 0:0.1:10;
Alternately,
<MASK>
12
Example 4-1. Continuation. We can inspect various values of t.
>> t(1:5)
ans =
<MASK>
13
<MASK>
>> v = 9.8*t;This command generates 101 values of v corresponding to the 101 values of t. It can be plotted by the command>> plot(t, v)The result is a “raw” plot but various labels can be added as will be shown on the next slide.
<MASK>
Example 4-1. Continuation.
A horizontal label is provided.>> xlabel(‘Time, seconds’)A vertical label is provided.>> ylabel(‘Velocity, meters/second’)A title is provided.>> title(‘Figure 4-3. Velocity of falling
object of Example 4-1 with grid.’)A grid is added.>> grid
<MASK>
17
<MASK>
21( ) 0.25f t t
Assume 101-point t vector is in memory.>> f1 = 0.25*t.*t; or>> f1 = 0.25*t.^2:>> plot(t, f1)>> xlabel(‘Time, seconds’)>> ylabel(‘Force, newtons’)>> title(‘Figure 4-4. Force as a function of
time in Example 4-2.’)>> grid
<MASK>
19
Example 4-3. A force in newtons (N) is given below. Plot the function.
22 ( ) 25 0.25f t t
Assume 101-point t-vector is in memory.>> f2 = 25+0.25*t.^2; >> plot(t, f2)>> xlabel(‘Time, seconds’)>> ylabel(‘Force, newtons’)>> title(‘Figure 4-6. Second force as initially
obtained in Example 4-3.’)>> grid
20
<MASK>
Plot is modified by the command
<MASK>
22
23
Multiple Plots on Same Graph
The two functions f1 and f2 of the previous two examples can be plotted on the same graph by the command>> plot(t, f1, t, f2)The command gtext(‘label’) allows a label to placed on a graph using crosshairs. The resulting functions are shown on the next slide.
24
<MASK>
ky Cx
<MASK>
' ' 'y mx b
<MASK>
Example 4-5. Plot the 2nd degree function below on a log-log graph.
2y x>> x = logspace(-1, 1, 100);>> y = x.^2;>> loglog(x, y)A grid and additional labeling were provided and the curve is shown on the next slide.
27
<MASK>
Bar and Stem Plots
<MASK>
Command for a stem plot:
>> stem (x, y)
<MASK>
>> year = 1993:2002;>> sales = [ the 10 values in the text];>> bar(year, sales)The graph with additional labeling is shown on the next slide.
<MASK>
>> stem (year, sales)
The plot with additional labeling is shown on the next slide.
<MASK>
<UNMASK>
# 1 chapter 4 curve plotting with matlab matlab provides some very powerful features for plotting and...
32
1 Chapter 4 Curve Plotting with MATLAB MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
Post on 14-Dec-2015
244 views
Category:
## Documents
TRANSCRIPT
1
Chapter 4Curve Plotting with MATLAB
MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
<MASK>
MATLAB has the capability to generate plots of many types. This includes linear plots, line plots, logarithmic plots on both scales, logarithmic plots on one scale, stem plots, bar graphs, and three-dimensional plots. We will be using these capabilities throughout the text, so the present development is intended as an introduction, with many operations to follow in later chapters.
3
Vector Lengths
A very important fact that should be emphasized at the outset is that to plot one vector against another, the vectors must have the same number of elements. One can plot either a column vector or a row vector versus either a column vector or a row vector provided they have the same number of values.
4
Different Vector Lengths
If the vectors have different lengths, it is possible to use a portion of the longer one as one of the variables. For example, suppose y has 200 values and x has 120 values. One could define y1 by the following command:>> y1 = y(1:120)The variable y1 now has the same number of points as x and the two could be plotted together.
<MASK>
In the two-dimensional plotting commands, the horizontal axis will be referred to as the x-axis and the vertical axis will be referred to as the y-axis. However, the actual variables can be labeled with any quantities. It is only in the plot commands that x and y are used.
6
Creating a Linear Array
Whenever a plot is to be created from an equation, and linear plots for both the dependent and independent variables are desired, the most convenient way to achieve the result is to create a linear array or vector for the values of the independent variable. MATLAB offers a number of different commands that can be used for this purpose. For this explanation, assume that the independent variable is x.
7
Command for Linear Array
>> x = x1:xstep:x2where x1=beginning point, x2=final point, and xstep=step size. Assuming that the final point coincides with an integer multiple of xstep, the number of points N is
2 1 1step
x xN
x
<MASK>
Alternate Command for Linear Array
>> x = linspace(x1, x2, N)where x1=beginning point, x2=final point, and N=number of points. The name linspace represents “linear spacing”. Again, the number of points N is
<MASK>
x xN
x
9
<MASK>
9.8v tUse MATLAB to plot the velocity over a time interval from 0 to 10 s.
10
Example 4-1. Continuation.
It should be emphasized that this is a simple linear equation with a vertical intercept of 0 so we actually need only two points to plot the curve. However, our purpose is to learn how to use MATLAB for plotting and we will utilize far more points than necessary as a learning process.
11
Example 4-1. Continuation.
A time step of 0.1 s will be selected.
>> t = 0:0.1:10;
Alternately,
>> t = linspace(0,10,101);
12
Example 4-1. Continuation. We can inspect various values of t.
>> t(1:5)
ans =
0 0.1000 0.2000 0.3000 0.4000
13
<MASK>
>> v = 9.8*t;This command generates 101 values of v corresponding to the 101 values of t. It can be plotted by the command>> plot(t, v)The result is a “raw” plot but various labels can be added as will be shown on the next slide.
<MASK>
15
Example 4-1. Continuation.
A horizontal label is provided.>> xlabel(‘Time, seconds’)A vertical label is provided.>> ylabel(‘Velocity, meters/second’)A title is provided.>> title(‘Figure 4-3. Velocity of falling
object of Example 4-1 with grid.’)A grid is added.>> grid
<MASK>
17
Example 4-2. A force in newtons (N) is given below. Plot the function.
21( ) 0.25f t t
Assume 101-point t vector is in memory.>> f1 = 0.25*t.*t; or>> f1 = 0.25*t.^2:>> plot(t, f1)>> xlabel(‘Time, seconds’)>> ylabel(‘Force, newtons’)>> title(‘Figure 4-4. Force as a function of
time in Example 4-2.’)>> grid
18
19
Example 4-3. A force in newtons (N) is given below. Plot the function.
22 ( ) 25 0.25f t t
Assume 101-point t-vector is in memory.>> f2 = 25+0.25*t.^2; >> plot(t, f2)>> xlabel(‘Time, seconds’)>> ylabel(‘Force, newtons’)>> title(‘Figure 4-6. Second force as initially
obtained in Example 4-3.’)>> grid
20
21
<MASK>
Plot is modified by the command
>> axis([0 10 0 50])
22
23
Multiple Plots on Same Graph
The two functions f1 and f2 of the previous two examples can be plotted on the same graph by the command>> plot(t, f1, t, f2)The command gtext(‘label’) allows a label to placed on a graph using crosshairs. The resulting functions are shown on the next slide.
24
<MASK>
ky Cx
10 10 10 10log log ( ) log logky Cx C k x
' ' 'y mx b
26
Example 4-5. Plot the 2nd degree function below on a log-log graph.
2y x>> x = logspace(-1, 1, 100);>> y = x.^2;>> loglog(x, y)A grid and additional labeling were provided and the curve is shown on the next slide.
27
28
Bar and Stem Plots
<MASK>
Command for a stem plot:
>> stem (x, y)
29
Example 4-6. The text contains the sales in thousands of dollars for a small business from 1993 through 2002. Construct a bar graph.
>> year = 1993:2002;>> sales = [ the 10 values in the text];>> bar(year, sales)The graph with additional labeling is shown on the next slide.
30
31
Example 4-7. Plot the data of the previous example using a stem plot.
Assume that the variables year and sales are still in memory. The command is
>> stem (year, sales)
The plot with additional labeling is shown on the next slide.
<MASK>
<UNMASK>
# 1 chapter 4 curve plotting with matlab matlab provides some very powerful features for plotting and...
32
1 Chapter 4 Curve Plotting with MATLAB MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
Post on 14-Dec-2015
244 views
Category:
## Documents
TRANSCRIPT
1
Chapter 4Curve Plotting with MATLAB
MATLAB provides some very powerful features for plotting and labeling curves. These operations can be performed as part of an overall mathematical analysis, or experimental data may be provided to the program for the primary purpose of plotting. Curves obtained from MATLAB plots can be exported to other programs for presentation purposes.
2
MATLAB has the capability to generate plots of many types. This includes linear plots, line plots, logarithmic plots on both scales, logarithmic plots on one scale, stem plots, bar graphs, and three-dimensional plots. We will be using these capabilities throughout the text, so the present development is intended as an introduction, with many operations to follow in later chapters.
3
Vector Lengths
A very important fact that should be emphasized at the outset is that to plot one vector against another, the vectors must have the same number of elements. One can plot either a column vector or a row vector versus either a column vector or a row vector provided they have the same number of values.
4
Different Vector Lengths
If the vectors have different lengths, it is possible to use a portion of the longer one as one of the variables. For example, suppose y has 200 values and x has 120 values. One could define y1 by the following command:>> y1 = y(1:120)The variable y1 now has the same number of points as x and the two could be plotted together.
5
The Variables x and y
In the two-dimensional plotting commands, the horizontal axis will be referred to as the x-axis and the vertical axis will be referred to as the y-axis. However, the actual variables can be labeled with any quantities. It is only in the plot commands that x and y are used.
6
Creating a Linear Array
Whenever a plot is to be created from an equation, and linear plots for both the dependent and independent variables are desired, the most convenient way to achieve the result is to create a linear array or vector for the values of the independent variable. MATLAB offers a number of different commands that can be used for this purpose. For this explanation, assume that the independent variable is x.
7
Command for Linear Array
>> x = x1:xstep:x2where x1=beginning point, x2=final point, and xstep=step size. Assuming that the final point coincides with an integer multiple of xstep, the number of points N is
2 1 1step
x xN
x
8
Alternate Command for Linear Array
>> x = linspace(x1, x2, N)where x1=beginning point, x2=final point, and N=number of points. The name linspace represents “linear spacing”. Again, the number of points N is
2 1 1step
x xN
x
9
Example 4-1. When air resistance can be ignored, the velocity (in m/s) of an object falling from rest is
9.8v tUse MATLAB to plot the velocity over a time interval from 0 to 10 s.
10
Example 4-1. Continuation.
It should be emphasized that this is a simple linear equation with a vertical intercept of 0 so we actually need only two points to plot the curve. However, our purpose is to learn how to use MATLAB for plotting and we will utilize far more points than necessary as a learning process.
11
Example 4-1. Continuation.
A time step of 0.1 s will be selected.
>> t = 0:0.1:10;
Alternately,
>> t = linspace(0,10,101);
12
Example 4-1. Continuation. We can inspect various values of t.
>> t(1:5)
ans =
0 0.1000 0.2000 0.3000 0.4000
13
Example 4-1. Continuation.
>> v = 9.8*t;This command generates 101 values of v corresponding to the 101 values of t. It can be plotted by the command>> plot(t, v)The result is a “raw” plot but various labels can be added as will be shown on the next slide.
14
15
Example 4-1. Continuation.
A horizontal label is provided.>> xlabel(‘Time, seconds’)A vertical label is provided.>> ylabel(‘Velocity, meters/second’)A title is provided.>> title(‘Figure 4-3. Velocity of falling
object of Example 4-1 with grid.’)A grid is added.>> grid
16
17
Example 4-2. A force in newtons (N) is given below. Plot the function.
21( ) 0.25f t t
Assume 101-point t vector is in memory.>> f1 = 0.25*t.*t; or>> f1 = 0.25*t.^2:>> plot(t, f1)>> xlabel(‘Time, seconds’)>> ylabel(‘Force, newtons’)>> title(‘Figure 4-4. Force as a function of
time in Example 4-2.’)>> grid
18
19
Example 4-3. A force in newtons (N) is given below. Plot the function.
22 ( ) 25 0.25f t t
Assume 101-point t-vector is in memory.>> f2 = 25+0.25*t.^2; >> plot(t, f2)>> xlabel(‘Time, seconds’)>> ylabel(‘Force, newtons’)>> title(‘Figure 4-6. Second force as initially
obtained in Example 4-3.’)>> grid
20
21
Example 4-3. Continuation.
Plot is modified by the command
>> axis([0 10 0 50])
22
23
Multiple Plots on Same Graph
The two functions f1 and f2 of the previous two examples can be plotted on the same graph by the command>> plot(t, f1, t, f2)The command gtext(‘label’) allows a label to placed on a graph using crosshairs. The resulting functions are shown on the next slide.
24
25
Log-Log Plots
ky Cx
10 10 10 10log log ( ) log logky Cx C k x
' ' 'y mx b
26
Example 4-5. Plot the 2nd degree function below on a log-log graph.
2y x>> x = logspace(-1, 1, 100);>> y = x.^2;>> loglog(x, y)A grid and additional labeling were provided and the curve is shown on the next slide.
27
28
Bar and Stem Plots
Command for a bar plot:>> bar (x, y)
Command for a stem plot:
>> stem (x, y)
29
Example 4-6. The text contains the sales in thousands of dollars for a small business from 1993 through 2002. Construct a bar graph.
>> year = 1993:2002;>> sales = [ the 10 values in the text];>> bar(year, sales)The graph with additional labeling is shown on the next slide.
30
31
Example 4-7. Plot the data of the previous example using a stem plot.
Assume that the variables year and sales are still in memory. The command is
>> stem (year, sales)
The plot with additional labeling is shown on the next slide.
32
| 5,689 | 6,699,407 |
# Given sinx=1/4 and cosx=3/4 what is sec2x ?
We are given that sin x = 1/4 and cos x = 3/4. We have to find sec 2x.
sec 2x = 1/ cos 2x
cos 2x = (cos x)^2 - (sin x)^2
=> (3/4)^2 - (1/4)^2
=> (9 - 1)/ 16
=> 8/ 16
=> 1/2
So sec 2x = 2
Approved by eNotes Editorial Team
Posted on
| 137 | 6,699,408 |
<MASK>
Letting N(n) give the nth nonagonal number and T(n) the nth triangular number,
${7N(n) + 3 = T(7n - 3)}.$
<MASK>
<UNMASK>
# Nonagonal number
<MASK>
1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364, 6666, 6975, 7291, 7614, 7944, 8281, 8625, 8976, 9334, 9699. (sequence A001106 in OEIS)
<MASK>
Letting N(n) give the nth nonagonal number and T(n) the nth triangular number,
${7N(n) + 3 = T(7n - 3)}.$
<MASK>
<UNMASK>
# Nonagonal number
<MASK>
1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364, 6666, 6975, 7291, 7614, 7944, 8281, 8625, 8976, 9334, 9699. (sequence A001106 in OEIS)
<MASK>
Letting N(n) give the nth nonagonal number and T(n) the nth triangular number,
${7N(n) + 3 = T(7n - 3)}.$
<MASK>
$x = \frac{\sqrt{56n+25}+5}{14}.$
<MASK>
<UNMASK>
# Nonagonal number
<MASK>
The first few nonagonal numbers are:
1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364, 6666, 6975, 7291, 7614, 7944, 8281, 8625, 8976, 9334, 9699. (sequence A001106 in OEIS)
The parity of nonagonal numbers follows the pattern odd-odd-even-even.
Letting N(n) give the nth nonagonal number and T(n) the nth triangular number,
${7N(n) + 3 = T(7n - 3)}.$
<MASK>
$x = \frac{\sqrt{56n+25}+5}{14}.$
<MASK>
<UNMASK>
# Nonagonal number
A nonagonal number or enneagonal number is a polygonal number that represents a nonagon. The nonagonal number for n is given by the formula:
$\frac {n(7n - 5)}{2}.$
The first few nonagonal numbers are:
1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364, 6666, 6975, 7291, 7614, 7944, 8281, 8625, 8976, 9334, 9699. (sequence A001106 in OEIS)
The parity of nonagonal numbers follows the pattern odd-odd-even-even.
Letting N(n) give the nth nonagonal number and T(n) the nth triangular number,
${7N(n) + 3 = T(7n - 3)}.$
## Test for nonagonal numbers
$x = \frac{\sqrt{56n+25}+5}{14}.$
If x is an integer, then n is the xth nonagonal number. If x is not an integer, then n is not nonagonal.
| 1,248 | 6,699,409 |
It is currently 23 Jan 2018, 09:42
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# If 3^m=81
Author Message
TAGS:
Moderator
Joined: 18 Apr 2015
Posts: 2684
Followers: 39
Kudos [?]: 387 [0], given: 1773
If 3^m=81 [#permalink] 08 Oct 2017, 00:49
Expert's post
00:00
Question Stats:
0% (00:00) correct 0% (00:00) wrong based on 0 sessions
If 3^m = 81 then m^3 =
A. 9
B. 16
C. 27
D. 54
E. 64
[Reveal] Spoiler: OA
_________________
Senior Manager
Joined: 20 Apr 2016
Posts: 323
Followers: 1
Kudos [?]: 202 [0], given: 55
Re: If 3^m=81 [#permalink] 09 Oct 2017, 01:45
Carcass wrote:
If 3^m = 81 then m^3 =
A. 9
B. 16
C. 27
D. 54
E. 64
we can re write as -
3^m = 81 = 3^4[/m]
Therefore m = 3
Now
m^3 = 4^3 =64
_________________
If you found this post useful, please let me know by pressing the Kudos Button
Director
Joined: 03 Sep 2017
Posts: 522
Followers: 0
Kudos [?]: 231 [0], given: 66
Re: If 3^m=81 [#permalink] 09 Oct 2017, 07:00
pranab01 wrote:
Carcass wrote:
If 3^m = 81 then m^3 =
A. 9
B. 16
C. 27
D. 54
E. 64
we can re write as -
3^m = 81 = 3^4
Therefore m = 3
Now
m^3 = 4^3 =64
There is a typo.
Rewriting the equation as 3^m = 3^4, it results that m=4.
Then, what follow is right. m^3 = 4^3 =64.
Re: If 3^m=81 [#permalink] 09 Oct 2017, 07:00
Display posts from previous: Sort by
| 668 | 6,699,410 |
Search a number
4472 = 231343
BaseRepresentation
bin1000101111000
320010122
41011320
5120342
632412
716016
oct10570
96118
104472
1133a6
122708
132060
1418b6
1514d2
hex1178
<MASK>
It is an interprime number because it is at equal distance from previous prime (4463) and next prime (4481).
<MASK>
It is a plaindrome in base 16.
<MASK>
4472 is a wasteful number, since it uses less digits than its factorization.
<MASK>
The sum of its prime factors is 62 (or 58 counting only the distinct ones).
<MASK>
<UNMASK>
Search a number
4472 = 231343
BaseRepresentation
bin1000101111000
320010122
41011320
5120342
632412
716016
oct10570
96118
104472
1133a6
122708
132060
1418b6
1514d2
hex1178
<MASK>
The previous prime is 4463. The next prime is 4481. The reversal of 4472 is 2744.
It is an interprime number because it is at equal distance from previous prime (4463) and next prime (4481).
<MASK>
It is a plaindrome in base 16.
It is a self number, because there is not a number n which added to its sum of digits gives 4472.
<MASK>
It is an amenable number.
It is a practical number, because each smaller number is the sum of distinct divisors of 4472, and also a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (4620).
<MASK>
4472 is a wasteful number, since it uses less digits than its factorization.
<MASK>
The sum of its prime factors is 62 (or 58 counting only the distinct ones).
The product of its digits is 224, while the sum is 17.
<MASK>
<UNMASK>
Search a number
4472 = 231343
BaseRepresentation
bin1000101111000
320010122
41011320
5120342
632412
716016
oct10570
96118
104472
1133a6
122708
132060
1418b6
1514d2
hex1178
<MASK>
The previous prime is 4463. The next prime is 4481. The reversal of 4472 is 2744.
It is an interprime number because it is at equal distance from previous prime (4463) and next prime (4481).
<MASK>
It is a plaindrome in base 16.
It is a self number, because there is not a number n which added to its sum of digits gives 4472.
<MASK>
It is an inconsummate number, since it does not exist a number n which divided by its sum of digits gives 4472.
It is an unprimeable number.
<MASK>
It is an amenable number.
It is a practical number, because each smaller number is the sum of distinct divisors of 4472, and also a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (4620).
<MASK>
4472 is a wasteful number, since it uses less digits than its factorization.
<MASK>
The sum of its prime factors is 62 (or 58 counting only the distinct ones).
The product of its digits is 224, while the sum is 17.
<MASK>
Adding to 4472 its sum of digits (17), we get a square (4489 = 672).
<MASK>
<UNMASK>
Search a number
4472 = 231343
BaseRepresentation
bin1000101111000
320010122
41011320
5120342
632412
716016
oct10570
96118
104472
1133a6
122708
132060
1418b6
1514d2
hex1178
4472 has 16 divisors (see below), whose sum is σ = 9240. Its totient is φ = 2016.
The previous prime is 4463. The next prime is 4481. The reversal of 4472 is 2744.
It is an interprime number because it is at equal distance from previous prime (4463) and next prime (4481).
It is a Smith number, since the sum of its digits (17) coincides with the sum of the digits of its prime factors.
It is a plaindrome in base 16.
It is a self number, because there is not a number n which added to its sum of digits gives 4472.
<MASK>
It is an inconsummate number, since it does not exist a number n which divided by its sum of digits gives 4472.
It is an unprimeable number.
<MASK>
It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 83 + ... + 125.
It is an amenable number.
It is a practical number, because each smaller number is the sum of distinct divisors of 4472, and also a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (4620).
4472 is an abundant number, since it is smaller than the sum of its proper divisors (4768).
<MASK>
4472 is a wasteful number, since it uses less digits than its factorization.
<MASK>
The sum of its prime factors is 62 (or 58 counting only the distinct ones).
The product of its digits is 224, while the sum is 17.
<MASK>
Adding to 4472 its sum of digits (17), we get a square (4489 = 672).
Subtracting from 4472 its reverse (2744), we obtain a cube (1728 = 123).
The spelling of 4472 in words is "four thousand, four hundred seventy-two", and thus it is an iban number.
<UNMASK>
Search a number
4472 = 231343
BaseRepresentation
bin1000101111000
320010122
41011320
5120342
632412
716016
oct10570
96118
104472
1133a6
122708
132060
1418b6
1514d2
hex1178
4472 has 16 divisors (see below), whose sum is σ = 9240. Its totient is φ = 2016.
The previous prime is 4463. The next prime is 4481. The reversal of 4472 is 2744.
It is an interprime number because it is at equal distance from previous prime (4463) and next prime (4481).
It is a Smith number, since the sum of its digits (17) coincides with the sum of the digits of its prime factors.
It is a plaindrome in base 16.
It is a self number, because there is not a number n which added to its sum of digits gives 4472.
It is a congruent number.
It is an inconsummate number, since it does not exist a number n which divided by its sum of digits gives 4472.
It is an unprimeable number.
4472 is an untouchable number, because it is not equal to the sum of proper divisors of any number.
It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 83 + ... + 125.
It is an amenable number.
It is a practical number, because each smaller number is the sum of distinct divisors of 4472, and also a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (4620).
4472 is an abundant number, since it is smaller than the sum of its proper divisors (4768).
It is a pseudoperfect number, because it is the sum of a subset of its proper divisors.
4472 is a wasteful number, since it uses less digits than its factorization.
4472 is an evil number, because the sum of its binary digits is even.
The sum of its prime factors is 62 (or 58 counting only the distinct ones).
The product of its digits is 224, while the sum is 17.
The square root of 4472 is about 66.8730139892. The cubic root of 4472 is about 16.4753227686.
Adding to 4472 its sum of digits (17), we get a square (4489 = 672).
Subtracting from 4472 its reverse (2744), we obtain a cube (1728 = 123).
The spelling of 4472 in words is "four thousand, four hundred seventy-two", and thus it is an iban number.
| 1,907 | 6,699,411 |
Home > Percent Error > What Is The Equation Used To Calculate Percentage Error
# What Is The Equation Used To Calculate Percentage Error
## Contents
Solve for the measured or observed value.Note due to the absolute value in the actual equation (above) there are two solutions. You measure the sides of the cube to find the volume and weigh it to find its mass. The difference between two measurements is called a variation in the measurements. Solve for percent error Solve for the actual value. news
Looking at the measuring device from a left or right angle will give an incorrect value. 3. The absolute error of the measurement shows how large the error actually is, while the relative error of the measurement shows how large the error is in relation to the correct Answer this question Flag as... Trending Are hammers edible? 21 answers Round to 2 significant figures? 5 answers Why is it that when things get wet they get darker, even though water is clear? 6 answers
## Percent Error Calculator
The precision is said to be the same as the smallest fractional or decimal division on the scale of the measuring instrument. Reply ↓ Leave a Reply Cancel reply Search for: Get the Science Notes Newsletter Get Projects Free in Email Top Posts & Pages Printable Periodic Tables List of Electron Configurations of Simply multiply the result, 0.1, by 100. Tolerance intervals: Error in measurement may be represented by a tolerance interval (margin of error).
The precision of a measuring instrument is determined by the smallest unit to which it can measure. 2. Flag as duplicate Thanks! Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework Solutions, Worksheets, Help, Data and Information for Engineers, browse this site Reply ↓ Todd Helmenstine Post authorJanuary 28, 2016 at 2:15 pm Thanks for pointing that out.
Ways of Expressing Error in Measurement: 1. Percent Error Worksheet The percent of error is found by multiplying the relative error by 100%. b.) the relative error in the measured length of the field. How to Calculate Here is the way to calculate a percentage error: Step 1: Calculate the error (subtract one value form the other) ignore any minus sign.
## Percent Error Chemistry
When weighed on a defective scale, he weighed 38 pounds. (a) What is the percent of error in measurement of the defective scale to the nearest tenth? (b) If Millie, the https://answers.yahoo.com/question/index?qid=20081015164014AA1BwJE In many situations, the true values are unknown. Percent Error Calculator Yes No Not Helpful 0 Helpful 0 Unanswered Questions How can I find the value of capital a-hypothetical? Can Percent Error Be Negative How would you prepare 1.5 litres of 0.257 molar pottasium bromide from solid pottasium bromide?
In this case, the real value is 10 and the estimated value is 9. navigate to this website Tips Some teachers like the percent error to be rounded to a certain point; most people will be satisfied with the percent error rounded to three significant digits. Please select a newsletter. Any links or some thing would be great.? Negative Percent Error
The theoreticalvalue (using physics formulas)is 0.64 seconds. The difference between the actual and experimental value is always the absolute value of the difference. |Experimental-Actual|/Actualx100 so it doesn't matter how you subtract. Percent error or percentage error expresses as a percentage the difference between an approximate or measured value and an exact or known value. More about the author We will be working with relative error.
Comparing Approximate to Exact "Error": Subtract Approximate value from Exact value. What Is A Good Percent Error Formulas for percent error and human error? About Todd HelmenstineTodd Helmenstine is the physicist/mathematician who creates most of the images and PDF files found on sciencenotes.org.
## Ex:-1/10 = -0.1 4 Find the absolute value of the result.
And we can use Percentage Error to estimate the possible error when measuring. Did you mean ? While both situations show an absolute error of 1 cm., the relevance of the error is very different. Significant Figures Definition Chemistry You can only upload files of type PNG, JPG, or JPEG.
Ex: 10 - 9 = 1 3 Divide the result by the real number. Flag as... Any measurements within this range are "tolerated" or perceived as correct. http://maxspywareremover.com/percent-error/what-is-percentage-error-equation.php However if percent error is equal to 100 percent or -100 percent, then there is only one calculated solution and one solution of infinity.
Math CalculatorsScientificFractionPercentageTimeTriangleVolumeNumber SequenceMore Math CalculatorsFinancial | Weight Loss | Math | Pregnancy | Other about us | sitemap © 2008 - 2016 calculator.net Error in Measurement Topic Index | Algebra Index The result of the difference is positive and therefore the percent error is positive. Repeat the same measure several times to get a good average value. 4. If you want to know how to calculate percentage error, all you need to know is the approximate and exact value and you'll be on your way.
This will convert the answer into percent form. Whether error is positive or negative is important. Absolute Error: Absolute error is simply the amount of physical error in a measurement. Write an Article 138 ⌂HomeMailSearchNewsSportsFinanceCelebrityWeatherAnswersFlickrMobileMore⋁PoliticsMoviesMusicTVGroupsStyleBeautyTechShopping Yahoo Answers 👤 Sign in ✉ Mail ⚙ Help Account Info Help Suggestions Send Feedback Answers Home All Categories Arts & Humanities Beauty & Style Business
| 1,215 | 6,699,412 |
<MASK>
<UNMASK>
<MASK>
by
<MASK>
(1) The x-intercept is the point at which the parabola crosses the x-axis. Using Graphing Calculator 3.5, we should trace along the graph and find the point where the y-value equals 0.
<MASK>
Now we translate the graph of vertically so that the y-coordinate can be -5.125. In fact, the graph should be shifted down by 2.25 units which are the difference of the y-coordinates (5.125 - 2.875= 2.25). Then the graph of has the same vertex as the graph of .
<UNMASK>
Translations of a parabola
by
<MASK>
(1) The x-intercept is the point at which the parabola crosses the x-axis. Using Graphing Calculator 3.5, we should trace along the graph and find the point where the y-value equals 0.
We have two x-intercepts, (0.85078106, 0) and (-2.3507811, 0), of the graph.
(2) The y-intercept is the point at which the parabola crosses the y-axis. Using Graphing Calculator 3.5, we should trace along the graph and find the point where the x-value equals 0.
<MASK>
The graph above shows that the vertex of the parabola is (-0.75, -0.5125).
<MASK>
Therefore the x-coordinate of the vertex of the parabola of is always negative. The animation below shows an example for -5 < n < 0.75.
<MASK>
To sum up, for a < 0.75 and b > 5.125 has the vertex of the graph in the second quadrant.
(III) Change the equation to produce a graph concave down that shares the same vertex.
<MASK>
Now we translate the graph of vertically so that the y-coordinate can be -5.125. In fact, the graph should be shifted down by 2.25 units which are the difference of the y-coordinates (5.125 - 2.875= 2.25). Then the graph of has the same vertex as the graph of .
<UNMASK>
Translations of a parabola
by
Hee Jung Kim
<MASK>
(1) The x-intercept is the point at which the parabola crosses the x-axis. Using Graphing Calculator 3.5, we should trace along the graph and find the point where the y-value equals 0.
We have two x-intercepts, (0.85078106, 0) and (-2.3507811, 0), of the graph.
(2) The y-intercept is the point at which the parabola crosses the y-axis. Using Graphing Calculator 3.5, we should trace along the graph and find the point where the x-value equals 0.
<MASK>
The graph above shows that the vertex of the parabola is (-0.75, -0.5125).
<MASK>
Therefore the x-coordinate of the vertex of the parabola of is always negative. The animation below shows an example for -5 < n < 0.75.
<MASK>
To sum up, for a < 0.75 and b > 5.125 has the vertex of the graph in the second quadrant.
(III) Change the equation to produce a graph concave down that shares the same vertex.
<MASK>
We need to translate the graph of to get the same vertex as the one of . To get the x-coordinate of -0.75 from the x-coordinate of 0.75, the graph should be horizontally shifted to the left by 1.5 units. Therefore is the graph which has the same x-coordinate as the x-coordinate of the vertex of .
Now we translate the graph of vertically so that the y-coordinate can be -5.125. In fact, the graph should be shifted down by 2.25 units which are the difference of the y-coordinates (5.125 - 2.875= 2.25). Then the graph of has the same vertex as the graph of .
<UNMASK>
Translations of a parabola
by
Hee Jung Kim
Let's graph the function .
The graph is a parabola which opens upward with following features:
(1) The x-intercept is the point at which the parabola crosses the x-axis. Using Graphing Calculator 3.5, we should trace along the graph and find the point where the y-value equals 0.
We have two x-intercepts, (0.85078106, 0) and (-2.3507811, 0), of the graph.
(2) The y-intercept is the point at which the parabola crosses the y-axis. Using Graphing Calculator 3.5, we should trace along the graph and find the point where the x-value equals 0.
(3) One of important features of a parabola is the vertex which is defined by the highest point or the lowest point of the parabola. If a parabola opens upward, the y-value of the vertex is the minimum value of the function. If a parabola opens downward, the y-value of the vertex is the maximum value.
The graph above shows that the vertex of the parabola is (-0.75, -0.5125).
(4) We can easily see that the graph is symmetric with respect to the vertical line x = -0.75 which is called the axis of symmetry.
(I) Now we consider as .When we overlay a new graph replacing each (x - 0) by (x - 4),
we find that the graph is the same as , except that is horizontally shifted to the right by 4 units along the x-axis. Therefore, the x-intercepts, the x-coordinate of the vertex, and the axis of symmetry are all shifted to the right by 4 units along the x-axis. Note that there is no change in the y-coordinate of the vertex. In general, is a parabola which horizontally shifted to the right (a > 0) or to the left (a < 0) by a units along the x-axis from the graph of .
(II) Then how can we change the equation to move the vertex of the graph into the second quadrant? The points (x, y) consists of the real values x and y such that x < 0 and y > 0. The keys to the answer are the vertex of the parabola is (-0.75, -5.125) and translations. First, using the horizontal shift of the graph along the x-axis, we can make the x-coordinate of the vertex negative. The x-coordinate of the vertex of the parabola of is 0.
Therefore the x-coordinate of the vertex of the parabola of is always negative. The animation below shows an example for -5 < n < 0.75.
Second, using the vertical shift of the graph along the y-axis, we can make the y-coordinate of the vertex positive. The y-coordinate of the vertex of the parabola is 0.
Therefore the x-coordinate of the vertex of the parabola of is always positive. The animation below shows an example for 5.125 < n < 8.
To sum up, for a < 0.75 and b > 5.125 has the vertex of the graph in the second quadrant.
(III) Change the equation to produce a graph concave down that shares the same vertex.
Since the concavity depends on the sign of the coefficient of (concave up for positive, negative for concave down), we can try to change the sign of . However, the vertex of the parabola is (-0.75, -0.5125) and the vertex of the parabola is (0.75, -2.875).
We need to translate the graph of to get the same vertex as the one of . To get the x-coordinate of -0.75 from the x-coordinate of 0.75, the graph should be horizontally shifted to the left by 1.5 units. Therefore is the graph which has the same x-coordinate as the x-coordinate of the vertex of .
Now we translate the graph of vertically so that the y-coordinate can be -5.125. In fact, the graph should be shifted down by 2.25 units which are the difference of the y-coordinates (5.125 - 2.875= 2.25). Then the graph of has the same vertex as the graph of .
| 1,789 | 6,699,413 |
SBI PO & Clerk Exam: How to Solve Pie Chart Questions in DI
Updated : Jul 13, 2016, 19:00
By : Ashish
We are providing you Important Short Tricks on Pie Chart Questions in DI which are usually asked in SBI PO Exams. Use these below given short cuts to solve questions within minimum time. These shortcuts will be very helpful for your upcoming All Banking Exam.
To make the chapter easy for you all, we are providing you all some SBI PO Exam: How to Solve Pie Chart Questions in DI which will surely make the chapter easy for you all.
Pie Charts
Introduction
Pie charts are specific types of data presentation where the data is represented in the form of a circle. In a pie chart, a circle is divided into various sections or segments such that each sector or segment represents a certain proportion or percentage of the total. In this, the circle is divided into sectors either percent wise or degree-wise. In percent-wise division, the total area of the chart is taken to be 100% and in degree wise division, the total area of the chart is taken to be 360o.
Important Point on Pie Chart:-
1. Understanding the various headings of DI table/graph/chart is very important.
2. Data Interpretation depends upon the type of questions asked.
3. Some questions are solved via reasoning process.
4. And solving some questions helps solving the other questions.
5. Try to use short tricks on Pie chart question .
This same process applies to every type of DI. In this article, we are covering the next type of Data Presentation, i.e. Pie Chart.
Sample Question
Directions: Study the following pie chart and answer the questions that follows:
Total Number of Teachers = 6400
(Uttar Bihar Gramin Bank 2012)
Question 1: If one-thirty sixth of the number of teachers from university F is professors and the salary of each professor is Rs 96000, what will be the total salary of all the professors together from university F?
[1] Rs 307.2 lakh
[2] 32.64 lakh
[3] Rs 3.072 lakh
[4] 3.264 lakh
[5] None of these
Solution:-
Number of teachers from university F = 18% of 6400 = 1152
1/36 of 1152 = 32
Total salary = 32*96000 = 3072000 = 30.72 lakh. Answer [5] is correct. (Note the tricky options [1] and [2])
Short Tricks:-
Total salary = (6400*18*1*96000)/100*36 = 30.72 Lakhs
Question 2: Difference between the total number of teachers in university A, B and C together and the total number of teachers in university D, E and F together is exactly equal to the number of teachers in which university?
[1] A
[2] B
[3] C
[4] D
[5] F
(You don’t even have to calculate the number of teachers. Just presence of mind is needed.)
Solution:-
Number of teachers in university A, B and C = 11+17+19 = 47%
Number of teachers in university D, E and F = 6+29+18 = 53%
Difference = 6% = University D. Answer [4] is correct.
Question 3: What is the average of teachers in university A, C, D and F together?
[1] 854
[2] 3546
[3] 3456
[4] 874
[5] None of these
Solution:-
Short Tricks:-
11+19+6+18 = 54%. Average = 54/4 % = [54/400]*6400 = 54*16 = 864. Answer [5] is correct.
Question 4: If twenty five percent of the number of teachers in university C is female, what is the number of male teachers in university C?
[1] 922
[2] 911
[3] 924
[4] 912
[5] None of these
Solution:-
Number of teachers in university C = 19% of 6400 = 19*64 = 1216
25% of this is female. Hence remaining 75% is male.
Number of male teachers = 75% of 1216 = [3/4]*1216 = 912. Option [4] is correct.
Short Tricks:-
Number of male teachers = (6400*19*75)/100*100 = 912
Question 5: Number of teachers in university B is approximately what percent of the total number of teachers in university D and E together?
[1] 55%
[2] 59%
[3] 49%
[4] 45%
[5] 65%
Solution:-
Short Tricks:-
Just solve the percentages.
University B = 17%. University D+E = 6+29 = 35%
Required percentage = [17/35]*100 = approx. 49%. Answer is [3]
Thanks
Jul 13Bank & Insurance
Posted by:
Ashish is a management professional with more than 4 years of experience as Mentor in Education sector. Currently working as Community Manager of Teaching exams category at Gradeup. He helps to provide quality content and solves the doubt of aspirants preparing for the exams. His email address is [email protected].
Member since Nov 2015
| 1,169 | 6,699,414 |
<MASK>
<UNMASK>
<MASK>
Try just $\displaystyle x = k\pi$
So $\displaystyle cos(k\pi)cosh(y) = \pm cosh(y) = cos(2)$
Case when k is odd:
cosh(y) = -cos(2)
Case when k is even:
cosh(y) = cos(2)
<MASK>
Originally Posted by Glitch
Try just $\displaystyle x = k\pi$ So $\displaystyle cos(k\pi)cosh(y) = \pm cosh(y) = cos(2)$
<MASK>
But
<MASK>
<UNMASK>
<MASK>
My attempt:
cos(z) = cos(x)cosh(y) - isin(x)sinh(y) = cos(2)
<MASK>
[1] is zero when $\displaystyle x = k\pi$ or y = 0
Substitute into [2]:
$\displaystyle cos(k\pi)cosh(0) \ne cos(2)$
<MASK>
Try just $\displaystyle x = k\pi$
So $\displaystyle cos(k\pi)cosh(y) = \pm cosh(y) = cos(2)$
Case when k is odd:
cosh(y) = -cos(2)
Case when k is even:
cosh(y) = cos(2)
Therefore, y = $\displaystyle cosh^-1(cos(2))$
<MASK>
But the solution is $\displaystyle \pm(2 + 2k\pi)$ :/
<MASK>
Originally Posted by Glitch
Try just $\displaystyle x = k\pi$ So $\displaystyle cos(k\pi)cosh(y) = \pm cosh(y) = cos(2)$
We have
$\displaystyle x=k\pi \Rightarrow \cos (k\pi)\cosh y=(-1)^k \cosh y$
But
$\displaystyle \cosh y \geq 1\; (\forall y\in\mathbb{R})\textrm{\;and\;}\cos 2\neq \pm 1$
<MASK>
<UNMASK>
<MASK>
• Apr 30th 2011, 09:01 PM
Glitch
Another trigonometric equation problem
The question:
Find all solutions $\displaystyle z \in C$ of cos(z) = cos(2)
My attempt:
cos(z) = cos(x)cosh(y) - isin(x)sinh(y) = cos(2)
<MASK>
[1] is zero when $\displaystyle x = k\pi$ or y = 0
Substitute into [2]:
$\displaystyle cos(k\pi)cosh(0) \ne cos(2)$
Try just y = 0:
So cos(x)cosh(0) = cos(x) which we want to equal cos(2), thus $\displaystyle x = \pm2$
Try just $\displaystyle x = k\pi$
So $\displaystyle cos(k\pi)cosh(y) = \pm cosh(y) = cos(2)$
Case when k is odd:
cosh(y) = -cos(2)
Case when k is even:
cosh(y) = cos(2)
Therefore, y = $\displaystyle cosh^-1(cos(2))$
So, $\displaystyle z = \pm 2 + icosh^{-1}(cos(2))$
But the solution is $\displaystyle \pm(2 + 2k\pi)$ :/
<MASK>
Originally Posted by Glitch
Try just $\displaystyle x = k\pi$ So $\displaystyle cos(k\pi)cosh(y) = \pm cosh(y) = cos(2)$
We have
$\displaystyle x=k\pi \Rightarrow \cos (k\pi)\cosh y=(-1)^k \cosh y$
But
$\displaystyle \cosh y \geq 1\; (\forall y\in\mathbb{R})\textrm{\;and\;}\cos 2\neq \pm 1$
<MASK>
<UNMASK>
# Another trigonometric equation problem
• Apr 30th 2011, 09:01 PM
Glitch
Another trigonometric equation problem
The question:
Find all solutions $\displaystyle z \in C$ of cos(z) = cos(2)
My attempt:
cos(z) = cos(x)cosh(y) - isin(x)sinh(y) = cos(2)
So we need:
sin(x)sinh(y) = 0 [1]
cos(x)cosh(y) = cos(2) [2]
[1] is zero when $\displaystyle x = k\pi$ or y = 0
Substitute into [2]:
$\displaystyle cos(k\pi)cosh(0) \ne cos(2)$
Try just y = 0:
So cos(x)cosh(0) = cos(x) which we want to equal cos(2), thus $\displaystyle x = \pm2$
Try just $\displaystyle x = k\pi$
So $\displaystyle cos(k\pi)cosh(y) = \pm cosh(y) = cos(2)$
Case when k is odd:
cosh(y) = -cos(2)
Case when k is even:
cosh(y) = cos(2)
Therefore, y = $\displaystyle cosh^-1(cos(2))$
So, $\displaystyle z = \pm 2 + icosh^{-1}(cos(2))$
But the solution is $\displaystyle \pm(2 + 2k\pi)$ :/
What am I doing wrong? Thanks.
• Apr 30th 2011, 11:45 PM
FernandoRevilla
Quote:
Originally Posted by Glitch
Try just $\displaystyle x = k\pi$ So $\displaystyle cos(k\pi)cosh(y) = \pm cosh(y) = cos(2)$
We have
$\displaystyle x=k\pi \Rightarrow \cos (k\pi)\cosh y=(-1)^k \cosh y$
But
$\displaystyle \cosh y \geq 1\; (\forall y\in\mathbb{R})\textrm{\;and\;}\cos 2\neq \pm 1$
So, we only obtain solutions for y=0.
| 1,338 | 6,699,415 |
<MASK>
Customized
for You
<MASK>
Practice
Pays
<MASK>
86% (01:35) correct 14% (01:37) wrong based on 44 sessions
<MASK>
(B) 2
<MASK>
(D) 7
<MASK>
01 Feb 2018, 01:13
Bunuel wrote:
729
5X3
+ 9X1
_____
2,2X3
<MASK>
(A) 0
(B) 2
<MASK>
(D) 7
<MASK>
<UNMASK>
<MASK>
Customized
for You
<MASK>
Practice
Pays
we will pick new questions that match your level based on your Timer History
<MASK>
Math Expert
Joined: 02 Sep 2009
Posts: 58396
<MASK>
01 Feb 2018, 00:17
00:00
<MASK>
86% (01:35) correct 14% (01:37) wrong based on 44 sessions
HideShow timer Statistics
<MASK>
(B) 2
(C) 3
(D) 7
<MASK>
01 Feb 2018, 01:13
Bunuel wrote:
729
5X3
+ 9X1
_____
2,2X3
<MASK>
(A) 0
(B) 2
<MASK>
(D) 7
<MASK>
<UNMASK>
<MASK>
It is currently 21 Oct 2019, 11:04
<MASK>
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
<MASK>
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
<MASK>
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 58396
Show Tags
01 Feb 2018, 00:17
00:00
<MASK>
86% (01:35) correct 14% (01:37) wrong based on 44 sessions
HideShow timer Statistics
<MASK>
In the addition calculation above, the number X must be
(A) 0
(B) 2
(C) 3
(D) 7
(E) 9
_________________
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3335
Location: India
GPA: 3.12
Re: In the addition calculation above, the number X must be [#permalink]
<MASK>
01 Feb 2018, 01:13
Bunuel wrote:
729
5X3
+ 9X1
_____
2,2X3
<MASK>
(A) 0
(B) 2
(C) 3
(D) 7
(E) 9
Since the first row has a carry over of 1, 1+2+X+X = 1X
The reason for the sum to be 1X and not X is we need a carry over of 1 to get the final value of 22X3.
<MASK>
<UNMASK>
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 21 Oct 2019, 11:04
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
In the addition calculation above, the number X must be
Author Message
TAGS:
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 58396
Show Tags
01 Feb 2018, 00:17
00:00
Difficulty:
15% (low)
Question Stats:
86% (01:35) correct 14% (01:37) wrong based on 44 sessions
HideShow timer Statistics
729
5X3
+ 9X1
_____
2,2X3
In the addition calculation above, the number X must be
(A) 0
(B) 2
(C) 3
(D) 7
(E) 9
_________________
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3335
Location: India
GPA: 3.12
Re: In the addition calculation above, the number X must be [#permalink]
<MASK>
01 Feb 2018, 01:13
Bunuel wrote:
729
5X3
+ 9X1
_____
2,2X3
In the addition calculation above, the number X must be
(A) 0
(B) 2
(C) 3
(D) 7
(E) 9
Since the first row has a carry over of 1, 1+2+X+X = 1X
The reason for the sum to be 1X and not X is we need a carry over of 1 to get the final value of 22X3.
This is only possible when X=7 as 3 + 7 + 7 = 17.
In order to double check if we substitute 7 in place of X, we will have 729 + 573 + 971 which will give us a sum of 2273(Option D)
_________________
You've got what it takes, but it will take everything you've got
Re: In the addition calculation above, the number X must be [#permalink] 01 Feb 2018, 01:13
Display posts from previous: Sort by
<UNMASK>
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 21 Oct 2019, 11:04
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
In the addition calculation above, the number X must be
Author Message
TAGS:
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 58396
Show Tags
01 Feb 2018, 00:17
00:00
Difficulty:
15% (low)
Question Stats:
86% (01:35) correct 14% (01:37) wrong based on 44 sessions
HideShow timer Statistics
729
5X3
+ 9X1
_____
2,2X3
In the addition calculation above, the number X must be
(A) 0
(B) 2
(C) 3
(D) 7
(E) 9
_________________
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3335
Location: India
GPA: 3.12
Re: In the addition calculation above, the number X must be [#permalink]
Show Tags
01 Feb 2018, 01:13
Bunuel wrote:
729
5X3
+ 9X1
_____
2,2X3
In the addition calculation above, the number X must be
(A) 0
(B) 2
(C) 3
(D) 7
(E) 9
Since the first row has a carry over of 1, 1+2+X+X = 1X
The reason for the sum to be 1X and not X is we need a carry over of 1 to get the final value of 22X3.
This is only possible when X=7 as 3 + 7 + 7 = 17.
In order to double check if we substitute 7 in place of X, we will have 729 + 573 + 971 which will give us a sum of 2273(Option D)
_________________
You've got what it takes, but it will take everything you've got
Re: In the addition calculation above, the number X must be [#permalink] 01 Feb 2018, 01:13
Display posts from previous: Sort by
| 1,835 | 6,699,416 |
<MASK>
A Le Monde mathematical puzzle with Alice and Bob:
<MASK>
```reward=function(tokens){
gain=0
if (max(tokens)>0){
#takes one token off
for (i in (1:5)[tokens>0]){
gain=max(gain,1-reward(tokens-((1:5)==i)))
if (gain==1) break()}
#create another singleton
if (max(tokens[-1])>1){
for (i in (2:5)[tokens[-1]>1]){
gain=max(gain,1-reward(c(tokens[1]+1,tokens[-1]-((2:5)==i))))
if (gain==1) break()}}}
return(gain)}
```
<MASK>
as all sets have to vanish at one point or another so order should not matter. However, with the starting values provided in the puzzle, two weeks of computation on our local cluster did produce nothing, as there are too many cases to examine! (The exact solution is that Alice cannot win the game if Bob plays in an optimal manner.)
<MASK>
```gain=function(mine,yours,none){
fine=none
if (length(mine)>0)
fine=none[apply(abs(outer(mine,none,"-")),
2,min)>1]
if (length(fine)>0){
rwrd=0
for (i in 1:length(fine))
rwrd=max(rwrd,1-gain(yours,c(mine,fine[i]),
none[none!=fine[i]]))
return(rwrd)}
return(0)}
```
<MASK>
Posted in Books, Kids, R, Statistics, University life with tags , , , on April 1, 2015 by xi'an
A recursive programming Le Monde mathematical puzzle:
<MASK>
```sol=lowa=plura[plura<100]
for (i in 3:6){
sli=plura[(plura>10^(i-1))&(plura<10^i)]
ace=sli-10^(i-1)*(sli%/%10^(i-1))
lowa=sli[apply(outer(ace,lowa,FUN="=="),
1,max)==1]
lowa=sort(unique(lowa))
sol=c(sol,lowa)}
```
which leads to the output
<MASK>
if (awale[i+1]==1){
bwale=awale
bwale[c(i,i+1)]=0
best=max(best,1-topA(bwale))
}
}}
return(best)
}
<MASK>
```[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
<pre>```
<MASK>
<UNMASK>
<MASK>
A Le Monde mathematical puzzle with Alice and Bob:
<MASK>
```reward=function(tokens){
gain=0
if (max(tokens)>0){
#takes one token off
for (i in (1:5)[tokens>0]){
gain=max(gain,1-reward(tokens-((1:5)==i)))
if (gain==1) break()}
#create another singleton
if (max(tokens[-1])>1){
for (i in (2:5)[tokens[-1]>1]){
gain=max(gain,1-reward(c(tokens[1]+1,tokens[-1]-((2:5)==i))))
if (gain==1) break()}}}
return(gain)}
```
<MASK>
as all sets have to vanish at one point or another so order should not matter. However, with the starting values provided in the puzzle, two weeks of computation on our local cluster did produce nothing, as there are too many cases to examine! (The exact solution is that Alice cannot win the game if Bob plays in an optimal manner.)
<MASK>
Posted in Books, Kids, Statistics, University life with tags , , , on October 9, 2015 by xi'an
<MASK>
In the ‘Og tradition, this calls for a recurrent R code:
```game=function(n=17,col=1,tak=rep(0,n)){
frei=rew=0*tak
# stopping rule
if (sum(tak==col)==0){
frei=(tak==0)}else{
for (i in (1:n)[tak!=-col])
frei[i]=(min(abs((1:n)[tak==col]-i))>1)}
# left positions
if (sum(frei)>0){
for (i in (1:n)[frei==1]){
prop=tak;prop[i]=col
rew[i]=1-game(n=n,col=-col,tak=prop)}}
# outcome of best choice
return(max(rew))}
```
While I did not run the rudimentary recursive function for n=17, I got a zero return from n=2 till n=12, meaning that the starting player is always going to lose if the other player plays optimally.
<MASK>
An game-theoretic Le Monde mathematical puzzle:
<MASK>
```gain=function(mine,yours,none){
fine=none
if (length(mine)>0)
fine=none[apply(abs(outer(mine,none,"-")),
2,min)>1]
if (length(fine)>0){
rwrd=0
for (i in 1:length(fine))
rwrd=max(rwrd,1-gain(yours,c(mine,fine[i]),
none[none!=fine[i]]))
return(rwrd)}
return(0)}
```
<MASK>
Posted in Books, Kids, R, Statistics, University life with tags , , , on April 1, 2015 by xi'an
A recursive programming Le Monde mathematical puzzle:
Given n tokens with 10≤n≤25, Alice and Bob play the following game: the first player draws an integer1≤m≤6 at random. This player can then take 1≤r≤min(2m,n) tokens. The next player is then free to take 1≤s≤min(2r,n-r) tokens. The player taking the last tokens is the winner. There is a winning strategy for Alice if she starts with m=3 and if Bob starts with m=2. Deduce the value of n.
<MASK>
outcome=(n<2*m+1)
if (n>2*m){
for (i in 1:(2*m))
outcome=max(outcome,1-optim(n-i,i))
}
return(outcome)
}
```
<MASK>
```sol=lowa=plura[plura<100]
for (i in 3:6){
sli=plura[(plura>10^(i-1))&(plura<10^i)]
ace=sli-10^(i-1)*(sli%/%10^(i-1))
lowa=sli[apply(outer(ace,lowa,FUN="=="),
1,max)==1]
lowa=sort(unique(lowa))
sol=c(sol,lowa)}
```
which leads to the output
```> subs=rep(0,16)
> for (n in 10:25) subs[n-9]=optim(n,3)
> for (n in 10:25) if (subs[n-9]==1) subs[n-9]=1-optim(n,2)
> subs
[1] 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
> (10:25)[subs==1]
[1] 18
```
<MASK>
For N≤18, N balls are placed in N consecutive holes. Two players, Alice and Bob, consecutively take two balls at a time provided those balls are in contiguous holes. The loser is left with orphaned balls. What is the values of N such that Bob can win, no matter what is Alice’s strategy?
<MASK>
if (awale[i+1]==1){
bwale=awale
bwale[c(i,i+1)]=0
best=max(best,1-topA(bwale))
}
}}
return(best)
}
<MASK>
```[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
<pre>```
(brute-force) answering the question that N=5,9,15 are the values where Alice has no winning strategy if Bob plays in an optimal manner. (The case N=5 is obvious as there always remains two adjacent 1’s once Alice removed any adjacent pair. The case N=9 can also be shown to be a lost cause by enumeration of Alice’s options.)
<UNMASK>
<MASK>
A Le Monde mathematical puzzle with Alice and Bob:
Alice and Bob play a game with 100 tokens set in ten piles of 1, 9 piles of 2, 8 piles of 3, 7 piles of 4, and 4 piles of 5. They each take a token in turn, either to remove it from the game, or to create a new pile of one, provided this token is taken from a pile with at least two remaining tokens. The winner is the one left with the last token. If Alice starts, who is the winner?
<MASK>
```reward=function(tokens){
gain=0
if (max(tokens)>0){
#takes one token off
for (i in (1:5)[tokens>0]){
gain=max(gain,1-reward(tokens-((1:5)==i)))
if (gain==1) break()}
#create another singleton
if (max(tokens[-1])>1){
for (i in (2:5)[tokens[-1]>1]){
gain=max(gain,1-reward(c(tokens[1]+1,tokens[-1]-((2:5)==i))))
if (gain==1) break()}}}
return(gain)}
```
<MASK>
as all sets have to vanish at one point or another so order should not matter. However, with the starting values provided in the puzzle, two weeks of computation on our local cluster did produce nothing, as there are too many cases to examine! (The exact solution is that Alice cannot win the game if Bob plays in an optimal manner.)
## Le Monde puzzle [#930]
Posted in Books, Kids, Statistics, University life with tags , , , on October 9, 2015 by xi'an
On a linear board of length 17, Alice and Bob set alternatively red and blue tokens. Two tokens of the same colour cannot sit next to one another. Devise a winning strategy for the first player.
In the ‘Og tradition, this calls for a recurrent R code:
```game=function(n=17,col=1,tak=rep(0,n)){
frei=rew=0*tak
# stopping rule
if (sum(tak==col)==0){
frei=(tak==0)}else{
for (i in (1:n)[tak!=-col])
frei[i]=(min(abs((1:n)[tak==col]-i))>1)}
# left positions
if (sum(frei)>0){
for (i in (1:n)[frei==1]){
prop=tak;prop[i]=col
rew[i]=1-game(n=n,col=-col,tak=prop)}}
# outcome of best choice
return(max(rew))}
```
While I did not run the rudimentary recursive function for n=17, I got a zero return from n=2 till n=12, meaning that the starting player is always going to lose if the other player plays optimally.
<MASK>
Posted in Books, Kids, Statistics, University life with tags , , on May 8, 2015 by xi'an
An game-theoretic Le Monde mathematical puzzle:
A two-person game consists in choosing an integer N and for each player to successively pick a number in {1,…,N} under the constraint that a player cannot pick a number next to a number this player has already picked. Is there a winning strategy for either player and for all values of N?
for which I simply coded a recursive optimal strategy function:
```gain=function(mine,yours,none){
fine=none
if (length(mine)>0)
fine=none[apply(abs(outer(mine,none,"-")),
2,min)>1]
if (length(fine)>0){
rwrd=0
for (i in 1:length(fine))
rwrd=max(rwrd,1-gain(yours,c(mine,fine[i]),
none[none!=fine[i]]))
return(rwrd)}
return(0)}
```
which returned a zero gain, hence no winning strategy for all values of N except 1.
<MASK>
## Le Monde puzzle [#905]
Posted in Books, Kids, R, Statistics, University life with tags , , , on April 1, 2015 by xi'an
A recursive programming Le Monde mathematical puzzle:
Given n tokens with 10≤n≤25, Alice and Bob play the following game: the first player draws an integer1≤m≤6 at random. This player can then take 1≤r≤min(2m,n) tokens. The next player is then free to take 1≤s≤min(2r,n-r) tokens. The player taking the last tokens is the winner. There is a winning strategy for Alice if she starts with m=3 and if Bob starts with m=2. Deduce the value of n.
Although I first wrote a brute force version of the following code, a moderate amount of thinking leads to conclude that the person given n remaining token and an adversary choice of m tokens such that 2m≥n always win by taking the n remaining tokens:
<MASK>
outcome=(n<2*m+1)
if (n>2*m){
for (i in 1:(2*m))
outcome=max(outcome,1-optim(n-i,i))
}
return(outcome)
}
```
<MASK>
```sol=lowa=plura[plura<100]
for (i in 3:6){
sli=plura[(plura>10^(i-1))&(plura<10^i)]
ace=sli-10^(i-1)*(sli%/%10^(i-1))
lowa=sli[apply(outer(ace,lowa,FUN="=="),
1,max)==1]
lowa=sort(unique(lowa))
sol=c(sol,lowa)}
```
which leads to the output
```> subs=rep(0,16)
> for (n in 10:25) subs[n-9]=optim(n,3)
> for (n in 10:25) if (subs[n-9]==1) subs[n-9]=1-optim(n,2)
> subs
[1] 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
> (10:25)[subs==1]
[1] 18
```
<MASK>
## Le Monde puzzle [#860]
<MASK>
A Le Monde mathematical puzzle that connects to my awalé post of last year:
For N≤18, N balls are placed in N consecutive holes. Two players, Alice and Bob, consecutively take two balls at a time provided those balls are in contiguous holes. The loser is left with orphaned balls. What is the values of N such that Bob can win, no matter what is Alice’s strategy?
I solved this puzzle by the following R code that works recursively on N by eliminating all possible adjacent pairs of balls and checking whether or not there is a winning strategy for the other player.
<MASK>
if (awale[i+1]==1){
bwale=awale
bwale[c(i,i+1)]=0
best=max(best,1-topA(bwale))
}
}}
return(best)
}
for (N in 2:18) print(topA(rep(1,N)))
```
<MASK>
```[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
<pre>```
(brute-force) answering the question that N=5,9,15 are the values where Alice has no winning strategy if Bob plays in an optimal manner. (The case N=5 is obvious as there always remains two adjacent 1’s once Alice removed any adjacent pair. The case N=9 can also be shown to be a lost cause by enumeration of Alice’s options.)
<UNMASK>
## Le Monde puzzle [#950]
Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , on March 10, 2016 by xi'an
A Le Monde mathematical puzzle with Alice and Bob:
Alice and Bob play a game with 100 tokens set in ten piles of 1, 9 piles of 2, 8 piles of 3, 7 piles of 4, and 4 piles of 5. They each take a token in turn, either to remove it from the game, or to create a new pile of one, provided this token is taken from a pile with at least two remaining tokens. The winner is the one left with the last token. If Alice starts, who is the winner?
I just wrote a most rudimentary recursive R function
```reward=function(tokens){
gain=0
if (max(tokens)>0){
#takes one token off
for (i in (1:5)[tokens>0]){
gain=max(gain,1-reward(tokens-((1:5)==i)))
if (gain==1) break()}
#create another singleton
if (max(tokens[-1])>1){
for (i in (2:5)[tokens[-1]>1]){
gain=max(gain,1-reward(c(tokens[1]+1,tokens[-1]-((2:5)==i))))
if (gain==1) break()}}}
return(gain)}
```
<MASK>
```reward=function(tokens){
#clean up piles with single token
tokens[1]=tokens[1]+sum((tokens[-1]==1))
tokens[-1][tokens[-1]==1]=0
if (max(tokens[-1])==0){
#end: no choice left
gain=1*(tokens[1]%%2==1)
}else{
gain=0
#all piles have to disappear, order should not matter
i=min((2:5)[tokens[-1]>1])
#set one token appart
gain=max(gain,1-reward(tokens-((1:5)==i)))
#create another singleton
gain=max(gain,1-reward(c(tokens[1]+1,tokens[-1]-((2:5)==i))))}
return(gain)}
```
as all sets have to vanish at one point or another so order should not matter. However, with the starting values provided in the puzzle, two weeks of computation on our local cluster did produce nothing, as there are too many cases to examine! (The exact solution is that Alice cannot win the game if Bob plays in an optimal manner.)
## Le Monde puzzle [#930]
Posted in Books, Kids, Statistics, University life with tags , , , on October 9, 2015 by xi'an
On a linear board of length 17, Alice and Bob set alternatively red and blue tokens. Two tokens of the same colour cannot sit next to one another. Devise a winning strategy for the first player.
In the ‘Og tradition, this calls for a recurrent R code:
```game=function(n=17,col=1,tak=rep(0,n)){
frei=rew=0*tak
# stopping rule
if (sum(tak==col)==0){
frei=(tak==0)}else{
for (i in (1:n)[tak!=-col])
frei[i]=(min(abs((1:n)[tak==col]-i))>1)}
# left positions
if (sum(frei)>0){
for (i in (1:n)[frei==1]){
prop=tak;prop[i]=col
rew[i]=1-game(n=n,col=-col,tak=prop)}}
# outcome of best choice
return(max(rew))}
```
While I did not run the rudimentary recursive function for n=17, I got a zero return from n=2 till n=12, meaning that the starting player is always going to lose if the other player plays optimally.
## Le Monde puzzle [#910]
Posted in Books, Kids, Statistics, University life with tags , , on May 8, 2015 by xi'an
An game-theoretic Le Monde mathematical puzzle:
A two-person game consists in choosing an integer N and for each player to successively pick a number in {1,…,N} under the constraint that a player cannot pick a number next to a number this player has already picked. Is there a winning strategy for either player and for all values of N?
for which I simply coded a recursive optimal strategy function:
```gain=function(mine,yours,none){
fine=none
if (length(mine)>0)
fine=none[apply(abs(outer(mine,none,"-")),
2,min)>1]
if (length(fine)>0){
rwrd=0
for (i in 1:length(fine))
rwrd=max(rwrd,1-gain(yours,c(mine,fine[i]),
none[none!=fine[i]]))
return(rwrd)}
return(0)}
```
which returned a zero gain, hence no winning strategy for all values of N except 1.
<MASK>
Meaning that the starting player is always the loser!
## Le Monde puzzle [#905]
Posted in Books, Kids, R, Statistics, University life with tags , , , on April 1, 2015 by xi'an
A recursive programming Le Monde mathematical puzzle:
Given n tokens with 10≤n≤25, Alice and Bob play the following game: the first player draws an integer1≤m≤6 at random. This player can then take 1≤r≤min(2m,n) tokens. The next player is then free to take 1≤s≤min(2r,n-r) tokens. The player taking the last tokens is the winner. There is a winning strategy for Alice if she starts with m=3 and if Bob starts with m=2. Deduce the value of n.
Although I first wrote a brute force version of the following code, a moderate amount of thinking leads to conclude that the person given n remaining token and an adversary choice of m tokens such that 2m≥n always win by taking the n remaining tokens:
```optim=function(n,m){
outcome=(n<2*m+1)
if (n>2*m){
for (i in 1:(2*m))
outcome=max(outcome,1-optim(n-i,i))
}
return(outcome)
}
```
eliminating solutions which dividers are not solutions themselves:
```sol=lowa=plura[plura<100]
for (i in 3:6){
sli=plura[(plura>10^(i-1))&(plura<10^i)]
ace=sli-10^(i-1)*(sli%/%10^(i-1))
lowa=sli[apply(outer(ace,lowa,FUN="=="),
1,max)==1]
lowa=sort(unique(lowa))
sol=c(sol,lowa)}
```
which leads to the output
```> subs=rep(0,16)
> for (n in 10:25) subs[n-9]=optim(n,3)
> for (n in 10:25) if (subs[n-9]==1) subs[n-9]=1-optim(n,2)
> subs
[1] 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
> (10:25)[subs==1]
[1] 18
```
Ergo, the number of tokens is 18!
## Le Monde puzzle [#860]
Posted in Books, Kids, R with tags , , , , on April 4, 2014 by xi'an
A Le Monde mathematical puzzle that connects to my awalé post of last year:
For N≤18, N balls are placed in N consecutive holes. Two players, Alice and Bob, consecutively take two balls at a time provided those balls are in contiguous holes. The loser is left with orphaned balls. What is the values of N such that Bob can win, no matter what is Alice’s strategy?
I solved this puzzle by the following R code that works recursively on N by eliminating all possible adjacent pairs of balls and checking whether or not there is a winning strategy for the other player.
```topA=function(awale){
# return 1 if current player can win, 0 otherwise
best=0
if (max(awale[-1]*awale[-N])==1){
#there are adjacent balls remaining
for (i in (1:(N-1))[awale[1:(N-1)]==1]){
if (awale[i+1]==1){
bwale=awale
bwale[c(i,i+1)]=0
best=max(best,1-topA(bwale))
}
}}
return(best)
}
for (N in 2:18) print(topA(rep(1,N)))
```
which returns the solution
```[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
<pre>```
(brute-force) answering the question that N=5,9,15 are the values where Alice has no winning strategy if Bob plays in an optimal manner. (The case N=5 is obvious as there always remains two adjacent 1’s once Alice removed any adjacent pair. The case N=9 can also be shown to be a lost cause by enumeration of Alice’s options.)
<UNMASK>
## Le Monde puzzle [#950]
Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , on March 10, 2016 by xi'an
A Le Monde mathematical puzzle with Alice and Bob:
Alice and Bob play a game with 100 tokens set in ten piles of 1, 9 piles of 2, 8 piles of 3, 7 piles of 4, and 4 piles of 5. They each take a token in turn, either to remove it from the game, or to create a new pile of one, provided this token is taken from a pile with at least two remaining tokens. The winner is the one left with the last token. If Alice starts, who is the winner?
I just wrote a most rudimentary recursive R function
```reward=function(tokens){
gain=0
if (max(tokens)>0){
#takes one token off
for (i in (1:5)[tokens>0]){
gain=max(gain,1-reward(tokens-((1:5)==i)))
if (gain==1) break()}
#create another singleton
if (max(tokens[-1])>1){
for (i in (2:5)[tokens[-1]>1]){
gain=max(gain,1-reward(c(tokens[1]+1,tokens[-1]-((2:5)==i))))
if (gain==1) break()}}}
return(gain)}
```
where token represents the number of remaining sets with 1, 2, 3, 4, and 5 tolkiens. With the suggested values, (10,18,24,28,20), the R code takes too long on my machine! Or even overnight on our server. So as usual I thought a bit more about it and started cutting at unnecessary bits, reaching the faster recursive function
```reward=function(tokens){
#clean up piles with single token
tokens[1]=tokens[1]+sum((tokens[-1]==1))
tokens[-1][tokens[-1]==1]=0
if (max(tokens[-1])==0){
#end: no choice left
gain=1*(tokens[1]%%2==1)
}else{
gain=0
#all piles have to disappear, order should not matter
i=min((2:5)[tokens[-1]>1])
#set one token appart
gain=max(gain,1-reward(tokens-((1:5)==i)))
#create another singleton
gain=max(gain,1-reward(c(tokens[1]+1,tokens[-1]-((2:5)==i))))}
return(gain)}
```
as all sets have to vanish at one point or another so order should not matter. However, with the starting values provided in the puzzle, two weeks of computation on our local cluster did produce nothing, as there are too many cases to examine! (The exact solution is that Alice cannot win the game if Bob plays in an optimal manner.)
## Le Monde puzzle [#930]
Posted in Books, Kids, Statistics, University life with tags , , , on October 9, 2015 by xi'an
On a linear board of length 17, Alice and Bob set alternatively red and blue tokens. Two tokens of the same colour cannot sit next to one another. Devise a winning strategy for the first player.
In the ‘Og tradition, this calls for a recurrent R code:
```game=function(n=17,col=1,tak=rep(0,n)){
frei=rew=0*tak
# stopping rule
if (sum(tak==col)==0){
frei=(tak==0)}else{
for (i in (1:n)[tak!=-col])
frei[i]=(min(abs((1:n)[tak==col]-i))>1)}
# left positions
if (sum(frei)>0){
for (i in (1:n)[frei==1]){
prop=tak;prop[i]=col
rew[i]=1-game(n=n,col=-col,tak=prop)}}
# outcome of best choice
return(max(rew))}
```
While I did not run the rudimentary recursive function for n=17, I got a zero return from n=2 till n=12, meaning that the starting player is always going to lose if the other player plays optimally.
## Le Monde puzzle [#910]
Posted in Books, Kids, Statistics, University life with tags , , on May 8, 2015 by xi'an
An game-theoretic Le Monde mathematical puzzle:
A two-person game consists in choosing an integer N and for each player to successively pick a number in {1,…,N} under the constraint that a player cannot pick a number next to a number this player has already picked. Is there a winning strategy for either player and for all values of N?
for which I simply coded a recursive optimal strategy function:
```gain=function(mine,yours,none){
fine=none
if (length(mine)>0)
fine=none[apply(abs(outer(mine,none,"-")),
2,min)>1]
if (length(fine)>0){
rwrd=0
for (i in 1:length(fine))
rwrd=max(rwrd,1-gain(yours,c(mine,fine[i]),
none[none!=fine[i]]))
return(rwrd)}
return(0)}
```
which returned a zero gain, hence no winning strategy for all values of N except 1.
```> gain(NULL,NULL,1)
[1] 1
> gain(NULL,NULL,1:2)
[1] 0
> gain(NULL,NULL,1:3)
[1] 0
> gain(NULL,NULL,1:4)
[1] 0
```
Meaning that the starting player is always the loser!
## Le Monde puzzle [#905]
Posted in Books, Kids, R, Statistics, University life with tags , , , on April 1, 2015 by xi'an
A recursive programming Le Monde mathematical puzzle:
Given n tokens with 10≤n≤25, Alice and Bob play the following game: the first player draws an integer1≤m≤6 at random. This player can then take 1≤r≤min(2m,n) tokens. The next player is then free to take 1≤s≤min(2r,n-r) tokens. The player taking the last tokens is the winner. There is a winning strategy for Alice if she starts with m=3 and if Bob starts with m=2. Deduce the value of n.
Although I first wrote a brute force version of the following code, a moderate amount of thinking leads to conclude that the person given n remaining token and an adversary choice of m tokens such that 2m≥n always win by taking the n remaining tokens:
```optim=function(n,m){
outcome=(n<2*m+1)
if (n>2*m){
for (i in 1:(2*m))
outcome=max(outcome,1-optim(n-i,i))
}
return(outcome)
}
```
eliminating solutions which dividers are not solutions themselves:
```sol=lowa=plura[plura<100]
for (i in 3:6){
sli=plura[(plura>10^(i-1))&(plura<10^i)]
ace=sli-10^(i-1)*(sli%/%10^(i-1))
lowa=sli[apply(outer(ace,lowa,FUN="=="),
1,max)==1]
lowa=sort(unique(lowa))
sol=c(sol,lowa)}
```
which leads to the output
```> subs=rep(0,16)
> for (n in 10:25) subs[n-9]=optim(n,3)
> for (n in 10:25) if (subs[n-9]==1) subs[n-9]=1-optim(n,2)
> subs
[1] 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
> (10:25)[subs==1]
[1] 18
```
Ergo, the number of tokens is 18!
## Le Monde puzzle [#860]
Posted in Books, Kids, R with tags , , , , on April 4, 2014 by xi'an
A Le Monde mathematical puzzle that connects to my awalé post of last year:
For N≤18, N balls are placed in N consecutive holes. Two players, Alice and Bob, consecutively take two balls at a time provided those balls are in contiguous holes. The loser is left with orphaned balls. What is the values of N such that Bob can win, no matter what is Alice’s strategy?
I solved this puzzle by the following R code that works recursively on N by eliminating all possible adjacent pairs of balls and checking whether or not there is a winning strategy for the other player.
```topA=function(awale){
# return 1 if current player can win, 0 otherwise
best=0
if (max(awale[-1]*awale[-N])==1){
#there are adjacent balls remaining
for (i in (1:(N-1))[awale[1:(N-1)]==1]){
if (awale[i+1]==1){
bwale=awale
bwale[c(i,i+1)]=0
best=max(best,1-topA(bwale))
}
}}
return(best)
}
for (N in 2:18) print(topA(rep(1,N)))
```
which returns the solution
```[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
[1] 1
[1] 1
[1] 0
[1] 1
[1] 1
[1] 1
<pre>```
(brute-force) answering the question that N=5,9,15 are the values where Alice has no winning strategy if Bob plays in an optimal manner. (The case N=5 is obvious as there always remains two adjacent 1’s once Alice removed any adjacent pair. The case N=9 can also be shown to be a lost cause by enumeration of Alice’s options.)
| 7,930 | 6,699,417 |
<MASK>
Question 2:
Use theory of relativity to solve the first question.
<MASK>
<UNMASK>
<MASK>
A stick of 1m in length travels at v = 1/2 c along its axis away from the observer.
<MASK>
Question 2:
Use theory of relativity to solve the first question.
<MASK>
Now I'm not sure how to calculate the perceived length. I guess I have to find out where the photon of the far end has to be emitted relative to the front end, but how would I do that?
Regarding 2:
I'm not sure about the approach. I know how to use Lorentz transformation to get the new length, but would I include the effect calculated for question 1 as well or not?
for question 2, you can use lorentz position transformations to find this but it is more difficult than to use the length contraction relation (if you have studied it). Ultimately, though, they will reduce to the same thing, and you will find: $$l = l_{o}\sqrt{1-\frac{v^{2}}{c^{2}}}$$ If you use the lorentz transofmrations, be sure to remember that some relations must cancel ($$\Delta ? = 0$$ -- i leave the question mark for you to fill in) based on the way you must make measurements.
<UNMASK>
<MASK>
A stick of 1m in length travels at v = 1/2 c along its axis away from the observer.
<MASK>
Question 2:
Use theory of relativity to solve the first question.
Regarding 1:
The light emitted by the far end of the stick has to be emitted a little bit sooner so that it reaches the observer at the same time as light from the front end of the stick. Since light travels at c towards the observer and the stick travels at 1/2 c away, it's necessary to know how fast 1/2 c (the difference between the speed of light and the speed of the stick) needs to cover 1m. The time it takes for this determines how much later the front end has to emit a photon so that it reaches the observer at the same time as a photon emitted by the far end.
<MASK>
Now I'm not sure how to calculate the perceived length. I guess I have to find out where the photon of the far end has to be emitted relative to the front end, but how would I do that?
Regarding 2:
I'm not sure about the approach. I know how to use Lorentz transformation to get the new length, but would I include the effect calculated for question 1 as well or not?
for question 2, you can use lorentz position transformations to find this but it is more difficult than to use the length contraction relation (if you have studied it). Ultimately, though, they will reduce to the same thing, and you will find: $$l = l_{o}\sqrt{1-\frac{v^{2}}{c^{2}}}$$ If you use the lorentz transofmrations, be sure to remember that some relations must cancel ($$\Delta ? = 0$$ -- i leave the question mark for you to fill in) based on the way you must make measurements.
<UNMASK>
## Length Contraction
A stick of 1m in length travels at v = 1/2 c along its axis away from the observer.
Question 1:
Show that the observer perceives the length of the stick to be shorter without theory of relativity. Calculate the length as perceived by him if he calculates it by the difference in length between both ends which have been photographed at the same time.
Question 2:
Use theory of relativity to solve the first question.
Regarding 1:
The light emitted by the far end of the stick has to be emitted a little bit sooner so that it reaches the observer at the same time as light from the front end of the stick. Since light travels at c towards the observer and the stick travels at 1/2 c away, it's necessary to know how fast 1/2 c (the difference between the speed of light and the speed of the stick) needs to cover 1m. The time it takes for this determines how much later the front end has to emit a photon so that it reaches the observer at the same time as a photon emitted by the far end.
<MASK>
Now I'm not sure how to calculate the perceived length. I guess I have to find out where the photon of the far end has to be emitted relative to the front end, but how would I do that?
Regarding 2:
I'm not sure about the approach. I know how to use Lorentz transformation to get the new length, but would I include the effect calculated for question 1 as well or not?
for question 2, you can use lorentz position transformations to find this but it is more difficult than to use the length contraction relation (if you have studied it). Ultimately, though, they will reduce to the same thing, and you will find: $$l = l_{o}\sqrt{1-\frac{v^{2}}{c^{2}}}$$ If you use the lorentz transofmrations, be sure to remember that some relations must cancel ($$\Delta ? = 0$$ -- i leave the question mark for you to fill in) based on the way you must make measurements.
<UNMASK>
## Length Contraction
A stick of 1m in length travels at v = 1/2 c along its axis away from the observer.
Question 1:
Show that the observer perceives the length of the stick to be shorter without theory of relativity. Calculate the length as perceived by him if he calculates it by the difference in length between both ends which have been photographed at the same time.
Question 2:
Use theory of relativity to solve the first question.
Regarding 1:
The light emitted by the far end of the stick has to be emitted a little bit sooner so that it reaches the observer at the same time as light from the front end of the stick. Since light travels at c towards the observer and the stick travels at 1/2 c away, it's necessary to know how fast 1/2 c (the difference between the speed of light and the speed of the stick) needs to cover 1m. The time it takes for this determines how much later the front end has to emit a photon so that it reaches the observer at the same time as a photon emitted by the far end.
Is that correct so far?
Now I'm not sure how to calculate the perceived length. I guess I have to find out where the photon of the far end has to be emitted relative to the front end, but how would I do that?
Regarding 2:
I'm not sure about the approach. I know how to use Lorentz transformation to get the new length, but would I include the effect calculated for question 1 as well or not?
for question 2, you can use lorentz position transformations to find this but it is more difficult than to use the length contraction relation (if you have studied it). Ultimately, though, they will reduce to the same thing, and you will find: $$l = l_{o}\sqrt{1-\frac{v^{2}}{c^{2}}}$$ If you use the lorentz transofmrations, be sure to remember that some relations must cancel ($$\Delta ? = 0$$ -- i leave the question mark for you to fill in) based on the way you must make measurements.
| 1,565 | 6,699,418 |
# Different forces from different inertial frames?
I've been having trouble understanding this equation here derived in the picture given below. If I change reference frames to one which is moving at a velocity u with respect to the first frame, then the term $$v$$ in the equation would change to $$(v-u)$$ however as far as I can tell the $$\frac{dm}{dt}$$ can't change. Does this means these two observers would measure different values of force? So far, I've been told that all inertial reference frames measure the same value for a given force.
Also isn't $$F=\frac{dp}{dt}$$ only valid for systems with mass that is not varying with time? As they said in here: Second law of Newton for variable mass systems
If so, then why is the text considering $$\frac{dp}{dt}$$ to be the force?
• Yes the velocity is constant and I don't really see how what you wrote above resolves the issue. Jul 19, 2019 at 10:32
• sorry I write it again with $P=(M+m(t))\,v$ and $F=dP/dt$ you get $dP/dt=dm/dt\,v$ this is your result ?
– Eli
Jul 19, 2019 at 10:40
• Edited question with link Jul 19, 2019 at 10:49
• $F=\frac{dp}{dt}$ is always valid in an inertial frame. $F=ma$ is only valid if $m$ does not change with time. You can consider the freight car and the hopper separately, but there is no horizontal force on the hopper because the sand falls vertically w.r.t. the hopper - the horizontal momentum of the sand (in any intertial frame) does not change until it hits the freight car. Jul 19, 2019 at 13:03
• The linked answer looks fine. In the frame of reference in which the hopper is stationary, a force must be applied to the sand to increase its velocity from zero to the velocity of the freight car. In the frame of reference in which the freight car is stationary, the same force must be applied to the sand to increase its velocity from an initial negative value (the hopper is now going backwards) to zero. In either case the force is $v \frac{dm}{dt}$ which is derived from $\frac{d}{dt}(mv)$ with constant $v$ and changing $m$. Jul 19, 2019 at 15:39
| 544 | 6,699,419 |
# Singular Sets of Algebraic Curves and Surfaces
If we are given a curve f(x,y)=0 in the plane, or a surface f(t,x,y)=0 in 3-space, there are many points on the curve that have a well-defined, unique tangent line, and many points on the surface that have a well-defined unique tangent plane. See the pictures below for some examples
### The nephroid with one of its tangent lines, and the surface F(t,x,y)=0 with one of its tangent planes
If we want to compute the equations for these tangent lines or planes, we can use a fact we learned in multivariable calculus. Since the curve is the level set of the value 0 for the function f(x,y) or f(x,y,z), the tangent line/plane is perpendicular to the gradient vector (df/dx,df/dy) or (df/dx,df/dy,df/dz) evaluated at that point. Since we already know one point that we want the tangent line/plane to go through, this gives enough information to compute the tangent line or plane's equation, as long as the gradient is not the zero vector at that point.
On the other hand, points on the curve/surface where the gradient is the zero vector do not have a well-defined tangent line/plane. These points are highlighted in the pictures below
### Singular points on the nephroid, and on the surface F(t,x,y)=0
These points are called the singular set of the curve/surface, and are defined by the (polynomial) equations
f(x,y)=0
d/dx f(x,y)=0
d/dy f(x,y)=0
for an algebraic curve f(x,y)=0, and by the equations
F(t,x,y)=0
d/dx F(t,x,y)=0
d/dy F(t,x,y)=0
d/dt F(t,x,y)=0
for an algebraic surface F(t,x,y)=0. If one has a method for solving polynomial equations, then one can find the solutions to the above equations and compute the singular set of your curve/surface. For example, the Maple computations shown below give the singularities of the nephroid and the graph surface F(t,x,y)=0.
```
4 2 2 2 2 2 4 2 4 6
nephroid := 12 x y - 4 - 15 y + 12 x - 24 y x + 12 y x - 12 y + 4 y
4 6
- 12 x + 4 x
2 2 2 2 2 2
F := ((y - t) (2 - t ) + t (4 - t )) - t x (4 - t )
> solve({nephroid=0,diff(nephroid,x)=0,diff(nephroid,y)=0},{x,y});
2
{y = 0, x = 1}, {y = 0, x = -1}, {x = 0, y = RootOf(1 + 2 _Z )}
> solve({F=0,diff(F,x)=0,diff(F,y)=0,diff(F,t)=0},{t,x,y});
t
{x = x, y = 0, t = 0}, {t = t, x = 0, y = 2 --------},
2
- 2 + t
2 2
{x = 0, y = - 1/2 RootOf(2 + _Z ), t = RootOf(2 + _Z )}
```
| 789 | 6,699,420 |
+0
# Evaluate without a calculator:
0
623
2
Evaluate without a calculator:
Guest Apr 29, 2015
#1
+18934
+13
$$\mathbf{12\binom{3}{3} + 11\binom{4}{3} + 10\binom{5}{3} + \cdots + 2\binom{13}{3} +\binom{14}{3} = \;?}$$
$$\small{\text{ \begin{array}{l} \mathbf{ 12\binom{3}{3} + 11\binom{4}{3} + 10\binom{5}{3} + 9\binom{6}{3} + 8\binom{7}{3} + 7\binom{8}{3} + 6\binom{9}{3} + 5\binom{10}{3} + 4\binom{11}{3} + 3\binom{12}{3} + 2\binom{13}{3} + 1\binom{14}{3} } \\ \\ = \mathbf{ \binom{4}{4} + \binom{5}{4} + \binom{6}{4} + \binom{7}{4} + \binom{8}{4} + \binom{9}{4} + \binom{10}{4} + \binom{11}{4} + \binom{12}{4} + \binom{13}{4} + \binom{14}{4} + \binom{15}{4} }\\\\ = \mathbf{ \binom{16}{5} }\\\\ = \mathbf{ \dfrac{16!}{5!\cdot 11!} }\\\\ = \mathbf{ \dfrac{12\cdot 13\cdot 14\cdot 15\cdot 16}{5!} }\\\\ = \mathbf{ \dfrac{12\cdot 13\cdot 14\cdot 15\cdot 16}{1\cdot 2\cdot 3\cdot 4\cdot 5} }\\\\ = \mathbf{ \left(\dfrac{12}{3}\right)\cdot 13 \cdot \left(\dfrac{14}{2}\right)\cdot \left(\dfrac{15}{5}\right)\cdot \left(\dfrac{16}{4}\right) }\\\\ = \mathbf{4 \cdot 13 \cdot 7 \cdot 3 \cdot 4}\\\\ = \mathbf{13 \cdot 16 \cdot 21}\\\\ = \mathbf{4368} \end{array} }}$$
see Hockey Stick Pattern: http://ptri1.tripod.com/
If a diagonal of numbers of any length is selected starting at any of the 1's bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself. If you don't understand that, look at the drawing.1+6+21+56 = 841+7+28+84+210+462+924 = 17161+12 = 13
heureka Apr 30, 2015
Sort:
#1
+18934
+13
$$\mathbf{12\binom{3}{3} + 11\binom{4}{3} + 10\binom{5}{3} + \cdots + 2\binom{13}{3} +\binom{14}{3} = \;?}$$
$$\small{\text{ \begin{array}{l} \mathbf{ 12\binom{3}{3} + 11\binom{4}{3} + 10\binom{5}{3} + 9\binom{6}{3} + 8\binom{7}{3} + 7\binom{8}{3} + 6\binom{9}{3} + 5\binom{10}{3} + 4\binom{11}{3} + 3\binom{12}{3} + 2\binom{13}{3} + 1\binom{14}{3} } \\ \\ = \mathbf{ \binom{4}{4} + \binom{5}{4} + \binom{6}{4} + \binom{7}{4} + \binom{8}{4} + \binom{9}{4} + \binom{10}{4} + \binom{11}{4} + \binom{12}{4} + \binom{13}{4} + \binom{14}{4} + \binom{15}{4} }\\\\ = \mathbf{ \binom{16}{5} }\\\\ = \mathbf{ \dfrac{16!}{5!\cdot 11!} }\\\\ = \mathbf{ \dfrac{12\cdot 13\cdot 14\cdot 15\cdot 16}{5!} }\\\\ = \mathbf{ \dfrac{12\cdot 13\cdot 14\cdot 15\cdot 16}{1\cdot 2\cdot 3\cdot 4\cdot 5} }\\\\ = \mathbf{ \left(\dfrac{12}{3}\right)\cdot 13 \cdot \left(\dfrac{14}{2}\right)\cdot \left(\dfrac{15}{5}\right)\cdot \left(\dfrac{16}{4}\right) }\\\\ = \mathbf{4 \cdot 13 \cdot 7 \cdot 3 \cdot 4}\\\\ = \mathbf{13 \cdot 16 \cdot 21}\\\\ = \mathbf{4368} \end{array} }}$$
see Hockey Stick Pattern: http://ptri1.tripod.com/
If a diagonal of numbers of any length is selected starting at any of the 1's bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself. If you don't understand that, look at the drawing.1+6+21+56 = 841+7+28+84+210+462+924 = 17161+12 = 13
heureka Apr 30, 2015
#2
+91773
0
Thanks Heureka,
That web site that you have referenced looks very interesting.
I know that there are a lot of features on Pascal's triangle that I am not aware of or that I do not retain in my memory. So thank you. I think I will store that site away in our reference material :)
Melody Apr 30, 2015
### 11 Online Users
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners. See details
| 1,590 | 6,699,421 |
Math Tricks | Mensuration Formulas | Qualitative | Aptitude
<MASK>
Time speed distance solved quantitive aptitudes
<MASK>
<UNMASK>
Math Tricks | Mensuration Formulas | Qualitative | Aptitude
Time and Speed quantitative aptitude of Competitive Maths
<MASK>
Time speed distance solved quantitive aptitudes
<MASK>
<UNMASK>
Math Tricks | Mensuration Formulas | Qualitative | Aptitude
Time and Speed quantitative aptitude of Competitive Maths
We have discussed in the last tutorial, the basic concept and shortcut formula of time speed and distance aptitude for competitive exams. The shortcut formula and aptitudes were basic levels, which are very important for those government and banking jobs aspirant, who newly started preparation for competitive math. But now, here we are grading up aptitude level which has a probability to ask for the competitive math. This chapter is also a very simple if you have the basic idea of this chapter, or you have read the opening tutorial of Time Speed and Distance aptitude tricks. Here, we will learn how to customize the shortcut formula and apply to solve a math problem of time speed and distance. Because, sometimes, we can not apply shortcut formula directly, so we need to apply the formula in a tricky way. Let's see few question and solution to observe, how we can apply shortcut formula in Time-Speed and Distance math?. Also, we promise you that you will love this math tricks at the end of this tutorial.
Time speed distance solved quantitive aptitudes
Aptitude 1: A traveller counted 37 trees in 2 minutes while he was travelling on the bus. If every two trees were 40-meters apart, then find the speed of the bus?
Solution: I want to notice you that the traveller covered the distance 36*40 metres in 2 minutes. Because the traveller covered first 40 metres and two trees. Just because of the distance between two trees is 40 m.
Therefore, we will get the speed of the bus by accepting the bus ran 40*36 =1440 meters in 120 seconds.
You knew the formula in the last tutorial:-
Speed = Distance/Time
So, now apply the value of Distance and time in the formula. Look at below.
Speed= 1440/120 meter/second.
= 12meter/second
=12*(18/5) km/h
=43.2 km/h.
Is not is simple math tricks friends?
quantitative aptitude 2: A train is thrice faster than a motorbike. If the bike can cover a certain distance in 1 hour and 30 minutes, then how long time will take to cover the same distance by the train?
Solution: Let's suppose, the speed of the motorbike is x km per minute. Therefore the speed of the train will be 3x km per hours.
The question is very simple. You can solve this problem within a few seconds without knowing the formula. But the problem-solving method will help you to solve different conditional problems. It may be the first object or second object.
<MASK>
Math Tricks of Mensuration Formulas, Triangle, Cube, Cuboid, Quadrilateral, Circle, Partnership, Pipe and cistern, Problem on Train and ages, Profit and Loss, simple-Compound interest, Simplification Solved Quantitative Aptitude, Time Speed, Distance, Time and work Age calculation, Average, Boat and Stream, Division rules LCM and HCF Shortcut Formulas.
Copyright © 2017-2018 |Newmathtricks.com | Terms of Use-Privacy Policy | About Us-Contact Us |
<UNMASK>
Math Tricks | Mensuration Formulas | Qualitative | Aptitude
Time and Speed quantitative aptitude of Competitive Maths
We have discussed in the last tutorial, the basic concept and shortcut formula of time speed and distance aptitude for competitive exams. The shortcut formula and aptitudes were basic levels, which are very important for those government and banking jobs aspirant, who newly started preparation for competitive math. But now, here we are grading up aptitude level which has a probability to ask for the competitive math. This chapter is also a very simple if you have the basic idea of this chapter, or you have read the opening tutorial of Time Speed and Distance aptitude tricks. Here, we will learn how to customize the shortcut formula and apply to solve a math problem of time speed and distance. Because, sometimes, we can not apply shortcut formula directly, so we need to apply the formula in a tricky way. Let's see few question and solution to observe, how we can apply shortcut formula in Time-Speed and Distance math?. Also, we promise you that you will love this math tricks at the end of this tutorial.
Time speed distance solved quantitive aptitudes
Aptitude 1: A traveller counted 37 trees in 2 minutes while he was travelling on the bus. If every two trees were 40-meters apart, then find the speed of the bus?
Solution: I want to notice you that the traveller covered the distance 36*40 metres in 2 minutes. Because the traveller covered first 40 metres and two trees. Just because of the distance between two trees is 40 m.
Therefore, we will get the speed of the bus by accepting the bus ran 40*36 =1440 meters in 120 seconds.
You knew the formula in the last tutorial:-
Speed = Distance/Time
So, now apply the value of Distance and time in the formula. Look at below.
Speed= 1440/120 meter/second.
= 12meter/second
=12*(18/5) km/h
=43.2 km/h.
Is not is simple math tricks friends?
quantitative aptitude 2: A train is thrice faster than a motorbike. If the bike can cover a certain distance in 1 hour and 30 minutes, then how long time will take to cover the same distance by the train?
Solution: Let's suppose, the speed of the motorbike is x km per minute. Therefore the speed of the train will be 3x km per hours.
The question is very simple. You can solve this problem within a few seconds without knowing the formula. But the problem-solving method will help you to solve different conditional problems. It may be the first object or second object.
<MASK>
Post a Comment
Math Tricks of Mensuration Formulas, Triangle, Cube, Cuboid, Quadrilateral, Circle, Partnership, Pipe and cistern, Problem on Train and ages, Profit and Loss, simple-Compound interest, Simplification Solved Quantitative Aptitude, Time Speed, Distance, Time and work Age calculation, Average, Boat and Stream, Division rules LCM and HCF Shortcut Formulas.
Copyright © 2017-2018 |Newmathtricks.com | Terms of Use-Privacy Policy | About Us-Contact Us |
<UNMASK>
Math Tricks | Mensuration Formulas | Qualitative | Aptitude
Time and Speed quantitative aptitude of Competitive Maths
We have discussed in the last tutorial, the basic concept and shortcut formula of time speed and distance aptitude for competitive exams. The shortcut formula and aptitudes were basic levels, which are very important for those government and banking jobs aspirant, who newly started preparation for competitive math. But now, here we are grading up aptitude level which has a probability to ask for the competitive math. This chapter is also a very simple if you have the basic idea of this chapter, or you have read the opening tutorial of Time Speed and Distance aptitude tricks. Here, we will learn how to customize the shortcut formula and apply to solve a math problem of time speed and distance. Because, sometimes, we can not apply shortcut formula directly, so we need to apply the formula in a tricky way. Let's see few question and solution to observe, how we can apply shortcut formula in Time-Speed and Distance math?. Also, we promise you that you will love this math tricks at the end of this tutorial.
Time speed distance solved quantitive aptitudes
Aptitude 1: A traveller counted 37 trees in 2 minutes while he was travelling on the bus. If every two trees were 40-meters apart, then find the speed of the bus?
Solution: I want to notice you that the traveller covered the distance 36*40 metres in 2 minutes. Because the traveller covered first 40 metres and two trees. Just because of the distance between two trees is 40 m.
Therefore, we will get the speed of the bus by accepting the bus ran 40*36 =1440 meters in 120 seconds.
You knew the formula in the last tutorial:-
Speed = Distance/Time
So, now apply the value of Distance and time in the formula. Look at below.
Speed= 1440/120 meter/second.
= 12meter/second
=12*(18/5) km/h
=43.2 km/h.
Is not is simple math tricks friends?
quantitative aptitude 2: A train is thrice faster than a motorbike. If the bike can cover a certain distance in 1 hour and 30 minutes, then how long time will take to cover the same distance by the train?
Solution: Let's suppose, the speed of the motorbike is x km per minute. Therefore the speed of the train will be 3x km per hours.
The question is very simple. You can solve this problem within a few seconds without knowing the formula. But the problem-solving method will help you to solve different conditional problems. It may be the first object or second object.
Share:
Post a Comment
Math Tricks of Mensuration Formulas, Triangle, Cube, Cuboid, Quadrilateral, Circle, Partnership, Pipe and cistern, Problem on Train and ages, Profit and Loss, simple-Compound interest, Simplification Solved Quantitative Aptitude, Time Speed, Distance, Time and work Age calculation, Average, Boat and Stream, Division rules LCM and HCF Shortcut Formulas.
Copyright © 2017-2018 |Newmathtricks.com | Terms of Use-Privacy Policy | About Us-Contact Us |
| 2,097 | 6,699,422 |
<MASK>
<UNMASK>
## two work problems!
<MASK>
<UNMASK>
## two work problems!
A circular swimming pool has a diameter of 16 meters, the sides are 3 meters high, and the depth of the water is 1.5 meters. The acceleration due to gravity is 9.8 and the density of water is 1000 .
How much work is required to:
(a) pump all of the water over the side?
(b) pump all of the water out of an outlet 2 m over the side?
<MASK>
<UNMASK>
## two work problems!
A circular swimming pool has a diameter of 16 meters, the sides are 3 meters high, and the depth of the water is 1.5 meters. The acceleration due to gravity is 9.8 and the density of water is 1000 .
How much work is required to:
(a) pump all of the water over the side?
(b) pump all of the water out of an outlet 2 m over the side?
I know how to find the answer to a, which is 2116800 pi Joules
But for B, what do they mean? Does 2 meter over the side mean the radius changes?
<MASK>
I know you have to do this in two parts and then add them. I am able to do an example when it is in ft and lbs but the whole kg and meter is confusing me.
<MASK>
<UNMASK>
## two work problems!
A circular swimming pool has a diameter of 16 meters, the sides are 3 meters high, and the depth of the water is 1.5 meters. The acceleration due to gravity is 9.8 and the density of water is 1000 .
How much work is required to:
(a) pump all of the water over the side?
(b) pump all of the water out of an outlet 2 m over the side?
I know how to find the answer to a, which is 2116800 pi Joules
But for B, what do they mean? Does 2 meter over the side mean the radius changes?
second problem:
A chain 69 meters long whose mass is 23 kilograms is hanging over the edge of a tall building and does not touch the ground. How much work is required to lift the top 2 meters of the chain to the top of the building? Use that the acceleration due to gravity is 9.8 meters per second squared.
I know you have to do this in two parts and then add them. I am able to do an example when it is in ft and lbs but the whole kg and meter is confusing me.
I know W=Fd, F=ma, so I did ma=9.8*23= 225.4
How can I do this problem without converting to ft and lbs? Thanks
| 576 | 6,699,423 |
Checking For Prime
<MASK>
but really we need only to go to half the numbers, upto n//2 + 1
<MASK>
then we can use it like that:
<MASK>
``````
<MASK>
``````primes = []
for i in range(10000):
prime = is_prime(i)
if prime:
primes.append(i)
<MASK>
the above checks the first time there is a difference of 100. modify the loop to check for the next 10 occurances.
<UNMASK>
Checking For Prime
<MASK>
prime numbers are surprisingly easy to check for. to check if a number is prime, we divide it by it’s factors. 1 is not prime
<MASK>
we take all numbers from 2 to the num and we start dividing by 2, 3, 4, 5, …
but really we need only to go to half the numbers, upto n//2 + 1
<MASK>
then we can use it like that:
<MASK>
``````
<MASK>
``````primes = []
for i in range(10000):
prime = is_prime(i)
if prime:
primes.append(i)
<MASK>
first_100_diff(primes)
<MASK>
the above checks the first time there is a difference of 100. modify the loop to check for the next 10 occurances.
<UNMASK>
Checking For Prime
<MASK>
prime numbers are surprisingly easy to check for. to check if a number is prime, we divide it by it’s factors. 1 is not prime
``````def is_prime(num):
if num > 1:
for i in range(2, num):
if num%i == 0:
return False
return True
<MASK>
we take all numbers from 2 to the num and we start dividing by 2, 3, 4, 5, …
but really we need only to go to half the numbers, upto n//2 + 1
<MASK>
``````
then we can use it like that:
<MASK>
``````
<MASK>
``````primes = []
for i in range(10000):
prime = is_prime(i)
if prime:
primes.append(i)
def first_100_diff(primes):
for primeA in primes:
for primeB in primes:
if abs(primeA - primeB) == 100:
print(primeA, primeB)
return
first_100_diff(primes)
<MASK>
the above checks the first time there is a difference of 100. modify the loop to check for the next 10 occurances.
<UNMASK>
Checking For Prime
programming problems
prime numbers are surprisingly easy to check for. to check if a number is prime, we divide it by it’s factors. 1 is not prime
``````def is_prime(num):
if num > 1:
for i in range(2, num):
if num%i == 0:
return False
return True
``````
we take all numbers from 2 to the num and we start dividing by 2, 3, 4, 5, …
but really we need only to go to half the numbers, upto n//2 + 1
<MASK>
``````
then we can use it like that:
``````for i in range(100):
prime = is_prime(i)
if prime: print(i)
``````
for primes upto 100. pretty easy.
## checking for the first time there is a difference of 100
``````primes = []
for i in range(10000):
prime = is_prime(i)
if prime:
primes.append(i)
def first_100_diff(primes):
for primeA in primes:
for primeB in primes:
if abs(primeA - primeB) == 100:
print(primeA, primeB)
return
first_100_diff(primes)
``````
the above checks the first time there is a difference of 100. modify the loop to check for the next 10 occurances.
<UNMASK>
Checking For Prime
programming problems
prime numbers are surprisingly easy to check for. to check if a number is prime, we divide it by it’s factors. 1 is not prime
``````def is_prime(num):
if num > 1:
for i in range(2, num):
if num%i == 0:
return False
return True
``````
we take all numbers from 2 to the num and we start dividing by 2, 3, 4, 5, …
but really we need only to go to half the numbers, upto n//2 + 1
``````def is_prime(num):
if num > 1:
for i in range(2, (num//2)+1):
if num%i == 0:
return False
return True
``````
then we can use it like that:
``````for i in range(100):
prime = is_prime(i)
if prime: print(i)
``````
for primes upto 100. pretty easy.
## checking for the first time there is a difference of 100
``````primes = []
for i in range(10000):
prime = is_prime(i)
if prime:
primes.append(i)
def first_100_diff(primes):
for primeA in primes:
for primeB in primes:
if abs(primeA - primeB) == 100:
print(primeA, primeB)
return
first_100_diff(primes)
``````
the above checks the first time there is a difference of 100. modify the loop to check for the next 10 occurances.
| 1,173 | 6,699,424 |
# What is Bigha Acre and Hectare and their value in square feet
In this article we know about What is Bigha Acre and Hectare and their value in square feet.And how we convert these land measurement unit in square feet and square yard.
we can watch this video
Many of Asian country like India, Nepal Pakistan ,Bhutan and Bangladesh in local language land measurement unit is Bigha, value of Bigha may be vary from area to area like in Uttar Pradesh, Punjab ,Haryana ,Himachal Pradesh ,Bihar Jharkhand States etc . But we try to explain Bigha, acre and hectare that is land measurement unit used in several Nation.
## How many square yard in one Bigha
How many square yard in one bigha and 1 Bigha = 44 ×44 square yard land
Hence 1 bigha = 1936 sq yard
How many square yard in one bigha Bigha
## How many square yard in one acre
How many square yard in one acre and 1Acre =44×110 square yard land
1 acre = 4840 sq yard
How many square yard in one acre Acre
## How many sq meter in 1 hectare?
Hectare is SI and international unit for land measurement that is equal to 100×100 square metre land and 10000 sq meter present in 1 hectare.
How many square metre in one hectare Hectare
### convert 1 Bigha into square yard and square feet
We know that one bigha is equal to 44 ×44 square yard
1Bigha =44× 44 yard2
1Bigha =1936 yard2
We know that 1yard2 =9 ft2
So we should convert square yard into square feet
1 Bigha =9×1936 square feet
1 Bigha = 17424 square feet
### How to Convert 1 acre into square yard and square feet
1 Acre= 44×110 square yard
1 Acre = 4840 square yard
And we know that
1 square yard = 9 ft2
So we convert square yard into square feet
1 Acre = 4840× 9 square feet
1 Acre = 43560 ft2
### how to convert 1 hectare into square feet
1 Hectare = 100×100 m2
1hectare = 10000m2
1m2 = 10.67 ft2
1 hectare = 10000×10.67 ft2
● important conversion unit:-
1) 1Bigha = 0.4 Acre
2) 1Acre = 2.5 Bigha
3) 1Bigha = 0.162 hectare
4) 1hectare =6.175 Bigha
5)1Acre =0.405 hectare
6) 1hectare =2.47 Acre
how many acre in 45 bigha
We know that 1bigha =0.4 Acre
45Bigha = 0.4×45 Acre
45 bigha = 18 Acre Ans.
how many Bigha in 10 acre
We know 1 acre =2.5 Bigha
10 acre = 2.5×10 bigha
10 Acre =25 Bigha Ans.
how many hectare in 50 Bigha
We know that 1 Bigha= 0.162 hectare
50 bigha =0.162 ×50 hectare
50 bigha = 8.10 hectare Ans.
how many acre in 10 hectare
We know 1hectare =2.47 acre
10 hectare = 2.47 ×10 acre
10 hectare = 24.7 Acre Ans.
how many hectare in 20 acre
We know that
1 Acre =0.405 Hectare
20 Acre = 0.405×20 hectare
20 Acre = 8.1 hectare Ans.
Now your turns:-If you are happy to see this post please like comment and share to your friend and if any query and questions about this topics please ask, your questions are most welcome
Now you can follow me on on Facebook and subscribe my youtubechannel
### 1 thought on “What is Bigha Acre and Hectare and their value in square feet”
1. Details in measurment of land in least count
| 992 | 6,699,425 |
Activities on Factoring Polynomials?
Factoring polynomials is one of the more difficult concepts of algebra because students must use a variety of different strategies to factor different types of equations. The use of math manipulati... Read More »
http://www.ehow.com/info_8155700_activities-factoring-polynomials.html
Top Q&A For: Activities on Factoring Polynomials
Directions on Factoring Polynomials?
Polynomials are algebraic equations containing two squared or two cubed numbers. Each kind of polynomial equation contains one squared or cubed algebraic letter, i.e. x or y, and one squared or cub... Read More »
http://www.ehow.com/how_8061654_directions-factoring-polynomials.html
How to Do Multiplying & Factoring Polynomials?
Polynomials are expressions containing variables and integers using only arithmetic operations and positive integer exponents between them. All polynomials have a factored form where the polynomial... Read More »
http://www.ehow.com/how_8092395_do-multiplying-factoring-polynomials.html
Tips on Factoring Polynomials?
Factoring is the rewriting of an expression to produce individual numbers that when worked out again, make the original expression. Polynomials are expressions with multiple numbers or variables. W... Read More »
http://www.ehow.com/info_8310119_tips-factoring-polynomials.html
Methods of Factoring Polynomials?
Polynomials contain two or more terms. Many can be factored, but there are prime polynomials that cannot. A factor of a number (or term) will divide the number exactly without leaving a remainder. ... Read More »
http://www.ehow.com/info_8111229_methods-factoring-polynomials.html
Related Questions
| 355 | 6,699,426 |
# Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this. 2x + 10y = 58 -14x + 8y = 140
91,718 questions, page 831
1. ## ALGEBRA 2 (LOGARITHMIC FUNCTIONS)
These are some I chose to practice but I have a hard time knowing how to do it. The answers are online but that doesn't help. These deal with logarithmic functions etc. PLEASE HELP! THANKYOU 1.)solve the equation: 3^(x+1)=27^(x+3) 2.) e^x=5 3.) 2^(3x)+9=25
asked by DINA on April 15, 2015
2. ## Math Check
Solve the inequality. So i have done this problem at least three times and I keep getting the same answer. But in the back of the book the correct answer for this problem is x
asked by Heather on August 25, 2010
3. ## Algebra
I'm studying for a test so I want to make sure I'm doing these problems right. Numerator (-5a exponent -2) Denominator b3 All to the -1 exponent For a= -4. And b=2 My answer I get is 1/10 is the right ? -16 with -1/2 exponent Would the answer be -4? 2-3
asked by Lexi on November 22, 2014
4. ## PHYSICS!!!
To simulate the extreme accelerations during launch, astronauts train in a large centrifuge. If the centrifuge diameter is 13.5m , what should be its rotation period to produce a centripetal acceleration of If the centrifuge diameter is 13.5m , what should
asked by Amy on September 14, 2013
5. ## Calculus
a rectangular box of given volume is to be "a" times long as it is wide. find the dimension for which it has the least total surface area ? L= aw Total surface area= 2LW + 2Lh + 2Wh = 2aw^2 + 2awh + 2wh But volume= lwh= 2aw^2 h or h= V/2aw^2 Total Surface
asked by Hero on December 10, 2006
6. ## Physics
The temperature of 2.0 g of helium is increased at constant volume by ΔT. What mass of oxygen can have its temperature increased by the same amount at constant volume using the same amount of heat? I know for constant volume I should use the equation:
asked by Joshua on December 2, 2015
7. ## college chemistry
A zinc-copper battery is constructed as follows: Zn | Zn+2(0.10 M) || Cu+2 (2.50 M)| Cu The mass of each electrode is 200.0 g. Each half cell contains 1.00 liter of solution. a) Calculate the cell potential when this battery is first connected. b)
asked by Stephanie on April 28, 2010
8. ## english essay
Life after College Caitlin Petre is a graduate from Stanford University. She graduated with a 3.90 GPA with a Master’s degree in Philosophy. One day after graduation, Petre went for an interview for a job and she was asked to fill out a W-4 form. When
asked by julie on November 28, 2007
9. ## to writ teacher
Life after College Caitlin Petre is a graduate from Stanford University. She graduated with a 3.90 GPA with a Master’s degree in Philosophy. One day after graduation, Petre went for an interview for a job and she was asked to fill out a W-4 form. When
asked by julie on November 28, 2007
10. ## chemistry
N2(g) + 3 F2(g) → 2 NF3(g) ΔH° = –264 kJ/mol ΔS° = –278 J/(mol∙K) a. Using the information provided above (only), calculate the maximum amount of non-PΔV work that can be accomplished through this reaction at a temperature of 500°C. b.
asked by chamy on April 21, 2012
11. ## math
since 120/500 is equal to 100 that means ...............i still don't get it......... please..... i need help.can you at least give me the answer please tbhis is hohnors math and for me its hard........ Tom,Ted,Tony,Terry worked together on a job.The job
asked by Katherine on November 9, 2009
12. ## Physics
An engineer has an odd-shaped 11.4 kg object and needs to find its rotational inertia about an axis through its center of mass. The object is supported on a wire stretched along the desired axis. The wire has a torsion constant κ = 0.428 N·m. If this
asked by Brent on May 16, 2012
13. ## English
Read each sentence identify and rewrite the subordinate clause 1. Adam studied for his science test because he wanted to do well. because he wanted to do well 2.You should turn the music down so that your brother can study. so that your brother can study
asked by Jerald on January 24, 2013
14. ## College Accounting
Your client is preparing financial statements to show the bank. You know that he has incurred a computer repair expense during the month, but you see no such expense on the books. When you question the client, he tells you that he has not received the
asked by Sheryl on February 14, 2010
15. ## physics
You let out a shout, echo comes back and reaches you in 1.4 seconds. if the speed of sound in air is 343 m/s, what is the distance to the cliff? a. need assitance in setting up problem? Let d be the distance to the cliff, and V be the speed of sound. The
asked by tim on September 3, 2006
16. ## Precalculus
Records from the buzz sawmill show that workers have 0.03 probability of losing a leg and a 0.04 probability of losing an arm in any one year. Armand leg casualty insurance company plans to offer insurance policies for the workers. The company will pay
asked by Anonymous on June 9, 2014
17. ## Chemistry
Fluorine reacts with oxygen to yield oxygen difluoride. 2 F2(g) + O2(g) 2 OF2(g) What is the value of K if the following concentrations are found at equilibrium: [O2]= 0.200 mol/L, [F2]=0.0100 mol/L, and [OF2]=0.0633 mol/L Chemistry - DrBob222, Thursday,
asked by Sally on March 2, 2012
18. ## Math
Issa is driving to the airport to catch a flight to Miami, and her arrival depends on traffic. If the traffic is light, then she can drive 60 mph and arrive at the airport 1 hour early. If traffic is heavy, she can drive 35 mph and arrive at the airport on
asked by Jane on July 10, 2012
19. ## Statistics: 82% Confidence Interval
I'm having some difficulty with this problem: Over the last 15 Major League Baseball seasons, the mean # of strikeouts by the American League leader is 258.5. Assuming that the # of strikeouts by the league leader is normally distributed & the standard
asked by melanie on June 22, 2008
20. ## English
Can you check these other sentences, too. Thank you. 1) Michael Brown believes that shows such as the “Funniest Home Videos” have a negative effect on the children. 2) They show children making fun of their parents while the TV hosts, along with the
asked by Henry2 on September 9, 2011
21. ## Calculus
1. The problem statement, all variables and given/known data a ball is thrown straight down from the top of a 220 feet building with an initital velocity of -22ft/s. What is the velocity of the ball after 3 seconds? What is the velocity of the ball after
asked by Audrey on February 3, 2012
1. The problem statement, all variables and given/known data a ball is thrown straight down from the top of a 220 feet building with an initital velocity of -22ft/s. What is the velocity of the ball after 3 seconds? What is the velocity of the ball after
asked by Desperate!!!! on February 3, 2012
Last year, Mark was 46 inches tall. This year, Mark's height is 3 inches less than Pter's height. Peter is 51 inches tall. How tall is Mark? Write an equation to represent the situation. Then solve the equation. Answer: 51-3=I I=48 inches tall I=number of
asked by Javonte on January 14, 2015
24. ## calculus
The differential equation below models the temperature of a 87°C cup of coffee in a 17°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 67°C. Solve the differential equation to find an expression for
asked by alex on April 27, 2017
25. ## Chemistry
Consider the isomerization equlibria of an alkene (C4H8) that co-exists as three isomers. (These 3 are in an equlibrium triangle) Reaction 1: cis-2-butene ⇔ trans-2-butene Reaction 2: cis-2-butene ⇔ 2-methylpropene Reaction 3: trans-2-butene ⇔
asked by Gleb on March 3, 2013
26. ## A comparison and contrast essay uses which of the
1. A comparison-and-Contrast essay uses which of the following elements? (Select all that apply) A. two subjects that are similar and different B. an opinion to change the readers mind C. supporting factual details D. the relationship between cause and
asked by Jason Sameous on November 9, 2016
27. ## STAT
Numerous studies have found that males report higher self-esteem than females, especially for adolescents (Kling, Hyde, Showers, & Buswell, 1999). Typical results show a mean self-esteem score of M 39.0 with SS 60.2 for a sample of n 10 male
asked by Nickie on February 10, 2013
28. ## Geometry/Trig
Hello, I would appreciate it if... 1. If you could give me information on finding the area of a circle inscribed in an equilateral triangle. 2. This problem: A running track is shaped like a rectangle with a semicircle on each of the shorter sides. The
asked by Lisa on October 10, 2006
29. ## Economy Essay Outline/Structure Help
I'm writing a synthesis paper about why we should utilize middle-out economics (basically economics that focus on the middle class) I'm very new to the topic so I don't want to spread myself too thin or dive into something I don't know too much about. I'm
asked by Tyler on August 10, 2014
30. ## Social Studies I NEED HELP PLEASE!!!
Which best describes how land ownership changed after Ukrainian independence? A. land ownership has stayed the same B. lands formerly owned by the state have been privatized. C. lands formerly owned by individuals have been seized by the state D.
asked by Sarah on April 27, 2015
31. ## CHemistry....Please double check DrBobb
What mass of ethylene glycol C2H6O2, the main component of antifreeze, must be added to 10.0 L water to produce a solution for use in a car's radiator that freezes at -23C? Assume density for water is exactly 1g/mL. First we find out the freezing point:
asked by Saira on January 24, 2009
32. ## chemistry
A pan full of hot salt water, NaCl(aq), is cooled and NaCl(s) precipitates. Explain why this happens. Chloride ions are more strongly attracted to the metal in the pan than the sodium ions at low temperatures. As the vapor pressure of the solution
asked by krystal on February 21, 2012
33. ## English Expressions
1. She did her part, and the native speaker did her part. They were cooperative and did their parts properly. 2. Teachers should make students say, act out and work together in class. 3. The quality of a teacher's voice is important in teaching. If she
asked by John on January 9, 2008
34. ## fractions
fraction word problem. please advise on how to work the problem Phillip and Jane had a pie each. Phippip ate 1/4 of his and Jane ate more of the pie than Phillip. Which fraction of his pie could Jane have eaten. Possible answers: 2/5 1/6 2/8 or 1/8 Please
asked by Suzanne on June 10, 2011
35. ## Physics
An electron is accelerated through a uniform electric field of magnitude 2.5x10^2 N/C with an initial speed of 1.2x10^6 m/s parallel to the electric field. a) Calculate the work done on the electron by the field when the electron has travelled 2.5 cm in
asked by Linn on April 21, 2009
36. ## Physics - important
An electron is accelerated through a uniform electric field of magnitude 2.5x10^2 N/C with an initial speed of 1.2x10^6 m/s parallel to the electric field. a) Calculate the work done on the electron by the field when the electron has travelled 2.5 cm in
asked by Linn on April 21, 2009
37. ## finance/math
A group of economics students gathered to study for a test on the money and banking system in the U.S. During a fast and furious brainstorm session, Jill scribbled down several key phrases she will use to study tomorrow. Unfortunately, in her haste, all
asked by kim on May 16, 2011
38. ## Chemistry
If you divide the molar mass of a compound by the empirical formula mass, what is the result? That division should be VERY close to a whole number. Usually, 0.9 to 1.1 or 1.9 to 2.1 (all depending upon the accuracy and precision of experimental data). That
asked by Chris on February 4, 2007
39. ## social studies and language arts
Read the following scenario: You are a second-grade teacher at Happy Valley Elementary School. You are teaching language arts and social studies. Your classroom of 21 students consists of 7 white students, 5 Latino students, 4 African American students, 3
asked by tammie on October 3, 2009
40. ## math
If cos(3 x)6Ó14 x e^2 y=0, find [ dy/dx] using implicit differentiation. In this problem, you should differentiate both sides of this equation with respect to x. First, using implicit differentiation, differentiate the left side of the equation with
asked by Zhao on September 30, 2012
41. ## Chemistry
Carminic acid,a naturally occuring red pigment extracted from the cochineal insect, contains only C,H,and O. It was commonly used as a dye in the first half of the nineteenth century. It is 53.66% C and 4.09% H by mass. A titration of the 0.3602 g sample
asked by Jake on August 27, 2007
42. ## Algebra 2
For the annual rate of change of +70%, find the corresponding growth or decay factor. 0.70 1.70 170 70 For the annual rate of change of -75%, find the corresponding growth or decay factor. -75 1.25 -0.75 0.25 I'm confused, please help? How can I solve
43. ## Science 10
I was given a question of "The ollowing apperatua was set up and left undisterbed for 24 hours. After this time, a change had taken place in the size of the dialysis tube. Describe what happend and why. Use the terms hypertonic, Hypotonice, and
asked by Mary on July 16, 2005
44. ## chemistry Dr BOB
Am I correct? From data below, calculate the total heat (J) needed to convert 0.539 mol gaseous ethanol at 300°C and 1 atm to liquid ethanol at 25.0°C and 1 atm. Boiling point at 1 atm 78.5°C c ethanol gas 1.43 J/g·°C c ethanol liquid 2.45 J/g·°C
asked by patrick on October 16, 2015
45. ## stats
4. Data from the 2008 General Social Survey (GSS) sample show that the mean number of children per respondent was 1.94 with a standard deviation of 1.70. A total of 2,020 people answered this question. (a) Estimate the population mean number of children
asked by taylor quenzler on July 26, 2012
46. ## Algebra
1) If N is the set of natural numbers that are factors of 18, represent this set in roster form. 2) Suppose U={1,2,3,4,5,6,7,8,9,10} is the universal set and A={1,4,5,9,10} What is A? 3) The wrestling team is holding a car wash. The teams goal is to raise
47. ## Algebra
1) If N is the set of natural numbers that are factors of 18, represent this set in roster form. 2) Suppose U={1,2,3,4,5,6,7,8,9,10} is the universal set and A={1,4,5,9,10} What is A? 3) The wrestling team is holding a car wash. The teams goal is to raise
asked by Asia Powell on March 13, 2014
48. ## Algebra
1) If N is the set of natural numbers that are factors of 18, represent this set in roster form. 2) Suppose U={1,2,3,4,5,6,7,8,9,10} is the universal set and A={1,4,5,9,10} What is A? 3) The wrestling team is holding a car wash. The teams goal is to raise
49. ## chemistry
From data below, calculate the total heat (J) needed to convert 0.539 mol gaseous ethanol at 300°C and 1 atm to liquid ethanol at 25.0°C and 1 atm. Boiling point at 1 atm 78.5°C c ethanol gas 1.43 J/g·°C c ethanol liquid 2.45 J/g·°C ÄH°vap 40.5
asked by patrick on October 15, 2015
50. ## chemistry
From data below, calculate the total heat (J) needed to convert 0.539 mol gaseous ethanol at 300°C and 1 atm to liquid ethanol at 25.0°C and 1 atm. Boiling point at 1 atm 78.5°C c ethanol gas 1.43 J/g·°C c ethanol liquid 2.45 J/g·°C ÄH°vap 40.5
asked by patrick on October 16, 2015
51. ## chemistry
From data below, calculate the total heat (J) needed to convert 0.539 mol gaseous ethanol at 300°C and 1 atm to liquid ethanol at 25.0°C and 1 atm. Boiling point at 1 atm 78.5°C c ethanol gas 1.43 J/g·°C c ethanol liquid 2.45 J/g·°C ÄH°vap 40.5
asked by patrick on October 15, 2015
52. ## chemistry
From data below, calculate the total heat (J) needed to convert 0.539 mol gaseous ethanol at 300°C and 1 atm to liquid ethanol at 25.0°C and 1 atm. Boiling point at 1 atm 78.5°C c ethanol gas 1.43 J/g·°C c ethanol liquid 2.45 J/g·°C ÄH°vap 40.5
asked by patrick on October 16, 2015
53. ## Physics
A block of mass 4 kg, which has an initial speed of 2 m/s at time t = 0, slides on a horizontal surface. Find the magnitude of the work that must be done on the block to bring it to rest. Answer in units of J. If a constant friction force of magnitude 5
54. ## chemistry
Given 20g of benzoic acid contaminated with 0.4g of salicytic acid. 1)Determine the volume of hot water required to dissolve the mass of acid. 2)Calculate the mass of benzoic acid which participates when the solution is cooled to 20°C. Deduce. 3)Calculate
asked by Jenny on October 21, 2015
55. ## trig is killing me...
the angles of elevation of a tower at two places due west of it are 63 degrees and 56 degrees. given that the foot of the tower and the two points are on ground level and the distance between the two points is 20 meters, find the height of the tower. You
asked by holly on December 7, 2006
56. ## maths
each of the students in a class writes a dirrerent 2 digit number on the whiteboard. the teacher claims that no matter what the students write, there will be at least three numbers on the whiteboard whose digits have the same sum. what is the smallest
asked by Emma on July 7, 2007
57. ## Physics >.<
A big truck has a mass of 5,000kg. It is moving down the road at a velocity of 15m/s. As it approaches a stop sign, the driver tries to stop, but the truck slides on ice and hits a car (mass = 500kg) that is stopped. After the collision, they move
asked by Devynn on December 10, 2014
58. ## Chemistry
You are handed a well insulated container holding about 100ml of water at a temp between 75 deg C and 100 deg C. you are given a thermometer with a maxmimum temp of 50 deg C. create an experiment to determine the temp of the water. i am really confused on
asked by Anonymous on October 18, 2016
59. ## Chemistry
I don't understand how to work out this problem, non the less where to begin with any calculations. Help!! Three of the strongest lines in the He+ ion spectrum are observed at the following wavelength: (1) 121.57 nm (2) 164.12 nm (3) 468.90 nm Find the
asked by Very Confused :( on November 18, 2010
60. ## Art
The United States supreme Courts architectural design symbolizes what values that are also shared with Greek culture? A. resistance and persistence B. creativity and love C. tranquility and peace D. balance and justice I feel like it's D or B 2.Realists
asked by Anonymous on March 11, 2015
61. ## physics
Use principles of force and kinematics to answer these: a. A 40 kg skater pushes on the 60 kg skater (initially at rest). As they collide, the 40 kg skater pushes on the 60 kg skater, applying a force 100 Newtons for 1.2 seconds. Find the velocity at which
asked by jen on November 1, 2010
62. ## Physics
Use principles of force and kinematics to answer these: a. A 40 kg skater pushes on the 60 kg skater (initially at rest). As they collide, the 40 kg skater pushes on the 60 kg skater, applying a force 100 Newtons for 1.2 seconds. Find the velocity at which
asked by jen on November 1, 2010
63. ## math
The driving distance between Manchester and London is 195 miles. Farris intends to travel from Manchester to London by coach. The coach will leave Manchester at 3:30pm Fairs assumes that the coach will travel at an average speed of 50 mph. A) Using his
asked by Jesss on September 29, 2018
64. ## statistics
1) One hundred clients of a registered tax agent fall into the categories of personal (25 male, 30 female), corporate (10 male, 15 female) and small business (8 male, 12 female). The agent selects one of the clients at random to begin work. Find the
asked by purna on August 13, 2014
65. ## series and parellel circuits
ok here we go ... its a series-parellel circuit with 5 resistors. 12 volt battery- wire goes up then one wire branches up at 45 deg with a 10 ohm resistor then down at a 45 deg with a 10 ohm resistor, a second wire branches down at 45 deg with a 20 ohm
asked by Dave on May 21, 2012
66. ## Math
Chris has a 10 foot long strip of red ribbon and a 15 foot long strip of blue ribbon. A] draw a diagram to show the relationship of the length of the red ribbon to the length of the blue ribbon. B] write a ratio to describe the relationship of the length
asked by Samantha on September 5, 2016
67. ## Alg II
I have a rectangle whose sides are in the proportion b:a, where b is the longer side. I draw a line parallel to a inside the rectangle in order to break it up into a square with sides of length a and a second rectangle. Amazingly, the sides of the new
asked by Peter on January 11, 2011
68. ## math
Jen, I will attempt to write the others out. #3)ADD: -57 + (-22) I got -79 #4)Subtract -2.3-(-8.8) I got -6.5 #7)perform indicated operation: -21 - (-9) I tried to underline ----------- -21 - (-21) I got -12 over o or undfined #8)Multiply:(4){-3/2} This
asked by barbara on July 18, 2010
69. ## Calculus
Integrate 1/sinx dx using the identity sinx=2(sin(x/2)cos(x/2)). I rewrote the integral to 1/2 ∫ 1/(sin(x/2)cos(x/2))dx, but I don't know how to continue. Thanks for the help. Calculus - Steve, Tuesday, January 12, 2016 at 12:45am 1/2 ∫
70. ## Int. Macro
Suppose the nation’s capital stock is equal to 3200 in 2011 and that a rise in the marginal product of capital raises the desired capital stock to 3800 in 2012. Suppose also that the desired capital stock remains at 3800 in subsequent years. Assume also
asked by robson on October 26, 2012
71. ## CRT 205
Crt 205 Vacuum Sales Digital Story Text Axia College Material Text of the Vacuum Sales Digital Story This text provides the script of the Vacuum Sales digital story. Use this text as an alternative or supplement to viewing the digital story itself. Script
asked by Unknown on October 19, 2010
72. ## AP chemistry
A certain complex of metal M is formulated as MCl3 ∙ 3 H2O . The coordination number of the complex is not known but is expected to be 4 or 6. (a) Would conductivity measurements provide information about the coordination number? (b) In using
asked by stark11 on September 22, 2015
73. ## Math
An open box contains 80cm^3 and is made from a square piece of tinplate with 3cm squares cut from each of its four corners. Find the dimensions of the original piece of tinplate. A=80 A=(length)(width)(height) since length and width are the same, and when
asked by Anonymous on October 7, 2010
74. ## Physics
An object is placed 30mm in front of a lens. An image of the object is located 90mm behind the lens. a) Is the lens converging or diverging? explain. b) What is the focal length of the lens? c) Draw a diagram with lens at x=0, locate the image. d) Is the
asked by Harly on May 30, 2011
75. ## Language Arts
1.)Knowing that the Latin root -equi- means "equal," choose the best meaning for the word equilateral as it is used in the sentence below. The box appears to be square, but I would have to measure its sides to prove that it is equilateral. A. equally
asked by Helen on November 28, 2016
76. ## SS
Well, i am going to ask them anyway! Where was the ancient region of Mesopotamia located? A. Africa B. Europe C. Middle East D. South America What was the world’s first system of writing called? A. Arabic B. cuneiform C. hieroglyphics D. Sanskrit Which
asked by Mary 💖💖 on October 29, 2018
77. ## MARKETING
1. Introduction – You need to work out what is the business scope of the company. That is what business are they in? 2. You will conduct an internal environment analysis – with a particular view to how well the company is placed to operate in another
asked by VISHNU on March 7, 2009
78. ## English Expressions
5. There was a white board, which seemed to be better than a blackboard in terms of health and convenience. 6. The native speaker threw a pack of sentence cards to each group so they started to do the group activity. 7. The use of power point was
asked by John on January 9, 2008
79. ## chemistry
you want to determine the amount of chlorophyll(Molecular weight=893.51g/mol)in a 25ml marine sample.before analysis by UV spectoscopy,2ml of sollution was centrifuged.0.5ml of the supernatant was diluted to 150ml with DI water.the diluted sample was
asked by ussy on December 10, 2014
80. ## English-Ms. Sue
Ms. Sue the A-z story I did on the Ugly duckling was a rough draft. Now I have to polish it, and these are the postive and negative comments I received from some classmates. Postive ones: -Cute -Descriptive -Good ending -Good grammar Negative ones: -No
asked by Sara on May 31, 2010
81. ## physics!
In its search for flying insects, a bat uses an echolocating system based on pulses of high frequency sound. These pulses are 2.0 ms in duration, have a frequency of 50 kHz, and an intensity level of 100 dB at 1.0 m from the bat’s mouth. Assume the bat
asked by Jamie on April 16, 2016
82. ## PHYSICS!!
In its search for flying insects, a bat uses an echolocating system based on pulses of high frequency sound. These pulses are 2.0 ms in duration, have a frequency of 50 kHz, and an intensity level of 100 dB at 1.0 m from the bat’s mouth. Assume the bat
asked by Ben on April 13, 2016
83. ## Chem
Manganese dioxide reacts with bromide ions to form manganese ions and bromate ions in an acidic aqueous solution. Write the balanced equation for this reaction. 1)Br- + MnO2 + 3H2O --> Mn2+ + BrO3- + 2H3O+ 2)Br- + 3MnO2 + 6H3O+ ---> 3Mn2+ + 9H2O + BrO3-
asked by Rob on December 12, 2007
84. ## COM/155
I wrote these 5 sentences for a assignment and would like some feeback on them as to what I may need to work on. 1.Our education plays (simple present tense) a major role in our future finances. 2.My financial planning class opened (simple past tense) my
asked by Brandy on November 19, 2010
85. ## physics
A trapeze artist, starting from rest, swings downward on the bar, lets go at the bottom of the swing, and falls freely to the net. An assistant, standing on the same platform as the trapeze artist, jumps from rest straight downward. Friction and air
asked by Papito on May 7, 2007
86. ## Statistics
A recruiter estimates that if you are hired to work for her company and you put in a full week at the commissioned sales representative position she is offering, you will make “$525 plus or minus$250, 80% of the time.” She adds, “It all depends on
asked by shuggins on June 11, 2015
87. ## chemistry
Determine the density of methane (CH4) in g/L at 24.7oC and 1.1 atm ideal gas eqn: P = (rho)RT, where: pressure: P = 1.1 atm specific gas constant: R, R = |R / M, where |R = 0.0821 L-atm / K-mol and, molecular weightCH4: M = 0.016 kg/mol R = 0.0821/0.016 =
asked by kellie on October 16, 2010
Employed persons 2006-07 in Australia: -Proffestionals 19.3% -Intermediate clerical and service workers 16.5% -Associated professionals 12.9% -Tradespeople and related work 12.7% -Elementary clerical and service workers 9.3% -Labourers and related workers
asked by Amethyst on February 8, 2012
89. ## Algebra
Celsius temperature readings can be converted to Fahrenheit readings using formula F=9/5c+32 What is the fahrenheit temp. that corresponds to each of the following Celsius temperatures -10, 0, 15, 1000? I will be happy to critique your work on this. You
asked by Jean on September 16, 2006
90. ## Physics
A simple pendulum of length 2.00m is made with a mass of 2.00kg. The mass has speed of 3.00 m/s when the pendulum is 30 degrees above its lowest position. -What is the maximum angle away from the lowest position the pendulum will reach? -What is the speed
asked by Alex on November 5, 2014
91. ## Physics
A simple pendulum of length 2.00m is made with a mass of 2.00kg. The mass has speed of 3.00 m/s when the pendulum is 30 degrees above its lowest position. -What is the maximum angle away from the lowest position the pendulum will reach? -What is the speed
asked by Alex on November 9, 2014
92. ## Biomechanics
If the volume of a material is conserved under loading (Volume without force = Volume with force) what is the material’s Poisson’s ratio? You can assume that the material is isotropic (it will behave the same in every direction). Hint: take a cube and
asked by Aaron on October 14, 2012
93. ## Math
I can't to get this problem. Perform the indicated operation 11 -6/7 -7 Can someone help me??? It's easier to perform the 11-7 first. 11-7 is 4. Now, take 4 of something. Cut one of them up into 7 pieces and take away 6 of the 7 pieces. What are you left
asked by Norma on May 12, 2007
94. ## Calculus
A festivals being planned. The planners need to enclose to adjacent 200 M^2 areas with fencing. They have budgeted $1000 for fencing. Fencing currently cost$10/meter. The diagram of the area is as follows: 1. Write an equation representing the total
asked by Roygbiv on April 6, 2016
95. ## math,help
can someone help me set this word problem up into equations so i can solve. I don't want the answer just if you could help me with how to set it up. thanks The base of ladder is 14feet away from the wall. The top of the ladder is 17feet from the floor.Find
asked by jasmine20 on April 20, 2007
96. ## Simple Interest
A \$4000 loan made at 11.75% is to b repaid in three equal payments, due 30, 90, and 150 days, respectively, after the date of the loan. Detemine the size of the payments. Let the amount of each payment be represented by x. P1= ______x__________=0.9904349x
asked by Thara on July 31, 2009
97. ## English expression
What sport do you like best? I like rope jumping best. What sports do you like best? I like cricket and dodgeball best. Which sport do you like better, billiards or squash? I like billiards better. Do you play rugby sometimes? Yes, I do. Do you play sports
asked by John on April 14, 2008
98. ## Physics
You and your friends are organizing a trip to Europe. Your plan is to rent a car and drive through the major European capitals. By consulting a map you estimate that you will cover a total distance of 5000 \rm km. Consider the euro-dollar exchange rate
asked by John on January 23, 2010
99. ## critique my essay
Hybrid English course is good but it has also bad sides. If a student is looking for flexible time and busy of work hybrid class is your choice. I enjoyed this class and my writing has improved. I found out what are my bad habits in writing, and I also
asked by grammar :( on July 13, 2011
100. ## Math
Robin and Emily are selling boxes of candy bars. Robin sells 36 boxes of candy and Emily sells 24 boxes of candy. Which expression, using the form a(b+c), represents the amount they sold together a. 2(12+24) b. 4(9+6) c. 24 (6 + 36) d. 12 (3 + 2) My answer
asked by Matt on October 4, 2016
| 8,668 | 6,699,427 |
The Fibonacci Roulette Strategy: Maximizing Wins Through Mathematical Patterns - CasinoBabes
# The Fibonacci Roulette Strategy: Maximizing Wins Through Mathematical Patterns
The Fibonacci Roulette Strategy: Maximizing Wins Through Mathematical Patterns
# The Fibonacci Roulette Strategy: Maximizing Wins Through Mathematical Patterns
By Walter Hemphill, Editor at CasinoBabes.net
When it comes to playing Roulette, many gamblers are constantly on the lookout for strategies that can increase their chances of winning. One such strategy that has gained popularity among players is the Fibonacci Roulette Strategy.
## Understanding The Fibonacci Sequence
The Fibonacci sequence is a mathematical pattern that appears in various natural phenomena, including the growth patterns of plants, the structure of galaxies, and even the stock market. In the world of Roulette, this sequence can be utilized to maximize wins.
The Fibonacci sequence starts with 0 and 1, with each subsequent number being the sum of the two preceding numbers. The sequence goes as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. The numbers in this sequence can be used as betting units when playing Roulette.
## The Fibonacci Roulette Strategy
The Fibonacci Roulette Strategy is a progressive betting system that involves placing bets based on the Fibonacci sequence. It is commonly used with even-money bets such as Red/Black, Odd/Even, or 1-18/19-36.
Here’s how the strategy works:
1. Start by betting one unit (which is equivalent to the table’s minimum bet).
2. If you win, repeat the same bet for the next round.
3. If you lose, move to the next number in the Fibonacci sequence and bet that amount.
4. Continue this pattern until you win, at which point you go back two numbers in the sequence and bet that amount.
5. If you reach the beginning of the sequence without winning, start again from the beginning.
By following this betting pattern, players can take advantage of winning streaks and minimize losses during losing streaks. The idea is that a win will cover all previous losses and leave the player with a profit.
## FAQs
### Q: Is the Fibonacci Roulette Strategy guaranteed to work?
A: No betting strategy can guarantee consistent wins in Roulette. The Fibonacci strategy is based on mathematical patterns and can be effective in certain situations, but it is not foolproof.
### Q: Can I use the Fibonacci Roulette Strategy on other casino games?
A: The Fibonacci strategy can be applied to other casino games that involve even-money bets, but its effectiveness may vary depending on the game’s rules and odds.
### Q: What bankroll do I need to use the Fibonacci Roulette Strategy?
A: It is recommended to start with a sufficient bankroll that allows you to go through multiple losing streaks without running out of funds. The larger your bankroll, the better chances you have of recovering from losses using this strategy.
### Q: Are there any risks associated with using the Fibonacci Roulette Strategy?
A: Like any betting system, the Fibonacci strategy carries the risk of losing money. It is important to set limits, manage your bankroll wisely, and understand that no betting system can guarantee success.
## Conclusion
The Fibonacci Roulette Strategy can be an interesting approach for players who enjoy utilizing mathematical patterns in their gameplay. However, it is important to remember that Roulette is ultimately a game of chance, and no strategy can guarantee consistent wins. The Fibonacci strategy can be a useful tool for maximizing wins during winning streaks and minimizing losses during losing streaks, but it should be used with caution and a clear understanding of its limitations.
8 months ago
0 91
8 months ago
0 95
8 months ago
0 87
| 767 | 6,699,428 |
<MASK>
<UNMASK>
<MASK>
For 3x - 2y = - 12 3x - 2(0) = - 12, so 3x = - 12, so x = - 4 The x-intercept is (- 4, 0) 3(0) - 2y = - 12, so - 2y = - 12, so y = -12/-2, so y = 6 The y-intercept is (0, 6)
<MASK>
<UNMASK>
<MASK>
For 3x + y = 9 3x + 0 = 9 so 3x = 9, so x = 3 The x-intercept is (3, 0) 3(0) + y = 9 so 0 + y = 9, so y = 9 The y-intercept is (0, 9)
For x + 4y = 8 x + 4(0) = 8, so x = 8 The x-intercept is (8, 0) 0 + 4y = 8, so 4y = 8, so y = 2 The y-intercept is (0, 2)
For 3x - 2y = - 12 3x - 2(0) = - 12, so 3x = - 12, so x = - 4 The x-intercept is (- 4, 0) 3(0) - 2y = - 12, so - 2y = - 12, so y = -12/-2, so y = 6 The y-intercept is (0, 6)
<MASK>
<UNMASK>
<MASK>
Explanation: Once you identify two points of a line, the line may be drawn on a coordinate graph. Finding the X and Y intercepts is fairly easy to do. The X-intercept will have an ordered pair of (number, 0) and the Y-intercept will have an ordered pair of (0, number). When given an equation, to find the X-intercept, simply substitute '0' for the 'y' and determine the value of 'x'. To find the Y-intercept, simply substitute '0' for the 'x' and determine the value of y. Observe these examples:
<MASK>
For 3x + y = 9 3x + 0 = 9 so 3x = 9, so x = 3 The x-intercept is (3, 0) 3(0) + y = 9 so 0 + y = 9, so y = 9 The y-intercept is (0, 9)
For x + 4y = 8 x + 4(0) = 8, so x = 8 The x-intercept is (8, 0) 0 + 4y = 8, so 4y = 8, so y = 2 The y-intercept is (0, 2)
For 3x - 2y = - 12 3x - 2(0) = - 12, so 3x = - 12, so x = - 4 The x-intercept is (- 4, 0) 3(0) - 2y = - 12, so - 2y = - 12, so y = -12/-2, so y = 6 The y-intercept is (0, 6)
<MASK>
<UNMASK>
# Find the X and Y Intercepts, Given the Equation - Set 1
Explanation: Once you identify two points of a line, the line may be drawn on a coordinate graph. Finding the X and Y intercepts is fairly easy to do. The X-intercept will have an ordered pair of (number, 0) and the Y-intercept will have an ordered pair of (0, number). When given an equation, to find the X-intercept, simply substitute '0' for the 'y' and determine the value of 'x'. To find the Y-intercept, simply substitute '0' for the 'x' and determine the value of y. Observe these examples:
Examples of finding the X and Y intercepts for the given equations:
For 3x + y = 9 3x + 0 = 9 so 3x = 9, so x = 3 The x-intercept is (3, 0) 3(0) + y = 9 so 0 + y = 9, so y = 9 The y-intercept is (0, 9)
For x + 4y = 8 x + 4(0) = 8, so x = 8 The x-intercept is (8, 0) 0 + 4y = 8, so 4y = 8, so y = 2 The y-intercept is (0, 2)
For 3x - 2y = - 12 3x - 2(0) = - 12, so 3x = - 12, so x = - 4 The x-intercept is (- 4, 0) 3(0) - 2y = - 12, so - 2y = - 12, so y = -12/-2, so y = 6 The y-intercept is (0, 6)
Directions: Each equation is written in Standard Form. For each equation, find the X and Y intercepts using the steps shown. Try to accomplish this by mental math, and use pencil and paper if needed. Strive first for high accuracy, and then strive to increase your speed. Good luck and I hope you enjoy the challenge!
...
| 1,333 | 6,699,429 |
# Fractions
<MASK>
At the end of the lesson, students will be able to identify fractions, the parts of a fraction, add/subtract like denominator fractions, and reduce fractions.
<MASK>
WB
05/23/2017
US
<MASK>
<UNMASK>
# Fractions
<MASK>
At the end of the lesson, students will be able to identify fractions, the parts of a fraction, add/subtract like denominator fractions, and reduce fractions.
<MASK>
### t Great way to reveiw fractions!
<MASK>
WB
05/23/2017
US
<MASK>
<UNMASK>
# Fractions
<MASK>
Our Fractions lesson plan teaches students how to identify fractions, the parts of a fraction, add/subtract like-denominator fractions, and reduce fractions. Students complete practice problems that include fractions in order to solidify their understanding of the lesson.
<MASK>
At the end of the lesson, students will be able to identify fractions, the parts of a fraction, add/subtract like denominator fractions, and reduce fractions.
State Educational Standards: LB.Math.Content.3.NF.A.1, LB.Math.Content.3.NF.A.3.A, LB.Math.Content.3.NF.A.3.B, LB.Math.Content.4.NF.A.1, LB.Math.Content.4.NF.B.3.A
Customer Reviews
4.8 Based on 5 Reviews
5 ★
80%
4
4 ★
20%
1
3 ★
0%
0
2 ★
0%
0
1 ★
0%
0
Write a Review
<MASK>
Filter Reviews:
A
01/04/2022
United States
### t Great way to reveiw fractions!
Great way to review fractions for my 4th grade intervention kiddos
<MASK>
### Fractions
<MASK>
CM
03/17/2020
US
<MASK>
Thank you for sharing this amazing resource. I particularly like the different learning activities because they include visuals and worksheets to reinforce the students’ knowledge of fractions. Each student learns differently and this lesson reach different learning styles.
<MASK>
WB
05/23/2017
US
<MASK>
<UNMASK>
# Fractions
\$1.95
Our Fractions lesson plan teaches students how to identify fractions, the parts of a fraction, add/subtract like-denominator fractions, and reduce fractions. Students complete practice problems that include fractions in order to solidify their understanding of the lesson.
## Description
Our Fractions lesson plan enables students to identify fractions, the parts of a fraction, add/subtract like-denominator fractions, and reduce fractions. Students are asked to complete an interactive activity in which they cut out circles with different numbers of equal parts and then compare how many are needed in various scenarios in order to help visualize fractions. Students are also asked to complete practice problems adding and subtracting fractions on their own.
At the end of the lesson, students will be able to identify fractions, the parts of a fraction, add/subtract like denominator fractions, and reduce fractions.
State Educational Standards: LB.Math.Content.3.NF.A.1, LB.Math.Content.3.NF.A.3.A, LB.Math.Content.3.NF.A.3.B, LB.Math.Content.4.NF.A.1, LB.Math.Content.4.NF.B.3.A
Customer Reviews
4.8 Based on 5 Reviews
5 ★
80%
4
4 ★
20%
1
3 ★
0%
0
2 ★
0%
0
1 ★
0%
0
Write a Review
Thank you for submitting a review!
Your input is very much appreciated. Share it with your friends so they can enjoy it too!
Filter Reviews:
A
01/04/2022
United States
### t Great way to reveiw fractions!
Great way to review fractions for my 4th grade intervention kiddos
DG
06/01/2020
US
### Fractions
It was great and my granddaughter loved it and the videos help reinforces that lessons.
CM
03/17/2020
US
### Fantastic Resource
Thank you for sharing this amazing resource. I particularly like the different learning activities because they include visuals and worksheets to reinforce the students’ knowledge of fractions. Each student learns differently and this lesson reach different learning styles.
LS
08/15/2017
US
<MASK>
WB
05/23/2017
US
### fractions
I like the educational materials I have downloaded. I have incorporated them into my lessons for practice and used as a resource.
<UNMASK>
# Fractions
\$1.95
Our Fractions lesson plan teaches students how to identify fractions, the parts of a fraction, add/subtract like-denominator fractions, and reduce fractions. Students complete practice problems that include fractions in order to solidify their understanding of the lesson.
## Description
Our Fractions lesson plan enables students to identify fractions, the parts of a fraction, add/subtract like-denominator fractions, and reduce fractions. Students are asked to complete an interactive activity in which they cut out circles with different numbers of equal parts and then compare how many are needed in various scenarios in order to help visualize fractions. Students are also asked to complete practice problems adding and subtracting fractions on their own.
At the end of the lesson, students will be able to identify fractions, the parts of a fraction, add/subtract like denominator fractions, and reduce fractions.
State Educational Standards: LB.Math.Content.3.NF.A.1, LB.Math.Content.3.NF.A.3.A, LB.Math.Content.3.NF.A.3.B, LB.Math.Content.4.NF.A.1, LB.Math.Content.4.NF.B.3.A
Customer Reviews
4.8 Based on 5 Reviews
5 ★
80%
4
4 ★
20%
1
3 ★
0%
0
2 ★
0%
0
1 ★
0%
0
Write a Review
Thank you for submitting a review!
Your input is very much appreciated. Share it with your friends so they can enjoy it too!
Filter Reviews:
A
01/04/2022
United States
### t Great way to reveiw fractions!
Great way to review fractions for my 4th grade intervention kiddos
DG
06/01/2020
US
### Fractions
It was great and my granddaughter loved it and the videos help reinforces that lessons.
CM
03/17/2020
US
### Fantastic Resource
Thank you for sharing this amazing resource. I particularly like the different learning activities because they include visuals and worksheets to reinforce the students’ knowledge of fractions. Each student learns differently and this lesson reach different learning styles.
LS
08/15/2017
US
### Great workbook
This is the perfect short workbook to cover or review the basics of fractions. It includes hands on activities as well as just worksheets.
WB
05/23/2017
US
### fractions
I like the educational materials I have downloaded. I have incorporated them into my lessons for practice and used as a resource.
| 1,491 | 6,699,430 |
# Find the maximum likelihood estimator for $\mu$
## Question
Let $X_1,X_2,...,X_n$ be independent continuous random variables with the probability density function, $f(x)= \frac{1}{\mu}e^{-\frac{x}{\mu}}$, if $x>0$ and $0$ otherwise, where $\mu>0$. Find the maximum likelihood estimator for $\mu$ and compute the variance of the maximum likelihood estimator for $\mu$.
I found the likelihood function to be $$l(\mu)=\prod_{i=1}^{n} \frac{1}{\mu}e^{-\frac{X_i}{\mu}}=\frac{1}{\mu^n}e^{-\frac{1}{\mu}\sum_{i=1}^{n}X_i}$$ and thus the log likelihood is $$L(\mu)=-n\log_e(\mu)-\frac{1}{\mu}\sum_{i=1}^{n}Xi.$$ So, $$\frac{\partial L(\mu)}{\partial \mu}=-\frac{n}{\mu}+\frac{1}{\mu^2}\sum_{i=1}^{n}X_i.$$ Equating this to $0$ gives $\frac{1}{\mu^2}\sum_{i=1}^{n}Xi=\frac{n}{\mu}$ and so $\bar X= \hat \mu$.
However, I am not sure that I wrote out the likelihood function correctly and I don't know how I would find the variance. I know $\text{Var}(X)=E(X^2)-(E(X))^2$ but I don't know how to calculate this using the MLE.
Your work is correct, although personally, I would write the full likelihood as \begin{align*} \mathcal L(\mu \mid \boldsymbol x) &= \prod_{i=1}^n \frac{1}{\mu} e^{-x_i/\mu} \mathbb 1 (x_i \ge 0) \\ &= \mu^{-n} \exp\left(-\frac{1}{\mu} \sum_{i=1}^n x_i\right) \mathbb 1 (x_{(1)} \ge 0) \\ &= \mu^{-n} e^{-n \bar x/\mu} \mathbb 1 (x_{(1)} \ge 0), \end{align*} where $\bar x$ is the sample mean; $$\mathbb 1(x_i \ge 0) = \begin{cases} 1, & x_i \ge 0 \\ 0, & \text{otherwise} \end{cases}$$ is an indicator function, and $x_{(1)} = \min_i x_i$ is the first (minimum) order statistic of the sample $\boldsymbol x = (x_1, x_2, \ldots, x_n)$. This however does not affect the validity of your computation of the MLE, which is indeed $\hat \mu = \bar x$, the sample mean. To calculate the variance of this estimator, we write $$\operatorname{Var}[\hat \mu] = \operatorname{Var}[\bar X] = \operatorname{Var}\left[\frac{1}{n} \sum_{i=1}^n X_i\right] \overset{\text{ind}}{=} \frac{1}{n^2} \sum_{i=1}^n \operatorname{Var}[X_i] \overset{\text{id}}{=} \frac{1}{n^2} \cdot n \operatorname{Var}[X] = \frac{\operatorname{Var}[X]}{n},$$ where "ind" over the equality sign indicates that the equality holds because the observations are independent; "id" over the equality sign indicates that equality holds because the observations are identically distributed; and $\operatorname{Var}[X]$ represents the variance of a single observation from this distribution. What is $\operatorname{Var}[X]$ in terms of the parameter $\mu$?
| 882 | 6,699,431 |
# A Comprehensive Guide to the Physics of Running on the Moon
Humans are going to live on the moon eventually. So how are we going to move around there?
One day humans will have a permanent presence on the moon. Right? One day it's going to happen. So, how are we going to live on the moon? And maybe a more important question—how are we going to move around there? In preparation for our lunar colony, let me look at three motions that we could do on the moon: jumping, running, and turning.
Let me note that this analysis is inspired by Andy Weir's recent novel Artemis. I'm not going to spoil the plot except to say there is a girl that moves around on the moon. Weir does a pretty nice job describing what would be different about moving on the moon as compared to the Earth.
What is different about the moon compared to the Earth? The biggest difference is the gravitational field on the surface. On the Earth, the field has a strength of 9.8 Newtons per kilogram (we use the symbol g for this). This means that a free falling object (no air resistance) would have a downward acceleration of 9.8 m/s2. On the moon, the gravitational field is about 1.6 N/kg, so that the vertical acceleration of an moon-object would be much less than one on Earth.
There is another important difference with the moon: It doesn't have any air. If you are a human jumping, that might not be a big deal; an Earth-bound jumping human doesn't move fast enough for air resistance to play a significant role. However, on the moon that same human would probably want to wear a spacesuit. This suit would both increase the effective mass and decrease the range of motion for a moving human. Oh, if there is a moon base there would probably be air inside of it so that you wouldn't have to wear a spacesuit unless you just thought it looked cool (it would).
Jumping on the moon
I will start with the easiest motion—jumping straight up. Let's say that during a normal human jump, a human pushes on the ground with some maximum force over some set distance. This distance is from the lowest position in pre-jump squat, up until the feet are no longer in contact with the ground.
Now for some starting values (you can change these if you like). I'm going to say this maximum jump force is three times the weight of the person (the weight on Earth) and the jump distance is 15 centimeters—that's just a guess. With these values, I cannot model the motion of a jumping human on Earth. I'll just calculate the total force as either the upward pushing force plus gravity while "in contact" with the ground or just gravity after that. It should be a fairly straightforward numerical calculation.
For a jumping human on the moon, I am going to make a few changes. Obviously the gravitational field will change—but also some other things. I'm going to assume the human is wearing a spacesuit, so this will increase the total mass (but not the max jumping force). Also, since a spacesuit is bulky, the jumping distance will also be smaller. OK, let's get to it. Here are two jumpers (moon and Earth). If you want the code for this calculation—here you go.
Here is what it would look like (using spherical humans for simplicity).
Also, here is a plot of the vertical position of both jumpers.
A few things to notice. First, the Earth jumper starts off with a faster speed. Why? Because the moon jumper has more mass (spacesuit and stuff). Second, the moon jumper both goes higher and stays off the ground for a much longer time because of the lower vertical acceleration.
But wait! How about a real video of a moon jump? Here is a video of John Young's famous "jump salute" during the Apollo 16 mission.
Pretty cool—but without a spacesuit, a human could probably jump even higher. Here is an old NASA film of a jumping human in simulated moon-gravity. NASA's method (very creative) to simulate moon-gravity is to have the a human suspended mostly horizontal by strings and then move on a mostly vertical surface.
Running on the moon
It's not really a spoiler, but one of the first scenes in the book Artemis has the main character (Jazz) out on the surface of the moon. For some reason (read the book), she starts running quite fast in her spacesuit. So, what would it be like to run on the moon?
Yes, there exists video of actual astronauts moving in a manner that could be considered "running"—but I still want to model this motion. I previously built a model of a running human and now I can just change some stuff to adapt it for the moon. Here is my previous post on a running human model. Some key points aspect of this model (remember, it's still just a model).
• A human is like a ball bouncing along the ground. It consists of two parts: contact with the ground and motion through the air.
• The part where the human is not in contact with the ground must last a minimum amount of time so that the human can switch feet from front to back.
• During the contact with the ground, the human can only exert some maximum force.
• The contact time with the ground decreases with linear running speed.
All of this together means that as the runner moves faster, a greater component of the pushing force must be applied in the vertical direction to get the human off the ground, since the contact time decreases. Eventually, the human reaches some maximum speed where all of the force is used in the vertical direction. You can check out my model running code here.
But what about running on the moon? The big difference is time. Since the gravitational field is small, the human will be in the air for a much larger time with a smaller vertical push force. This means that more of the max force can be used in the horizontal direction to increase the horizontal speed.
OK, but what about a plot? Here is my running model on both the Earth and the moon. I increased the mass of the moon-human to simulate a spacesuit and I also increased the "stride time" the human is off the ground to account for a bulky suit that would require more leg swinging time.
Here is a plot of the velocity as a function of time for these two runners.
The Earth-human gets to a speed of almost 10 m/s, but the moon-human easily can go over 15 m/s. But wait! It's even better. This is assuming the same kind of running style for both gravitational fields. However, on the moon it's very possible that there are more efficient running styles that take advantage of the low gravitational field.
It's probably not very surprising that people have already explored the idea of running in low gravity. Just check out this NASA test using the same "horizontal running" rig as in the jumping video.
Oh, there's also this interesting paper looking at theoretical and simulated running speeds on the moon—"The preferred walk to run transition speed in actual lunar gravity", from the Journal of Experimental Biology. For that study, they put actual humans on treadmills while in a plane flying in parabolic paths to create lower apparent weight. But really, who knows how it will really work until we get serious about being on the moon.
Running and turning
Running in a straight line might be fun for some short amount of time—but if you want to really maneuver around you are going to have to turn at some point. Would turning on the moon be different than on Earth? Of course. Let's consider a human running in a circle on the surface of the Earth. Here is a top and side view with a force diagram.
The key idea here is that you need a "sideways" force in order to make a turn. The direction of this turning force is towards the center of the circle you are turning in. Also, the magnitude of this force depends on the running speed and the size of the circle in the following manner.
So, faster running speed means a bigger force and a smaller radius (sharper turn) also means a bigger force. The force that pushes the human into a circular path is the frictional force between the feet and the ground. But of course you already know that—if you try taking a turn on low friction ice it doesn't work so well, does it?
Here's the last important point—the magnitude of the friction force is proportional to the force with which the ground pushes up on the human. In the case of maximum friction, the magnitude would be:
But what about the moon? What changes? The first thing is the gravitational force. With a lower gravitational force on the moon, there will also be a lower force of the ground pushing up on the human. This of course means that there will be a lower frictional force used for turning. Oh, add to this the fact that the human might be running faster and you get a big turning problem.
So, running on the moon is going to be different than running on the Earth. I'm sort of excited to see what cool tricks we can come up with to move around in this lower gravity environment. Oh, being on the moon would be cool too.
More Great WIRED Stories
| 1,915 | 6,699,432 |
In this chapter, we will develop certain techniques that help solve problems stated in words. These techniques involve rewriting problems in the form of symbols. For example, the stated problem
"Find a number which, when added lớn 3, yields 7"
may be written as:
3 + ? = 7, 3 + n = 7, 3 + x = 1
and so on, where the symbols ?, n, & x represent the number we want to find. We call such shorthand versions of stated problems equations, or symbolic sentences. Equations such as x + 3 = 7 are first-degree equations, since the variable has an exponent of 1. The terms lớn the left of an equals sign make up the left-hand member of the equation; those lớn the right cosplay the right-hand member. Thus, in the equation x + 3 = 7, the left-hand thành viên is x + 3 & the right-hand member is 7.
Bạn đang xem: Solve inequalities with step
SOLVING EQUATIONS
Equations may be true or false, just as word sentences may be true or false. The equation:
3 + x = 7
will be false if any number except 4 is substituted for the variable. The value of the variable for which the equation is true (4 in this example) is called the solution of the equation. We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable & determining the truth or falsity of the result.
Example 1 Determine if the value 3 is a solution of the equation
4x - 2 = 3x + 1
Solution We substitute the value 3 for x in the equation & see if the left-hand thành viên equals the right-hand member.
4(3) - 2 = 3(3) + 1
12 - 2 = 9 + 1
10 = 10
Ans. 3 is a solution.
The first-degree equations that we consider in this chapter have at most one solution. The solutions to many such equations can be determined by inspection.
Example 2 Find the solution of each equation by inspection.
a.x + 5 = 12b. 4 · x = -20
Solutions a. 7 is the solution since 7 + 5 = 12.b.-5 is the solution since 4(-5) = -20.
SOLVING EQUATIONS USING ADDITION và SUBTRACTION PROPERTIES
In Section 3.1 we solved some simple first-degree equations by inspection. However, the solutions of most equations are not immediately evident by inspection. Hence, we need some mathematical "tools" for solving equations.
EQUIVALENT EQUATIONS
Equivalent equations are equations that have identical solutions. Thus,
3x + 3 = x + 13, 3x = x + 10, 2x = 10, và x = 5
are equivalent equations, because 5 is the only solution of each of them. Notice in the equation 3x + 3 = x + 13, the solution 5 is not evident by inspection but in the equation x = 5, the solution 5 is evident by inspection. In solving any equation, we transform a given equation whose solution may not be obvious to lớn an equivalent equation whose solution is easily noted.
The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations.
If the same quantity is added to lớn or subtracted from both membersof an equation, the resulting equation is equivalent to the originalequation.
In symbols,
a - b, a + c = b + c, và a - c = b - c
are equivalent equations.
Example 1 Write an equation equivalent to
x + 3 = 7
by subtracting 3 from each member.
Solution Subtracting 3 from each thành viên yields
x + 3 - 3 = 7 - 3
or
x = 4
Notice that x + 3 = 7 và x = 4 are equivalent equations since the solution is the same for both, namely 4. The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation.
Example 2 Write an equation equivalent to
4x- 2-3x = 4 + 6
by combining lượt thích terms & then by adding 2 lớn each member.
Combining like terms yields
x - 2 = 10
Adding 2 to lớn each member yields
x-2+2 =10+2
x = 12
To solve an equation, we use the addition-subtraction property khổng lồ transform a given equation khổng lồ an equivalent equation of the size x = a, from which we can find the solution by inspection.
Example 3 Solve 2x + 1 = x - 2.
We want lớn obtain an equivalent equation in which all terms containing x are in one thành viên and all terms not containing x are in the other. If we first địa chỉ cửa hàng -1 to (or subtract 1 from) each member, we get
2x + 1- 1 = x - 2- 1
2x = x - 3
If we now add -x khổng lồ (or subtract x from) each member, we get
2x-x = x - 3 - x
x = -3
where the solution -3 is obvious.
The solution of the original equation is the number -3; however, the answer is often displayed in the size of the equation x = -3.
Since each equation obtained in the process is equivalent to lớn the original equation, -3 is also a solution of 2x + 1 = x - 2. In the above example, we can kiểm tra the solution by substituting - 3 for x in the original equation
2(-3) + 1 = (-3) - 2
-5 = -5
The symmetric property of equality is also helpful in the solution of equations. This property states
If a = b then b = a
This enables us to interchange the members of an equation whenever we please without having to lớn be concerned with any changes of sign. Thus,
If 4 = x + 2thenx + 2 = 4
If x + 3 = 2x - 5then2x - 5 = x + 3
If d = rtthenrt = d
There may be several different ways to apply the addition property above. Sometimes one method is better than another, và in some cases, the symmetric property of equality is also helpful.
Example 4 Solve 2x = 3x - 9.(1)
Solution If we first add -3x lớn each member, we get
2x - 3x = 3x - 9 - 3x
-x = -9
where the variable has a negative coefficient. Although we can see by inspection that the solution is 9, because -(9) = -9, we can avoid the negative coefficient by adding -2x và +9 khổng lồ each thành viên of Equation (1). In this case, we get
2x-2x + 9 = 3x- 9-2x+ 9
9 = x
from which the solution 9 is obvious. If we wish, we can write the last equation as x = 9 by the symmetric property of equality.
SOLVING EQUATIONS USING THE DIVISION PROPERTY
Consider the equation
3x = 12
The solution khổng lồ this equation is 4. Also, note that if we divide each thành viên of the equation by 3, we obtain the equations
whose solution is also 4. In general, we have the following property, which is sometimes called the division property.
If both members of an equation are divided by the same (nonzero)quantity, the resulting equation is equivalent to lớn the original equation.
In symbols,
are equivalent equations.
Example 1 Write an equation equivalent to
-4x = 12
by dividing each member by -4.
Solution Dividing both members by -4 yields
In solving equations, we use the above property to lớn produce equivalent equations in which the variable has a coefficient of 1.
Example 2 Solve 3y + 2y = 20.
We first combine like terms to lớn get
5y = 20
Then, dividing each member by 5, we obtain
In the next example, we use the addition-subtraction property và the division property khổng lồ solve an equation.
Example 3 Solve 4x + 7 = x - 2.
Solution First, we địa chỉ -x & -7 lớn each thành viên to get
4x + 7 - x - 7 = x - 2 - x - 1
Next, combining lượt thích terms yields
3x = -9
Last, we divide each thành viên by 3 to lớn obtain
SOLVING EQUATIONS USING THE MULTIPLICATION PROPERTY
Consider the equation
The solution lớn this equation is 12. Also, note that if we multiply each member of the equation by 4, we obtain the equations
whose solution is also 12. In general, we have the following property, which is sometimes called the multiplication property.
If both members of an equation are multiplied by the same nonzero quantity, the resulting equation Is equivalent to the original equation.
In symbols,
a = b và a·c = b·c (c ≠ 0)
are equivalent equations.
Example 1 Write an equivalent equation to
by multiplying each thành viên by 6.
Solution Multiplying each member by 6 yields
In solving equations, we use the above property khổng lồ produce equivalent equations that are không tính phí of fractions.
Example 2 Solve
Solution First, multiply each thành viên by 5 lớn get
Now, divide each member by 3,
Example 3 Solve
.
Solution First, simplify above the fraction bar to lớn get
Next, multiply each thành viên by 3 khổng lồ obtain
Last, dividing each member by 5 yields
FURTHER SOLUTIONS OF EQUATIONS
Now we know all the techniques needed khổng lồ solve most first-degree equations. There is no specific order in which the properties should be applied. Any one or more of the following steps listed on page 102 may be appropriate.
Steps to solve first-degree equations:Combine like terms in each member of an equation.Using the addition or subtraction property, write the equation with all terms containing the unknown in one thành viên and all terms not containing the unknown in the other.Combine lượt thích terms in each member.Use the multiplication property khổng lồ remove fractions.Use the division property lớn obtain a coefficient of 1 for the variable.
Example 1 Solve 5x - 7 = 2x - 4x + 14.
Solution First, we combine lượt thích terms, 2x - 4x, khổng lồ yield
5x - 7 = -2x + 14
Next, we showroom +2x and +7 to lớn each member and combine like terms to lớn get
5x - 7 + 2x + 7 = -2x + 14 + 2x + 1
7x = 21
Finally, we divide each thành viên by 7 to lớn obtain
In the next example, we simplify above the fraction bar before applying the properties that we have been studying.
Example 2 Solve
Solution First, we combine like terms, 4x - 2x, to get
Then we địa chỉ -3 to lớn each member and simplify
Next, we multiply each thành viên by 3 to lớn obtain
Finally, we divide each thành viên by 2 khổng lồ get
SOLVING FORMULAS
Equations that involve variables for the measures of two or more physical quantities are called formulas. We can solve for any one of the variables in a formula if the values of the other variables are known. We substitute the known values in the formula & solve for the unknown variable by the methods we used in the preceding sections.
Example 1 In the formula d = rt, find t if d = 24 & r = 3.
Solution We can solve for t by substituting 24 for d và 3 for r. That is,
d = rt
(24) = (3)t
8 = t
It is often necessary lớn solve formulas or equations in which there is more than one variable for one of the variables in terms of the others. We use the same methods demonstrated in the preceding sections.
Example 2 In the formula d = rt, solve for t in terms of r & d.
Xem thêm: Lớp Nào Có Mức Năng Lượng Thấp Nhất, Tra Cứu & Tìm Kiếm Đáp Án Của Câu Hỏi
Solution We may solve for t in terms of r and d by dividing both members by r to yield
from which, by the symmetric law,
In the above example, we solved for t by applying the division property khổng lồ generate an equivalent equation. Sometimes, it is necessary to apply more than one such property.
| 2,794 | 6,699,433 |
<MASK>
<UNMASK>
<MASK>
What I want is to do this: golf reflections
<MASK>
Would love to know how to compensate for this. Thanks!
<MASK>
The point that is reflecting off a wall according to those rules is not the center of the ball but the point on the ball where it touches the wall. Thats how you account for it. If you have a rectangle, you only need to use 4 points on the ball plus its center.
<MASK>
<UNMASK>
<MASK>
What I want is to do this: golf reflections
<MASK>
Would love to know how to compensate for this. Thanks!
<MASK>
Offset the wall by the radius of the ball. By doing this, you compensate for the radius of the ball, so thereafter, you can consider the ball to be a point, and do the computations shown in your reference.
<MASK>
The point that is reflecting off a wall according to those rules is not the center of the ball but the point on the ball where it touches the wall. Thats how you account for it. If you have a rectangle, you only need to use 4 points on the ball plus its center.
-
I am deleting the comments regarding the inappropriate username on this thread. The original username, and description, were not appropriate at all, and have been changed to something less offensive. They have been recorded in a previous deleted comment. – Eric Naslund May 6 '13 at 22:30
<UNMASK>
<MASK>
Problem: I'm struggling to compensate for the radius of a ball when reflecting it off a wall towards a target. (sorry I cannot yet post images)
What I want is to do this: golf reflections
<MASK>
Would love to know how to compensate for this. Thanks!
-
Offset the wall by the radius of the ball. By doing this, you compensate for the radius of the ball, so thereafter, you can consider the ball to be a point, and do the computations shown in your reference.
<MASK>
If you consider a golf ball to be spherical, the ball touches the wall when the center is one radius away. If you normally track the ball location with the center, you can move the walls one radius inward and have the proper moment of reflection. If you want to account for the dimples it is much harder. You need to account for the orientation of the ball, then.
-
So if I instead track the ball from the top and find the angle of which a point at the top of the ball must move in to hit R' (the reflected hole) and then shoot the center of the puck along that angle would I end up with the desired results? Results have varied for me It does make it in most of the time. – sinthetic May 6 '13 at 4:31
The point that is reflecting off a wall according to those rules is not the center of the ball but the point on the ball where it touches the wall. Thats how you account for it. If you have a rectangle, you only need to use 4 points on the ball plus its center.
-
I am deleting the comments regarding the inappropriate username on this thread. The original username, and description, were not appropriate at all, and have been changed to something less offensive. They have been recorded in a previous deleted comment. – Eric Naslund May 6 '13 at 22:30
<UNMASK>
# Reflecting a golfball off a wall to a hole and compensating for the balls radius
Problem: I'm struggling to compensate for the radius of a ball when reflecting it off a wall towards a target. (sorry I cannot yet post images)
What I want is to do this: golf reflections
but this does not compensate for the size of the ball. Any guidance would be helpful.
Would love to know how to compensate for this. Thanks!
-
Offset the wall by the radius of the ball. By doing this, you compensate for the radius of the ball, so thereafter, you can consider the ball to be a point, and do the computations shown in your reference.
-
Thanks this helped me get my head around the problem. This issue is I am not doing the physics, Its being run on a physics engine and collisions are taken place on the edge of the ball. I could do what you are saying and reduce the collider to a point inside the puck, but that would lead to a bunch of collision skips. (small colliders tend to miss quite often) – sinthetic May 6 '13 at 3:49
So in terms of this drawing, If i could figure out the place on the Original wall where the ball needs to reflect off of for this to work i would be set, I just don't know the math to accomplish it. – sinthetic May 6 '13 at 3:56
If you consider a golf ball to be spherical, the ball touches the wall when the center is one radius away. If you normally track the ball location with the center, you can move the walls one radius inward and have the proper moment of reflection. If you want to account for the dimples it is much harder. You need to account for the orientation of the ball, then.
-
So if I instead track the ball from the top and find the angle of which a point at the top of the ball must move in to hit R' (the reflected hole) and then shoot the center of the puck along that angle would I end up with the desired results? Results have varied for me It does make it in most of the time. – sinthetic May 6 '13 at 4:31
The point that is reflecting off a wall according to those rules is not the center of the ball but the point on the ball where it touches the wall. Thats how you account for it. If you have a rectangle, you only need to use 4 points on the ball plus its center.
-
I am deleting the comments regarding the inappropriate username on this thread. The original username, and description, were not appropriate at all, and have been changed to something less offensive. They have been recorded in a previous deleted comment. – Eric Naslund May 6 '13 at 22:30
| 1,271 | 6,699,434 |
# Homework Help: Fun question!
1. Jun 15, 2004
### 2Pac
160 newton box sits on 10 meter long frictionless plane inclined at an angke of 30 degrees to the horizontal. calculate the amount of work done in moving the box from the bottom to the top of the inclined plane.
2. Jun 15, 2004
### AKG
$$W = \vec{F}_{app} \cdot \Delta \vec{d}$$
$$W = (m \vec{g} \sin \theta ) \cdot \Delta \vec{d}$$
$$W = 800J$$
Also, I don't believe the path matters, so you can treat the problem as simply lifting the 160N box a height equivalent to the heigh of the ramp:
W = 160N x (10m)[sin(30$^{\circ}$)] = 800J
3. Jun 15, 2004
### 2Pac
thank you AKG. What is the deal with this formula. W=FD cos ? because i got 800J as well and everyone in my class said i was wrong.
4. Jun 15, 2004
### AKG
The dot product between two vectors, $\vec{u}$ and $\vec{v}$ is:
$$\vec{v} \cdot \vec{u} = |\vec{v}||\vec{u}|\cos \theta$$
where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$. And since the cosine function is symmetric about the y-axis, whether you measure from vector u to vector v, getting a positive angle, let's say, or measure from v to u, of course then getting a negative angle, it won't matter.
Now, Work is the dot product of the applied force and the direction of motion. The force you'd have to apply to push the block up the incline is obviously in the up-the-incline direction. That is also the direction of motion, so the angle between the two is zero, and cos(0) = 1. So, the work done is the magnitude of the applied force, times the magnitude of the displacement (times 1). The magnitude of displacement is 10m. If you draw the free-body diagram, you'll see that there's a normal force and gravitational force. Part of the gravitational force is counterbalanced by the normal force, and part of it would then have to be counterbalanced by the applied force, so you can calculate this applied force.
But what if you don't want to just counterbalance the applied force? What if you apply a greater force? The force-applied would be different, and you'd think you'd get a bigger result, right? No, because after some time you'd actually have to pull back on the object and apply a force opposite the direction of motion so that it stops at the end of the ramp. Or, maybe you'll only apply a force for the first 5 meters, then, having given it sufficient velocity, it slides to the top where it just stops. But in this case, although the applied force has increased (accelerating the block rather than just pushing it past equilibrium), the distance decreases. So no matter how you do it, if you bring it from rest at the bottom to rest at the top, the work will be the same.
Another way to calculate work is change in energy. At the bottom and top, it is at rest, so the kinetic energy change is zero. The gravitational potential energy changes, which is what I was getting at in the second approach where I said: "Also, I don't believe the path matters, so you can treat the problem as simply lifting the 160N box a height equivalent to the height of the ramp." Of course, I can't tell you why your class says you have it wrong. I'm pretty convinced I have it right...
5. Jun 15, 2004
### 2Pac
After reading that i am too. thank you again.
| 857 | 6,699,435 |
<MASK>
<UNMASK>
<MASK>
The circular garden measures 10 yards across. This means that the diameter of the garden is 10 yards. The perimeter of a circle is its circumference. The circumference $C$ of a circle can be found using the formula $C=2\pi{r}$ where r = radius. The radius of a circle is one-half of the circle's diameter. Thus, the radius of the garden is: $r = \frac{1}{2}(10) = 5$ yards Solve for the circumference of the circle using the formula above to obtain: $C=2\pi(5) = 10\pi \approx 31.4159$ yards Therefore, there is a need for around 31 yards of fencing.
<UNMASK>
## Thinking Mathematically (6th Edition)
The circular garden measures 10 yards across. This means that the diameter of the garden is 10 yards. The perimeter of a circle is its circumference. The circumference $C$ of a circle can be found using the formula $C=2\pi{r}$ where r = radius. The radius of a circle is one-half of the circle's diameter. Thus, the radius of the garden is: $r = \frac{1}{2}(10) = 5$ yards Solve for the circumference of the circle using the formula above to obtain: $C=2\pi(5) = 10\pi \approx 31.4159$ yards Therefore, there is a need for around 31 yards of fencing.
| 319 | 6,699,436 |
<MASK>
Posted by anujgarg on Sat, 05 Mar 2022 15:10:00 +0100
<MASK>
Common sorting algorithms include bubble sort, selection sort, insertion sort, merge sort, quick sort, Hill sort, heap sort, count sort, bucket sort and cardinal sort
## Bubble sorting
<MASK>
## Insert sort
<MASK>
## Shell Sort
Hill sort, also known as decreasing incremental sort algorithm, is a more efficient improved version of insertion sort. But Hill sorting is an unstable sorting algorithm.
<MASK>
void mergeSort(vector<int> & vec, int left, int right){
if(left < right){
int mid = (left + right) / 2;
mergeSort(vec, left, mid);
mergeSort(vec, mid + 1, right);
merge(vec, left, mid, mid + 1, right);
}
}
```
<MASK>
https://blog.csdn.net/weixin_41445507/article/details/90255906
<MASK>
void quickSort(vector<int> & vec, int left, int right){
if(left < right){
int pos = quick(vec, left, right);
quickSort(vec, left, pos - 1);
quickSort(vec, pos + 1, right);
}
}
```
<MASK>
Heap sort refers to a sort algorithm designed by using the data structure of heap. Heap is a structure similar to a complete binary tree, and satisfies the nature of heap at the same time: that is, the key value or index of a child node is always less than (or greater than) its parent node. Heap sort can be said to be a selective sort that uses the concept of heap to sort. There are two methods:
1. Large top heap: the value of each node is greater than or equal to the value of its child nodes, which is used for ascending arrangement in heap sorting algorithm;
2. Small top heap: the value of each node is less than or equal to the value of its child nodes, which is used for descending arrangement in heap sorting algorithm;
The average time complexity of heap sorting is Ο (nlogn).
<MASK>
if (right < size && nums[right] > nums[largest]) {
largest = right;
}
<MASK>
## Count sort
<MASK>
```void countingSort(vector<int> & nums) {
int maxValue = nums[0];
//Find maximum value
for(int i = 1; i < nums.size(); i++){
maxValue = max(maxValue, nums[i]);
}
<MASK>
Bucket sorting is an upgraded version of counting sorting. It makes use of the mapping relationship of the function. The key to efficiency lies in the determination of the mapping function. In order to make bucket sorting more efficient, we need to do these two things:
<MASK>
At the same time, for the sorting of elements in the bucket, it is very important to choose a comparative sorting algorithm for the performance.
<MASK>
Radix sorting is a non comparative integer sorting algorithm
<MASK>
• Cardinality sorting: allocate buckets according to each number of key value;
• Counting and sorting: only one key value is stored in each bucket;
• Bucket sorting: each bucket stores a certain range of values;
<MASK>
<UNMASK>
<MASK>
Posted by anujgarg on Sat, 05 Mar 2022 15:10:00 +0100
# Top ten sorting algorithms
Common sorting algorithms include bubble sort, selection sort, insertion sort, merge sort, quick sort, Hill sort, heap sort, count sort, bucket sort and cardinal sort
## Bubble sorting
<MASK>
## Insert sort
<MASK>
## Shell Sort
Hill sort, also known as decreasing incremental sort algorithm, is a more efficient improved version of insertion sort. But Hill sorting is an unstable sorting algorithm.
<MASK>
## Merge sort
```void merge(vector<int> & vec, int L1, int R1, int L2, int R2){
vector<int> temp; //Auxiliary array
int begin = L1;
while(L1 <= R1 && L2 <= R2){
if(vec[L1] < vec[L2]){
temp.push_back(vec[L1++]);
}else{
temp.push_back(vec[L2++]);
}
}
while(L1 <= R1){
temp.push_back(vec[L1++]);
}
while(L2 <= R2){
temp.push_back(vec[L2++]);
}
vec.assign(temp.begin(), temp.end());
}
void mergeSort(vector<int> & vec, int left, int right){
if(left < right){
int mid = (left + right) / 2;
mergeSort(vec, left, mid);
mergeSort(vec, mid + 1, right);
merge(vec, left, mid, mid + 1, right);
}
}
```
<MASK>
https://blog.csdn.net/weixin_41445507/article/details/90255906
<MASK>
void quickSort(vector<int> & vec, int left, int right){
if(left < right){
int pos = quick(vec, left, right);
quickSort(vec, left, pos - 1);
quickSort(vec, pos + 1, right);
}
}
```
<MASK>
Heap sort refers to a sort algorithm designed by using the data structure of heap. Heap is a structure similar to a complete binary tree, and satisfies the nature of heap at the same time: that is, the key value or index of a child node is always less than (or greater than) its parent node. Heap sort can be said to be a selective sort that uses the concept of heap to sort. There are two methods:
1. Large top heap: the value of each node is greater than or equal to the value of its child nodes, which is used for ascending arrangement in heap sorting algorithm;
2. Small top heap: the value of each node is less than or equal to the value of its child nodes, which is used for descending arrangement in heap sorting algorithm;
The average time complexity of heap sorting is Ο (nlogn).
```// Build large top reactor
void buildMaxHeap(vector<int> & nums) {
// Adjust the heap of each non leaf node
for (int i = nums.size() / 2; i >= 0; i--) {
heapify(nums, i);
}
}
// Heap adjustment, starting from the ith node
void heapify(vector<int> & nums, int i, int size) {
int left = 2 * i + 1; //Left child
int right = 2 * i + 2; //Right child
int largest = i;
//Find the maximum of the left and right children
if (left < size && nums[left] > nums[largest]) {
largest = left;
}
if (right < size && nums[right] > nums[largest]) {
largest = right;
}
//Replace with parent node
if (largest != i) {
swap(nums[i], nums[largest]);
heapify(nums, largest);
}
}
void heapSort(vector<int> & nums) {
buildMaxHeap(nums);
for (int i = nums.size() - 1; i > 0; i--) {
// Put the top node last each time
swap(nums[0], nums[i]);
heapify(nums, 0, i);
}
return nums;
}
```
## Count sort
<MASK>
```void countingSort(vector<int> & nums) {
int maxValue = nums[0];
//Find maximum value
for(int i = 1; i < nums.size(); i++){
maxValue = max(maxValue, nums[i]);
}
<MASK>
Bucket sorting is an upgraded version of counting sorting. It makes use of the mapping relationship of the function. The key to efficiency lies in the determination of the mapping function. In order to make bucket sorting more efficient, we need to do these two things:
<MASK>
At the same time, for the sorting of elements in the bucket, it is very important to choose a comparative sorting algorithm for the performance.
<MASK>
1. When is the slowest
<MASK>
## Cardinality sort
Radix sorting is a non comparative integer sorting algorithm
<MASK>
• Cardinality sorting: allocate buckets according to each number of key value;
• Counting and sorting: only one key value is stored in each bucket;
• Bucket sorting: each bucket stores a certain range of values;
Topics: C++ Algorithm
<UNMASK>
# Summary of top ten sorting algorithms
Posted by anujgarg on Sat, 05 Mar 2022 15:10:00 +0100
# Top ten sorting algorithms
Common sorting algorithms include bubble sort, selection sort, insertion sort, merge sort, quick sort, Hill sort, heap sort, count sort, bucket sort and cardinal sort
## Bubble sorting
When adjacent elements are equal, they do not exchange positions, so bubble sorting is a stable sorting.
<MASK>
## Select sort
<MASK>
## Insert sort
```void insertSort(vector<int> & nums){
for(int i = 1; i < nums.size(); i++){ // n - 1 cycle
int j;
int num = nums[i];
//Find the first element num[j] less than or equal to the current number forward, and move the subsequent elements backward by one bit
for(j = i - 1; j >= 0; j--){
if(num < nums[j]){ //Move back
nums[j + 1] = nums[j]; //Or swap (Num [J], Num [J + 1]);
}else{
break;
}
}
//Assign num to the next bit of num[j]
nums[j + 1] = num;
}
}
```
## Shell Sort
Hill sort, also known as decreasing incremental sort algorithm, is a more efficient improved version of insertion sort. But Hill sorting is an unstable sorting algorithm.
<MASK>
The basic idea of Hill sort is: first divide the whole record sequence to be sorted into several subsequences for direct insertion sort. When the records in the whole sequence are "basically orderly", then directly insert sort all records in turn.
```void shell_sort(vector<int> & nums) {
//Increment gap and gradually reduce the increment
for (int gap = nums.size() / 2; gap > 0; gap /= 2) {
//From the gap element, directly insert and sort the groups one by one
for (int i = gap; i < nums.size(); i++) {
for(int j = i; j >= gap && nums[j] < nums[j - gap]; j -= gap){
swap(nums[j], nums[j - gap]);
}
}
}
}
```
## Merge sort
```void merge(vector<int> & vec, int L1, int R1, int L2, int R2){
vector<int> temp; //Auxiliary array
int begin = L1;
while(L1 <= R1 && L2 <= R2){
if(vec[L1] < vec[L2]){
temp.push_back(vec[L1++]);
}else{
temp.push_back(vec[L2++]);
}
}
while(L1 <= R1){
temp.push_back(vec[L1++]);
}
while(L2 <= R2){
temp.push_back(vec[L2++]);
}
vec.assign(temp.begin(), temp.end());
}
void mergeSort(vector<int> & vec, int left, int right){
if(left < right){
int mid = (left + right) / 2;
mergeSort(vec, left, mid);
mergeSort(vec, mid + 1, right);
merge(vec, left, mid, mid + 1, right);
}
}
```
<MASK>
Why is quick sort called quick sort?
https://blog.csdn.net/weixin_41445507/article/details/90255906
https://blog.csdn.net/linfeng24/article/details/38429055
<MASK>
void quickSort(vector<int> & vec, int left, int right){
if(left < right){
int pos = quick(vec, left, right);
quickSort(vec, left, pos - 1);
quickSort(vec, pos + 1, right);
}
}
```
<MASK>
Heap sort refers to a sort algorithm designed by using the data structure of heap. Heap is a structure similar to a complete binary tree, and satisfies the nature of heap at the same time: that is, the key value or index of a child node is always less than (or greater than) its parent node. Heap sort can be said to be a selective sort that uses the concept of heap to sort. There are two methods:
1. Large top heap: the value of each node is greater than or equal to the value of its child nodes, which is used for ascending arrangement in heap sorting algorithm;
2. Small top heap: the value of each node is less than or equal to the value of its child nodes, which is used for descending arrangement in heap sorting algorithm;
The average time complexity of heap sorting is Ο (nlogn).
```// Build large top reactor
void buildMaxHeap(vector<int> & nums) {
// Adjust the heap of each non leaf node
for (int i = nums.size() / 2; i >= 0; i--) {
heapify(nums, i);
}
}
// Heap adjustment, starting from the ith node
void heapify(vector<int> & nums, int i, int size) {
int left = 2 * i + 1; //Left child
int right = 2 * i + 2; //Right child
int largest = i;
//Find the maximum of the left and right children
if (left < size && nums[left] > nums[largest]) {
largest = left;
}
if (right < size && nums[right] > nums[largest]) {
largest = right;
}
//Replace with parent node
if (largest != i) {
swap(nums[i], nums[largest]);
heapify(nums, largest);
}
}
void heapSort(vector<int> & nums) {
buildMaxHeap(nums);
for (int i = nums.size() - 1; i > 0; i--) {
// Put the top node last each time
swap(nums[0], nums[i]);
heapify(nums, 0, i);
}
return nums;
}
```
## Count sort
Counting sorting is not a comparative sorting, and the sorting speed is faster than any comparative sorting algorithm.
<MASK>
```void countingSort(vector<int> & nums) {
int maxValue = nums[0];
//Find maximum value
for(int i = 1; i < nums.size(); i++){
maxValue = max(maxValue, nums[i]);
}
<MASK>
for (int i = 0; i < maxValue + 1; i++) {
bucket[nums[i]] += 1;
}
int idx = 0;
for (int i = 0; i < maxValue + 1; i++) {
while(bucket[i] > 0) {
nums[idx++] = i;
bucket[i] -= 1;
}
}
}
```
<MASK>
Bucket sorting is an upgraded version of counting sorting. It makes use of the mapping relationship of the function. The key to efficiency lies in the determination of the mapping function. In order to make bucket sorting more efficient, we need to do these two things:
1. When there is enough extra space, try to increase the number of barrels
2. The mapping function used can evenly distribute the input N data into K buckets
At the same time, for the sorting of elements in the bucket, it is very important to choose a comparative sorting algorithm for the performance.
<MASK>
1. When is the slowest
<MASK>
## Cardinality sort
Radix sorting is a non comparative integer sorting algorithm
<MASK>
• Cardinality sorting: allocate buckets according to each number of key value;
• Counting and sorting: only one key value is stored in each bucket;
• Bucket sorting: each bucket stores a certain range of values;
Topics: C++ Algorithm
<UNMASK>
# Summary of top ten sorting algorithms
Posted by anujgarg on Sat, 05 Mar 2022 15:10:00 +0100
# Top ten sorting algorithms
Common sorting algorithms include bubble sort, selection sort, insertion sort, merge sort, quick sort, Hill sort, heap sort, count sort, bucket sort and cardinal sort
## Bubble sorting
When adjacent elements are equal, they do not exchange positions, so bubble sorting is a stable sorting.
<MASK>
## Select sort
<MASK>
## Insert sort
```void insertSort(vector<int> & nums){
for(int i = 1; i < nums.size(); i++){ // n - 1 cycle
int j;
int num = nums[i];
//Find the first element num[j] less than or equal to the current number forward, and move the subsequent elements backward by one bit
for(j = i - 1; j >= 0; j--){
if(num < nums[j]){ //Move back
nums[j + 1] = nums[j]; //Or swap (Num [J], Num [J + 1]);
}else{
break;
}
}
//Assign num to the next bit of num[j]
nums[j + 1] = num;
}
}
```
## Shell Sort
Hill sort, also known as decreasing incremental sort algorithm, is a more efficient improved version of insertion sort. But Hill sorting is an unstable sorting algorithm.
Hill sort is an improved method based on the following two properties of insertion sort:
<MASK>
The basic idea of Hill sort is: first divide the whole record sequence to be sorted into several subsequences for direct insertion sort. When the records in the whole sequence are "basically orderly", then directly insert sort all records in turn.
```void shell_sort(vector<int> & nums) {
//Increment gap and gradually reduce the increment
for (int gap = nums.size() / 2; gap > 0; gap /= 2) {
//From the gap element, directly insert and sort the groups one by one
for (int i = gap; i < nums.size(); i++) {
for(int j = i; j >= gap && nums[j] < nums[j - gap]; j -= gap){
swap(nums[j], nums[j - gap]);
}
}
}
}
```
## Merge sort
```void merge(vector<int> & vec, int L1, int R1, int L2, int R2){
vector<int> temp; //Auxiliary array
int begin = L1;
while(L1 <= R1 && L2 <= R2){
if(vec[L1] < vec[L2]){
temp.push_back(vec[L1++]);
}else{
temp.push_back(vec[L2++]);
}
}
while(L1 <= R1){
temp.push_back(vec[L1++]);
}
while(L2 <= R2){
temp.push_back(vec[L2++]);
}
vec.assign(temp.begin(), temp.end());
}
void mergeSort(vector<int> & vec, int left, int right){
if(left < right){
int mid = (left + right) / 2;
mergeSort(vec, left, mid);
mergeSort(vec, mid + 1, right);
merge(vec, left, mid, mid + 1, right);
}
}
```
<MASK>
Why is quick sort called quick sort?
https://blog.csdn.net/weixin_41445507/article/details/90255906
https://blog.csdn.net/linfeng24/article/details/38429055
<MASK>
void quickSort(vector<int> & vec, int left, int right){
if(left < right){
int pos = quick(vec, left, right);
quickSort(vec, left, pos - 1);
quickSort(vec, pos + 1, right);
}
}
```
<MASK>
Heap sort refers to a sort algorithm designed by using the data structure of heap. Heap is a structure similar to a complete binary tree, and satisfies the nature of heap at the same time: that is, the key value or index of a child node is always less than (or greater than) its parent node. Heap sort can be said to be a selective sort that uses the concept of heap to sort. There are two methods:
1. Large top heap: the value of each node is greater than or equal to the value of its child nodes, which is used for ascending arrangement in heap sorting algorithm;
2. Small top heap: the value of each node is less than or equal to the value of its child nodes, which is used for descending arrangement in heap sorting algorithm;
The average time complexity of heap sorting is Ο (nlogn).
```// Build large top reactor
void buildMaxHeap(vector<int> & nums) {
// Adjust the heap of each non leaf node
for (int i = nums.size() / 2; i >= 0; i--) {
heapify(nums, i);
}
}
// Heap adjustment, starting from the ith node
void heapify(vector<int> & nums, int i, int size) {
int left = 2 * i + 1; //Left child
int right = 2 * i + 2; //Right child
int largest = i;
//Find the maximum of the left and right children
if (left < size && nums[left] > nums[largest]) {
largest = left;
}
if (right < size && nums[right] > nums[largest]) {
largest = right;
}
//Replace with parent node
if (largest != i) {
swap(nums[i], nums[largest]);
heapify(nums, largest);
}
}
void heapSort(vector<int> & nums) {
buildMaxHeap(nums);
for (int i = nums.size() - 1; i > 0; i--) {
// Put the top node last each time
swap(nums[0], nums[i]);
heapify(nums, 0, i);
}
return nums;
}
```
## Count sort
Counting sorting is not a comparative sorting, and the sorting speed is faster than any comparative sorting algorithm.
<MASK>
```void countingSort(vector<int> & nums) {
int maxValue = nums[0];
//Find maximum value
for(int i = 1; i < nums.size(); i++){
maxValue = max(maxValue, nums[i]);
}
<MASK>
for (int i = 0; i < maxValue + 1; i++) {
bucket[nums[i]] += 1;
}
int idx = 0;
for (int i = 0; i < maxValue + 1; i++) {
while(bucket[i] > 0) {
nums[idx++] = i;
bucket[i] -= 1;
}
}
}
```
<MASK>
Bucket sorting is an upgraded version of counting sorting. It makes use of the mapping relationship of the function. The key to efficiency lies in the determination of the mapping function. In order to make bucket sorting more efficient, we need to do these two things:
1. When there is enough extra space, try to increase the number of barrels
2. The mapping function used can evenly distribute the input N data into K buckets
At the same time, for the sorting of elements in the bucket, it is very important to choose a comparative sorting algorithm for the performance.
<MASK>
1. When is the slowest
<MASK>
## Cardinality sort
Radix sorting is a non comparative integer sorting algorithm
<MASK>
• Cardinality sorting: allocate buckets according to each number of key value;
• Counting and sorting: only one key value is stored in each bucket;
• Bucket sorting: each bucket stores a certain range of values;
Topics: C++ Algorithm
<UNMASK>
# Summary of top ten sorting algorithms
Posted by anujgarg on Sat, 05 Mar 2022 15:10:00 +0100
# Top ten sorting algorithms
Common sorting algorithms include bubble sort, selection sort, insertion sort, merge sort, quick sort, Hill sort, heap sort, count sort, bucket sort and cardinal sort
## Bubble sorting
When adjacent elements are equal, they do not exchange positions, so bubble sorting is a stable sorting.
```void bubbleSort(vector<int> & nums){
// Cycle n-1 times
for(int i = 0; i < nums.size() - 1; i++){
// A maximum (small) value is found for each cycle, so the number of cycles per cycle decreases with the increase of i
for(int j = 0; j < nums.size() - i - 1; j++){
//In ascending order, change the large number to the back position
if(nums[j] > nums[j + 1]){
swap(nums[j], nums[j + 1]);
}
}
}
}
```
## Select sort
```void selectSort(vector<int> & nums){
for(int i = 0; i < nums.size() - 1; i++){ // Cycle n - 1
int minIdx = i; //Loop to find the subscript of the minimum value, and then exchange the minimum value with the starting value
for(int j = i; j < nums.size(); j++){
if(nums[minIdx] > nums[j]){
minIdx = j;
}
}
swap(nums[i], nums[minIdx]);
}
}
```
## Insert sort
```void insertSort(vector<int> & nums){
for(int i = 1; i < nums.size(); i++){ // n - 1 cycle
int j;
int num = nums[i];
//Find the first element num[j] less than or equal to the current number forward, and move the subsequent elements backward by one bit
for(j = i - 1; j >= 0; j--){
if(num < nums[j]){ //Move back
nums[j + 1] = nums[j]; //Or swap (Num [J], Num [J + 1]);
}else{
break;
}
}
//Assign num to the next bit of num[j]
nums[j + 1] = num;
}
}
```
## Shell Sort
Hill sort, also known as decreasing incremental sort algorithm, is a more efficient improved version of insertion sort. But Hill sorting is an unstable sorting algorithm.
Hill sort is an improved method based on the following two properties of insertion sort:
• Insertion sorting is efficient when operating on almost ordered data, that is, it can achieve the efficiency of linear sorting;
• However, insertion sorting is generally inefficient, because insertion sorting can only move data one bit at a time;
The basic idea of Hill sort is: first divide the whole record sequence to be sorted into several subsequences for direct insertion sort. When the records in the whole sequence are "basically orderly", then directly insert sort all records in turn.
```void shell_sort(vector<int> & nums) {
//Increment gap and gradually reduce the increment
for (int gap = nums.size() / 2; gap > 0; gap /= 2) {
//From the gap element, directly insert and sort the groups one by one
for (int i = gap; i < nums.size(); i++) {
for(int j = i; j >= gap && nums[j] < nums[j - gap]; j -= gap){
swap(nums[j], nums[j - gap]);
}
}
}
}
```
## Merge sort
```void merge(vector<int> & vec, int L1, int R1, int L2, int R2){
vector<int> temp; //Auxiliary array
int begin = L1;
while(L1 <= R1 && L2 <= R2){
if(vec[L1] < vec[L2]){
temp.push_back(vec[L1++]);
}else{
temp.push_back(vec[L2++]);
}
}
while(L1 <= R1){
temp.push_back(vec[L1++]);
}
while(L2 <= R2){
temp.push_back(vec[L2++]);
}
vec.assign(temp.begin(), temp.end());
}
void mergeSort(vector<int> & vec, int left, int right){
if(left < right){
int mid = (left + right) / 2;
mergeSort(vec, left, mid);
mergeSort(vec, mid + 1, right);
merge(vec, left, mid, mid + 1, right);
}
}
```
## Quick sort
Why is quick sort called quick sort?
https://blog.csdn.net/weixin_41445507/article/details/90255906
https://blog.csdn.net/linfeng24/article/details/38429055
```int quick(vector<int> & vec, int left, int right){
int temp = vec[left];
while(left < right){
//Find the first number on the left that is greater than or equal to temp
while(left < right && vec[left] < temp){
left += 1;
}
//Find the first number on the right that is less than or equal to temp
while(left < right && vec[right] > temp){
right -= 1;
}
//Exchange two numbers
swap(vec[left], vec[right]);
}
return left;
}
void quickSort(vector<int> & vec, int left, int right){
if(left < right){
int pos = quick(vec, left, right);
quickSort(vec, left, pos - 1);
quickSort(vec, pos + 1, right);
}
}
```
## Heap sort
Heap sort refers to a sort algorithm designed by using the data structure of heap. Heap is a structure similar to a complete binary tree, and satisfies the nature of heap at the same time: that is, the key value or index of a child node is always less than (or greater than) its parent node. Heap sort can be said to be a selective sort that uses the concept of heap to sort. There are two methods:
1. Large top heap: the value of each node is greater than or equal to the value of its child nodes, which is used for ascending arrangement in heap sorting algorithm;
2. Small top heap: the value of each node is less than or equal to the value of its child nodes, which is used for descending arrangement in heap sorting algorithm;
The average time complexity of heap sorting is Ο (nlogn).
```// Build large top reactor
void buildMaxHeap(vector<int> & nums) {
// Adjust the heap of each non leaf node
for (int i = nums.size() / 2; i >= 0; i--) {
heapify(nums, i);
}
}
// Heap adjustment, starting from the ith node
void heapify(vector<int> & nums, int i, int size) {
int left = 2 * i + 1; //Left child
int right = 2 * i + 2; //Right child
int largest = i;
//Find the maximum of the left and right children
if (left < size && nums[left] > nums[largest]) {
largest = left;
}
if (right < size && nums[right] > nums[largest]) {
largest = right;
}
//Replace with parent node
if (largest != i) {
swap(nums[i], nums[largest]);
heapify(nums, largest);
}
}
void heapSort(vector<int> & nums) {
buildMaxHeap(nums);
for (int i = nums.size() - 1; i > 0; i--) {
// Put the top node last each time
swap(nums[0], nums[i]);
heapify(nums, 0, i);
}
return nums;
}
```
## Count sort
Counting sorting is not a comparative sorting, and the sorting speed is faster than any comparative sorting algorithm.
Since the length of the array used for counting depends on the range of data in the array to be sorted (equal to the difference between the maximum and minimum values of the array to be sorted plus 1), counting sorting requires a lot of time and memory for arrays with large data range.
```void countingSort(vector<int> & nums) {
int maxValue = nums[0];
//Find maximum value
for(int i = 1; i < nums.size(); i++){
maxValue = max(maxValue, nums[i]);
}
vector<int> bucket(maxValue + 1, 0);
for (int i = 0; i < maxValue + 1; i++) {
bucket[nums[i]] += 1;
}
int idx = 0;
for (int i = 0; i < maxValue + 1; i++) {
while(bucket[i] > 0) {
nums[idx++] = i;
bucket[i] -= 1;
}
}
}
```
## Bucket sorting
Bucket sorting is an upgraded version of counting sorting. It makes use of the mapping relationship of the function. The key to efficiency lies in the determination of the mapping function. In order to make bucket sorting more efficient, we need to do these two things:
1. When there is enough extra space, try to increase the number of barrels
2. The mapping function used can evenly distribute the input N data into K buckets
At the same time, for the sorting of elements in the bucket, it is very important to choose a comparative sorting algorithm for the performance.
1. When is the fastest
When the input data can be evenly distributed to each bucket.
1. When is the slowest
When the input data is allocated to the same bucket.
## Cardinality sort
Radix sorting is a non comparative integer sorting algorithm
Cardinal sorting, counting sorting and bucket sorting all use the concept of bucket, but there are obvious differences in the use of bucket:
• Cardinality sorting: allocate buckets according to each number of key value;
• Counting and sorting: only one key value is stored in each bucket;
• Bucket sorting: each bucket stores a certain range of values;
Topics: C++ Algorithm
| 6,908 | 6,699,437 |
<MASK>
The motion of a conical pendulum can be analyzed using the principles of circular motion and Newton’s laws of motion. The centripetal force acting on the weight can be calculated using the formula F = mv²/r, where F is the force, m is the mass of the weight, v is the velocity, and r is the radius of the circular motion. The tension in the string or rod can be calculated using the formula T = mg cos θ + mv²/r, where T is the tension, m is the mass of the weight, g is the acceleration due to gravity, θ is the angle of inclination, and r is the radius of the circular motion.
The period of a conical pendulum is given by the formula T = 2π√(r/g)sin θ, where T is the period, r is the radius of the circular motion, g is the acceleration due to gravity, and θ is the angle of inclination. The period depends on the length of the string or rod, the mass of the weight, and the angle of inclination. The period of a conical pendulum is independent of the amplitude of the motion.
<MASK>
<UNMASK>
# What is a Conical Pendulum?
<MASK>
The motion of a conical pendulum can be analyzed using the principles of circular motion and Newton’s laws of motion. The centripetal force acting on the weight can be calculated using the formula F = mv²/r, where F is the force, m is the mass of the weight, v is the velocity, and r is the radius of the circular motion. The tension in the string or rod can be calculated using the formula T = mg cos θ + mv²/r, where T is the tension, m is the mass of the weight, g is the acceleration due to gravity, θ is the angle of inclination, and r is the radius of the circular motion.
The period of a conical pendulum is given by the formula T = 2π√(r/g)sin θ, where T is the period, r is the radius of the circular motion, g is the acceleration due to gravity, and θ is the angle of inclination. The period depends on the length of the string or rod, the mass of the weight, and the angle of inclination. The period of a conical pendulum is independent of the amplitude of the motion.
<MASK>
Conical pendulums are also used in amusement park rides, such as the Pirate Ship and the Kamikaze, to provide a swinging motion. In addition, conical pendulums are used in physics experiments to demonstrate the principles of circular motion, centripetal force, and uniform motion.
<MASK>
<UNMASK>
# What is a Conical Pendulum?
<MASK>
The motion of a conical pendulum can be analyzed using the principles of circular motion and Newton’s laws of motion. The centripetal force acting on the weight can be calculated using the formula F = mv²/r, where F is the force, m is the mass of the weight, v is the velocity, and r is the radius of the circular motion. The tension in the string or rod can be calculated using the formula T = mg cos θ + mv²/r, where T is the tension, m is the mass of the weight, g is the acceleration due to gravity, θ is the angle of inclination, and r is the radius of the circular motion.
The period of a conical pendulum is given by the formula T = 2π√(r/g)sin θ, where T is the period, r is the radius of the circular motion, g is the acceleration due to gravity, and θ is the angle of inclination. The period depends on the length of the string or rod, the mass of the weight, and the angle of inclination. The period of a conical pendulum is independent of the amplitude of the motion.
<MASK>
Conical pendulums are used in various applications, such as in gyroscopes, seismometers, and navigation devices. Gyroscopes use conical pendulums to measure the angular velocity of a rotating object. Seismometers use conical pendulums to detect seismic waves and measure the intensity of earthquakes. Navigation devices use conical pendulums to measure the acceleration of a moving vehicle and determine its position.
Conical pendulums are also used in amusement park rides, such as the Pirate Ship and the Kamikaze, to provide a swinging motion. In addition, conical pendulums are used in physics experiments to demonstrate the principles of circular motion, centripetal force, and uniform motion.
# Example of a Conical Pendulum in Action
<MASK>
<UNMASK>
# What is a Conical Pendulum?
A conical pendulum is a type of pendulum consisting of a weight or bob attached to the end of a string or rod that swings in a circular motion in a vertical plane. Unlike a simple pendulum, which moves in a plane perpendicular to the force of gravity, a conical pendulum moves in a plane that is tilted at an angle to the force of gravity. The rotation of the conical pendulum is due to the combination of the gravitational force and the tension in the string or rod.
The motion of a conical pendulum can be described as a combination of two types of motion: circular motion about the vertical axis and uniform motion in a horizontal plane. As the weight moves in a circular path, it experiences a centripetal force that keeps it on the circular path. At the same time, the tension in the string or rod provides the necessary force to keep the weight moving in a horizontal plane. The angle between the plane of the circular motion and the horizontal plane is known as the angle of inclination.
<MASK>
The motion of a conical pendulum can be analyzed using the principles of circular motion and Newton’s laws of motion. The centripetal force acting on the weight can be calculated using the formula F = mv²/r, where F is the force, m is the mass of the weight, v is the velocity, and r is the radius of the circular motion. The tension in the string or rod can be calculated using the formula T = mg cos θ + mv²/r, where T is the tension, m is the mass of the weight, g is the acceleration due to gravity, θ is the angle of inclination, and r is the radius of the circular motion.
The period of a conical pendulum is given by the formula T = 2π√(r/g)sin θ, where T is the period, r is the radius of the circular motion, g is the acceleration due to gravity, and θ is the angle of inclination. The period depends on the length of the string or rod, the mass of the weight, and the angle of inclination. The period of a conical pendulum is independent of the amplitude of the motion.
# Applications of Conical Pendulums
Conical pendulums are used in various applications, such as in gyroscopes, seismometers, and navigation devices. Gyroscopes use conical pendulums to measure the angular velocity of a rotating object. Seismometers use conical pendulums to detect seismic waves and measure the intensity of earthquakes. Navigation devices use conical pendulums to measure the acceleration of a moving vehicle and determine its position.
Conical pendulums are also used in amusement park rides, such as the Pirate Ship and the Kamikaze, to provide a swinging motion. In addition, conical pendulums are used in physics experiments to demonstrate the principles of circular motion, centripetal force, and uniform motion.
# Example of a Conical Pendulum in Action
One example of a conical pendulum in action is the Foucault pendulum, which is a pendulum that swings in a circular motion and rotates about its axis of suspension. The rotation of the Foucault pendulum is due to the rotation of the Earth. The motion of the Foucault pendulum demonstrates the rotation of the Earth and the effect of the Coriolis force. The Foucault pendulum is used in science museums and planetariums to provide a visual demonstration of the rotation of the Earth.
<UNMASK>
# What is a Conical Pendulum?
A conical pendulum is a type of pendulum consisting of a weight or bob attached to the end of a string or rod that swings in a circular motion in a vertical plane. Unlike a simple pendulum, which moves in a plane perpendicular to the force of gravity, a conical pendulum moves in a plane that is tilted at an angle to the force of gravity. The rotation of the conical pendulum is due to the combination of the gravitational force and the tension in the string or rod.
The motion of a conical pendulum can be described as a combination of two types of motion: circular motion about the vertical axis and uniform motion in a horizontal plane. As the weight moves in a circular path, it experiences a centripetal force that keeps it on the circular path. At the same time, the tension in the string or rod provides the necessary force to keep the weight moving in a horizontal plane. The angle between the plane of the circular motion and the horizontal plane is known as the angle of inclination.
# Mechanics of a Conical Pendulum
The motion of a conical pendulum can be analyzed using the principles of circular motion and Newton’s laws of motion. The centripetal force acting on the weight can be calculated using the formula F = mv²/r, where F is the force, m is the mass of the weight, v is the velocity, and r is the radius of the circular motion. The tension in the string or rod can be calculated using the formula T = mg cos θ + mv²/r, where T is the tension, m is the mass of the weight, g is the acceleration due to gravity, θ is the angle of inclination, and r is the radius of the circular motion.
The period of a conical pendulum is given by the formula T = 2π√(r/g)sin θ, where T is the period, r is the radius of the circular motion, g is the acceleration due to gravity, and θ is the angle of inclination. The period depends on the length of the string or rod, the mass of the weight, and the angle of inclination. The period of a conical pendulum is independent of the amplitude of the motion.
# Applications of Conical Pendulums
Conical pendulums are used in various applications, such as in gyroscopes, seismometers, and navigation devices. Gyroscopes use conical pendulums to measure the angular velocity of a rotating object. Seismometers use conical pendulums to detect seismic waves and measure the intensity of earthquakes. Navigation devices use conical pendulums to measure the acceleration of a moving vehicle and determine its position.
Conical pendulums are also used in amusement park rides, such as the Pirate Ship and the Kamikaze, to provide a swinging motion. In addition, conical pendulums are used in physics experiments to demonstrate the principles of circular motion, centripetal force, and uniform motion.
# Example of a Conical Pendulum in Action
One example of a conical pendulum in action is the Foucault pendulum, which is a pendulum that swings in a circular motion and rotates about its axis of suspension. The rotation of the Foucault pendulum is due to the rotation of the Earth. The motion of the Foucault pendulum demonstrates the rotation of the Earth and the effect of the Coriolis force. The Foucault pendulum is used in science museums and planetariums to provide a visual demonstration of the rotation of the Earth.
| 2,416 | 6,699,438 |
<MASK>
<UNMASK>
<MASK>
It is a cyclic number.
<MASK>
It is a congruent number.
<MASK>
It is a pernicious number, because its binary representation contains a prime number (11) of ones.
<MASK>
The square root of 2363167 is about 1537.2595746978. The cubic root of 2363167 is about 133.1981330919.
<MASK>
<UNMASK>
Search a number
2363167 is a prime number
BaseRepresentation
bin1001000000111100011111
311110001122201
421000330133
51101110132
6122352331
726041462
oct11007437
94401581
102363167
111374534
1295b6a7
13649831
144572d9
1531a2e7
hex240f1f
2363167 has 2 divisors, whose sum is σ = 2363168. Its totient is φ = 2363166.
<MASK>
2363167 is digitally balanced in base 2, because in such base it contains all the possibile digits an equal number of times.
<MASK>
It is a cyclic number.
<MASK>
It is an Ulam number.
It is a congruent number.
<MASK>
It is a pernicious number, because its binary representation contains a prime number (11) of ones.
<MASK>
2363167 is a deficient number, since it is larger than the sum of its proper divisors (1).
2363167 is an equidigital number, since it uses as much as digits as its factorization.
<MASK>
The product of its digits is 4536, while the sum is 28.
The square root of 2363167 is about 1537.2595746978. The cubic root of 2363167 is about 133.1981330919.
Adding to 2363167 its reverse (7613632), we get a palindrome (9976799).
<MASK>
The spelling of 2363167 in words is "two million, three hundred sixty-three thousand, one hundred sixty-seven".
<UNMASK>
Search a number
2363167 is a prime number
BaseRepresentation
bin1001000000111100011111
311110001122201
421000330133
51101110132
6122352331
726041462
oct11007437
94401581
102363167
111374534
1295b6a7
13649831
144572d9
1531a2e7
hex240f1f
2363167 has 2 divisors, whose sum is σ = 2363168. Its totient is φ = 2363166.
<MASK>
2363167 is digitally balanced in base 2, because in such base it contains all the possibile digits an equal number of times.
<MASK>
It is a cyclic number.
<MASK>
It is an Ulam number.
It is a congruent number.
<MASK>
It is a pernicious number, because its binary representation contains a prime number (11) of ones.
<MASK>
22363167 is an apocalyptic number.
2363167 is a deficient number, since it is larger than the sum of its proper divisors (1).
2363167 is an equidigital number, since it uses as much as digits as its factorization.
<MASK>
The product of its digits is 4536, while the sum is 28.
The square root of 2363167 is about 1537.2595746978. The cubic root of 2363167 is about 133.1981330919.
Adding to 2363167 its reverse (7613632), we get a palindrome (9976799).
<MASK>
The spelling of 2363167 in words is "two million, three hundred sixty-three thousand, one hundred sixty-seven".
<UNMASK>
Search a number
2363167 is a prime number
BaseRepresentation
bin1001000000111100011111
311110001122201
421000330133
51101110132
6122352331
726041462
oct11007437
94401581
102363167
111374534
1295b6a7
13649831
144572d9
1531a2e7
hex240f1f
2363167 has 2 divisors, whose sum is σ = 2363168. Its totient is φ = 2363166.
The previous prime is 2363159. The next prime is 2363171. The reversal of 2363167 is 7613632.
2363167 is digitally balanced in base 2, because in such base it contains all the possibile digits an equal number of times.
It is a strong prime.
It is a cyclic number.
It is not a de Polignac number, because 2363167 - 23 = 2363159 is a prime.
It is an Ulam number.
It is a congruent number.
It is not a weakly prime, because it can be changed into another prime (2363107) by changing a digit.
It is a pernicious number, because its binary representation contains a prime number (11) of ones.
It is a polite number, since it can be written as a sum of consecutive naturals, namely, 1181583 + 1181584.
It is an arithmetic number, because the mean of its divisors is an integer number (1181584).
22363167 is an apocalyptic number.
2363167 is a deficient number, since it is larger than the sum of its proper divisors (1).
2363167 is an equidigital number, since it uses as much as digits as its factorization.
2363167 is an odious number, because the sum of its binary digits is odd.
The product of its digits is 4536, while the sum is 28.
The square root of 2363167 is about 1537.2595746978. The cubic root of 2363167 is about 133.1981330919.
Adding to 2363167 its reverse (7613632), we get a palindrome (9976799).
It can be divided in two parts, 236 and 3167, that added together give a triangular number (3403 = T82).
The spelling of 2363167 in words is "two million, three hundred sixty-three thousand, one hundred sixty-seven".
| 1,353 | 6,699,439 |
<MASK>
<UNMASK>
## Algebra 1
<MASK>
<UNMASK>
## Algebra 1
The Pythagorean Theorem states that for a right triangle, $a^{2} +b^{2} = c^{2}$. In order to figure out if these sides can be a Pythagorean Triple, we plug them into the Pythagorean Theorem: $40^{2} + 41^{2} = 58^{2} \\\\ 3281=3364$ This is not true, so it is not a Pythagorean Triple.
| 116 | 6,699,440 |
<MASK>
$\left( \dfrac{dy}{dx} \right)^2 = \dfrac{x^2}{r^2 – x^2}$
<MASK>
<UNMASK>
<MASK>
From the figure, the area of the strip is
$dA = 2\pi \, x \, ds$
<MASK>
$\left( \dfrac{dy}{dx} \right)^2 = \dfrac{x^2}{r^2 – x^2}$
<MASK>
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \cos \theta\sqrt{\dfrac{1}{\cos^2 \theta}} \, d\theta$
<MASK>
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \, d\theta$
<MASK>
<UNMASK>
# Derivation of Formula for Total Surface Area of the Sphere by Integration
<MASK>
$A = 4\pi r^2$
From the figure, the area of the strip is
$dA = 2\pi \, x \, ds$
<MASK>
$\displaystyle A = 4\pi \int_0^r x \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$
From the figure,
$x^2 + y^2 = r^2$
<MASK>
$\left( \dfrac{dy}{dx} \right)^2 = \dfrac{x^2}{r^2 – x^2}$
Thus,
$\displaystyle A = 4\pi \int_0^r x \sqrt{1 + \dfrac{x^2}{r^2 – x^2}} \, dx$
<MASK>
$\displaystyle A = 4\pi \int_0^r x \sqrt{\dfrac{r^2}{r^2 – x^2}} \, dx$
<MASK>
Thus,
$\displaystyle A = 4\pi \int_0^{\pi/2} r \sin \theta \sqrt{\dfrac{r^2}{r^2 – r^2 \sin^2 \theta}} \, (r \cos \theta \, d\theta)$
$\displaystyle A = 4\pi \int_0^{\pi/2} r^2 \sin \theta \cos \theta\sqrt{\dfrac{r^2}{r^2(1 – \sin^2 \theta)}} \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \cos \theta\sqrt{\dfrac{1}{\cos^2 \theta}} \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \cos \theta \left( \dfrac{1}{\cos \theta} \right) \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \, d\theta$
$A = 4\pi r^2 \bigg[-\cos \theta \bigg]_0^{\pi/2}$
<MASK>
$A = 4\pi r^2$ okay!
<UNMASK>
# Derivation of Formula for Total Surface Area of the Sphere by Integration
<MASK>
Derivation of Formula for Total Surface Area of the Sphere by Integration
The total surface area of the sphere is four times the area of great circle. To know more about great circle, see properties of a sphere. Given the radius r of the sphere, the total surface area is
$A = 4\pi r^2$
From the figure, the area of the strip is
$dA = 2\pi \, x \, ds$
Where ds is the length of differential arc which is given by
$ds = \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx = \sqrt{1 + \left( \dfrac{dx}{dy} \right)^2} \, dy$
The total area of the sphere is equal to twice the sum of the differential area dA from 0 to r.
$\displaystyle A = 2 \left( \int_0^r 2\pi \, x \, ds \right)$
$\displaystyle A = 4\pi \int_0^r x \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$
From the figure,
$x^2 + y^2 = r^2$
$y = \sqrt{r^2 – x^2}$
<MASK>
$\dfrac{dy}{dx} = \dfrac{-x}{\sqrt{r^2 – x^2}}$
$\left( \dfrac{dy}{dx} \right)^2 = \dfrac{x^2}{r^2 – x^2}$
Thus,
$\displaystyle A = 4\pi \int_0^r x \sqrt{1 + \dfrac{x^2}{r^2 – x^2}} \, dx$
$\displaystyle A = 4\pi \int_0^r x \sqrt{\dfrac{(r^2 – x^2) + x^2}{r^2 – x^2}} \, dx$
$\displaystyle A = 4\pi \int_0^r x \sqrt{\dfrac{r^2}{r^2 – x^2}} \, dx$
Let
x = r sin θ
dx = r cos θ dθ
<MASK>
Thus,
$\displaystyle A = 4\pi \int_0^{\pi/2} r \sin \theta \sqrt{\dfrac{r^2}{r^2 – r^2 \sin^2 \theta}} \, (r \cos \theta \, d\theta)$
$\displaystyle A = 4\pi \int_0^{\pi/2} r^2 \sin \theta \cos \theta\sqrt{\dfrac{r^2}{r^2(1 – \sin^2 \theta)}} \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \cos \theta\sqrt{\dfrac{1}{\cos^2 \theta}} \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \cos \theta \left( \dfrac{1}{\cos \theta} \right) \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \, d\theta$
$A = 4\pi r^2 \bigg[-\cos \theta \bigg]_0^{\pi/2}$
<MASK>
$A = 4\pi r^2 \bigg[ -0 + 1 \bigg]$
$A = 4\pi r^2$ okay!
<UNMASK>
# Derivation of Formula for Total Surface Area of the Sphere by Integration
• Home
• Reviewers
• College Algebra
• Plane Trigonometry
• Plane Geometry
• Differential Calculus
• Engineering Mechanics
• Civil Engineering
• Blogs
• Forums
• Exams
• Courses
• Videos
• Help
Derivation of Formula for Total Surface Area of the Sphere by Integration
The total surface area of the sphere is four times the area of great circle. To know more about great circle, see properties of a sphere. Given the radius r of the sphere, the total surface area is
$A = 4\pi r^2$
From the figure, the area of the strip is
$dA = 2\pi \, x \, ds$
Where ds is the length of differential arc which is given by
$ds = \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx = \sqrt{1 + \left( \dfrac{dx}{dy} \right)^2} \, dy$
The total area of the sphere is equal to twice the sum of the differential area dA from 0 to r.
$\displaystyle A = 2 \left( \int_0^r 2\pi \, x \, ds \right)$
$\displaystyle A = 4\pi \int_0^r x \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$
From the figure,
$x^2 + y^2 = r^2$
$y = \sqrt{r^2 – x^2}$
$\dfrac{dy}{dx} = \dfrac{-2x}{2\sqrt{r^2 – x^2}}$
$\dfrac{dy}{dx} = \dfrac{-x}{\sqrt{r^2 – x^2}}$
$\left( \dfrac{dy}{dx} \right)^2 = \dfrac{x^2}{r^2 – x^2}$
Thus,
$\displaystyle A = 4\pi \int_0^r x \sqrt{1 + \dfrac{x^2}{r^2 – x^2}} \, dx$
$\displaystyle A = 4\pi \int_0^r x \sqrt{\dfrac{(r^2 – x^2) + x^2}{r^2 – x^2}} \, dx$
$\displaystyle A = 4\pi \int_0^r x \sqrt{\dfrac{r^2}{r^2 – x^2}} \, dx$
Let
x = r sin θ
dx = r cos θ dθ
When x = 0, θ = 0
When x = r, θ = π/2
Thus,
$\displaystyle A = 4\pi \int_0^{\pi/2} r \sin \theta \sqrt{\dfrac{r^2}{r^2 – r^2 \sin^2 \theta}} \, (r \cos \theta \, d\theta)$
$\displaystyle A = 4\pi \int_0^{\pi/2} r^2 \sin \theta \cos \theta\sqrt{\dfrac{r^2}{r^2(1 – \sin^2 \theta)}} \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \cos \theta\sqrt{\dfrac{1}{\cos^2 \theta}} \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \cos \theta \left( \dfrac{1}{\cos \theta} \right) \, d\theta$
$\displaystyle A = 4\pi r^2 \int_0^{\pi/2} \sin \theta \, d\theta$
$A = 4\pi r^2 \bigg[-\cos \theta \bigg]_0^{\pi/2}$
$A = 4\pi r^2 \bigg[-\cos \frac{1}{2}\pi + \cos 0 \bigg]$
$A = 4\pi r^2 \bigg[ -0 + 1 \bigg]$
$A = 4\pi r^2$ okay!
| 2,537 | 6,699,441 |
<MASK>
1 cm3 = 0.06762804511761 tbsp
To convert 19.3 cubic centimeters into tablespoons we have to multiply 19.3 by the conversion factor in order to get the volume amount from cubic centimeters to tablespoons. We can also form a simple proportion to calculate the result:
1 cm3 → 0.06762804511761 tbsp
19.3 cm3 → V(tbsp)
<MASK>
V(tbsp) = 1.3052212707699 tbsp
<MASK>
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 tablespoon is equal to 0.76615361884974 × 19.3 cubic centimeters.
<MASK>
## Conversion table
<MASK>
<UNMASK>
<MASK>
The conversion factor from cubic centimeters to tablespoons is 0.06762804511761, which means that 1 cubic centimeter is equal to 0.06762804511761 tablespoons:
1 cm3 = 0.06762804511761 tbsp
To convert 19.3 cubic centimeters into tablespoons we have to multiply 19.3 by the conversion factor in order to get the volume amount from cubic centimeters to tablespoons. We can also form a simple proportion to calculate the result:
1 cm3 → 0.06762804511761 tbsp
19.3 cm3 → V(tbsp)
<MASK>
V(tbsp) = 1.3052212707699 tbsp
<MASK>
19.3 cm3 → 1.3052212707699 tbsp
<MASK>
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 tablespoon is equal to 0.76615361884974 × 19.3 cubic centimeters.
Another way is saying that 19.3 cubic centimeters is equal to 1 ÷ 0.76615361884974 tablespoons.
<MASK>
For practical purposes we can round our final result to an approximate numerical value. We can say that nineteen point three cubic centimeters is approximately one point three zero five tablespoons:
<MASK>
## Conversion table
<MASK>
For quick reference purposes, below is the conversion table you can use to convert from cubic centimeters to tablespoons
cubic centimeters (cm3) tablespoons (tbsp)
20.3 cubic centimeters 1.373 tablespoons
21.3 cubic centimeters 1.44 tablespoons
22.3 cubic centimeters 1.508 tablespoons
23.3 cubic centimeters 1.576 tablespoons
24.3 cubic centimeters 1.643 tablespoons
25.3 cubic centimeters 1.711 tablespoons
26.3 cubic centimeters 1.779 tablespoons
27.3 cubic centimeters 1.846 tablespoons
28.3 cubic centimeters 1.914 tablespoons
29.3 cubic centimeters 1.982 tablespoons
<UNMASK>
## Conversion formula
The conversion factor from cubic centimeters to tablespoons is 0.06762804511761, which means that 1 cubic centimeter is equal to 0.06762804511761 tablespoons:
1 cm3 = 0.06762804511761 tbsp
To convert 19.3 cubic centimeters into tablespoons we have to multiply 19.3 by the conversion factor in order to get the volume amount from cubic centimeters to tablespoons. We can also form a simple proportion to calculate the result:
1 cm3 → 0.06762804511761 tbsp
19.3 cm3 → V(tbsp)
<MASK>
V(tbsp) = 19.3 cm3 × 0.06762804511761 tbsp
V(tbsp) = 1.3052212707699 tbsp
The final result is:
19.3 cm3 → 1.3052212707699 tbsp
We conclude that 19.3 cubic centimeters is equivalent to 1.3052212707699 tablespoons:
<MASK>
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 tablespoon is equal to 0.76615361884974 × 19.3 cubic centimeters.
Another way is saying that 19.3 cubic centimeters is equal to 1 ÷ 0.76615361884974 tablespoons.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that nineteen point three cubic centimeters is approximately one point three zero five tablespoons:
19.3 cm3 ≅ 1.305 tbsp
<MASK>
## Conversion table
### cubic centimeters to tablespoons chart
For quick reference purposes, below is the conversion table you can use to convert from cubic centimeters to tablespoons
cubic centimeters (cm3) tablespoons (tbsp)
20.3 cubic centimeters 1.373 tablespoons
21.3 cubic centimeters 1.44 tablespoons
22.3 cubic centimeters 1.508 tablespoons
23.3 cubic centimeters 1.576 tablespoons
24.3 cubic centimeters 1.643 tablespoons
25.3 cubic centimeters 1.711 tablespoons
26.3 cubic centimeters 1.779 tablespoons
27.3 cubic centimeters 1.846 tablespoons
28.3 cubic centimeters 1.914 tablespoons
29.3 cubic centimeters 1.982 tablespoons
<UNMASK>
## Conversion formula
The conversion factor from cubic centimeters to tablespoons is 0.06762804511761, which means that 1 cubic centimeter is equal to 0.06762804511761 tablespoons:
1 cm3 = 0.06762804511761 tbsp
To convert 19.3 cubic centimeters into tablespoons we have to multiply 19.3 by the conversion factor in order to get the volume amount from cubic centimeters to tablespoons. We can also form a simple proportion to calculate the result:
1 cm3 → 0.06762804511761 tbsp
19.3 cm3 → V(tbsp)
Solve the above proportion to obtain the volume V in tablespoons:
V(tbsp) = 19.3 cm3 × 0.06762804511761 tbsp
V(tbsp) = 1.3052212707699 tbsp
The final result is:
19.3 cm3 → 1.3052212707699 tbsp
We conclude that 19.3 cubic centimeters is equivalent to 1.3052212707699 tablespoons:
19.3 cubic centimeters = 1.3052212707699 tablespoons
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 tablespoon is equal to 0.76615361884974 × 19.3 cubic centimeters.
Another way is saying that 19.3 cubic centimeters is equal to 1 ÷ 0.76615361884974 tablespoons.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that nineteen point three cubic centimeters is approximately one point three zero five tablespoons:
19.3 cm3 ≅ 1.305 tbsp
<MASK>
## Conversion table
### cubic centimeters to tablespoons chart
For quick reference purposes, below is the conversion table you can use to convert from cubic centimeters to tablespoons
cubic centimeters (cm3) tablespoons (tbsp)
20.3 cubic centimeters 1.373 tablespoons
21.3 cubic centimeters 1.44 tablespoons
22.3 cubic centimeters 1.508 tablespoons
23.3 cubic centimeters 1.576 tablespoons
24.3 cubic centimeters 1.643 tablespoons
25.3 cubic centimeters 1.711 tablespoons
26.3 cubic centimeters 1.779 tablespoons
27.3 cubic centimeters 1.846 tablespoons
28.3 cubic centimeters 1.914 tablespoons
29.3 cubic centimeters 1.982 tablespoons
<UNMASK>
## Conversion formula
The conversion factor from cubic centimeters to tablespoons is 0.06762804511761, which means that 1 cubic centimeter is equal to 0.06762804511761 tablespoons:
1 cm3 = 0.06762804511761 tbsp
To convert 19.3 cubic centimeters into tablespoons we have to multiply 19.3 by the conversion factor in order to get the volume amount from cubic centimeters to tablespoons. We can also form a simple proportion to calculate the result:
1 cm3 → 0.06762804511761 tbsp
19.3 cm3 → V(tbsp)
Solve the above proportion to obtain the volume V in tablespoons:
V(tbsp) = 19.3 cm3 × 0.06762804511761 tbsp
V(tbsp) = 1.3052212707699 tbsp
The final result is:
19.3 cm3 → 1.3052212707699 tbsp
We conclude that 19.3 cubic centimeters is equivalent to 1.3052212707699 tablespoons:
19.3 cubic centimeters = 1.3052212707699 tablespoons
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 tablespoon is equal to 0.76615361884974 × 19.3 cubic centimeters.
Another way is saying that 19.3 cubic centimeters is equal to 1 ÷ 0.76615361884974 tablespoons.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that nineteen point three cubic centimeters is approximately one point three zero five tablespoons:
19.3 cm3 ≅ 1.305 tbsp
An alternative is also that one tablespoon is approximately zero point seven six six times nineteen point three cubic centimeters.
## Conversion table
### cubic centimeters to tablespoons chart
For quick reference purposes, below is the conversion table you can use to convert from cubic centimeters to tablespoons
cubic centimeters (cm3) tablespoons (tbsp)
20.3 cubic centimeters 1.373 tablespoons
21.3 cubic centimeters 1.44 tablespoons
22.3 cubic centimeters 1.508 tablespoons
23.3 cubic centimeters 1.576 tablespoons
24.3 cubic centimeters 1.643 tablespoons
25.3 cubic centimeters 1.711 tablespoons
26.3 cubic centimeters 1.779 tablespoons
27.3 cubic centimeters 1.846 tablespoons
28.3 cubic centimeters 1.914 tablespoons
29.3 cubic centimeters 1.982 tablespoons
| 2,204 | 6,699,442 |
# How to generate prime numbers?
I'm trying to create a formula to find a prime $$p$$ for some even $$n$$ and $$k\in\mathbb{N}$$
$$\Large p = 6n + 12k + 5$$
For example, choose an even number
$$n = 99999984$$
Applying the formula $$k=4$$ times
p = 599999909
p = 599999921
p = 599999933
p = 599999945
it finds a prime
p = 599999957
Question
Basically, I'm trying to find a good starting number, then determine any nearby prime within $$k$$ steps, rather than potentially starting in a prime free void and having to iterate to the next prime which could be millions of steps away?
For $$\large n<2^{26}$$, the formula is guaranteed to find a prime for $$k<80$$.
For $$\large n<2^{4096}$$, is it possible to compute a $$k$$ upper bound?
• There is a finite number of possibilities to try. An exhaustive search is theoretically possible, but far to large to be practical. Your search avoids numbers that have factors of $2$ or $3$, so the density of primes should be $3$ times higher than picking random numbers. I don't see why the coefficient on $k$ is $12$ instead of $6$. You could have $6n$ be just before a large prime gap. Commented Jul 25, 2022 at 3:00
There’s no easy way to guarantee a prime within some small number of steps in any linear recurrence. However, the Prime number Theorem states (approximately) that a random big $$n$$ is prime with probability $$1/log(n)$$, so you can approximate bounds.
In particular, since you are choosing numbers which are 5 mod 12, and all big primes are 1,5,7,or 11, then your are $$12/4=3$$ times as likely as a random number to be prime, so each number has probability $$3/log(n)$$. In general if your jump length is $$m$$ and you start at a relatively prime number, then $$\phi(m)$$ of the numbers are prime, so the chance of $$p$$ being prime is $$m/(\phi(m)log(p))$$.
Let’s say you have $$n$$ numbers each with probability $$q$$ of being prime. The exact estimate for the longest run seems tricky, but we can apply some heuristics to estimate it. The chance of a given starting point being at least $$d$$ without a prime is $$(1-q)^d$$. For this to occur once in the $$~n$$ starting positions means we want $$d$$ such that $$(1-q)^d=1/n$$ or $$d=-log(n)/log(1-q)$$.
Applying this to $$q=3/log(n)$$ and $$n=2^a$$ Longest wait is around $-log(2^a)/log(1-3/log(2^a))~a log(2)/(3/(log(2)a))~a^2*(ln(2))^2/3 For $$2^{26}$$, this gives an estimate of 110. For $$2^{4096}$$, this gives an estimate of 2.7 million. Note that these are estimates of the worst case. In practice, you expect to find a prime much much faster - within the first 30 thousand 99% of the time. Additionally, if you are selecting a jump length, the key thing is to maximize $$m/\phi(m)$$. You can do this by letting $$m$$ be a product of many primes like $$m=2*3*5*7*11*13*17*19*23$$. That’ll give a prime 6 times as likely as a random number (as opposed to 3 times that 12 gets you). In practice, just using 6 gets you most of the way there. • The idea of generating a prime 6 times as likely as a random number sounds great! I'm a maths newbie so what does your final suggested formula look like now? Thanks! Commented Jul 25, 2022 at 11:12 • My very approximate heuristic worst case scenario is$(\ln(2^a))^2/(m/\phi(m))$. For$2^{4096}\$ and my big m, that'd be around 1.4 million. This might be a large overestimate. Note that this is pretty irrelevant for nearly all purposes since you expect to take ln(2^4096)/6~500 tries on average and have a lower than 1% chance of taking more than 2500 tries where that probability is exponentially decreasing.
| 1,023 | 6,699,443 |
# 2.2 Centimeter to Inch
Convert 2.2 (two) Centimeters to Inches (cm to in) with our conversion calculator.
2.2 Centimeters to Inches equals 0.8661 in.
• Meter
• Kilometer
• Centimeter
• Millimeter
• Micrometer
• Nanometer
• Mile
• Yard
• Foot
• Inch
• Light Year
• Meter
• Kilometer
• Centimeter
• Millimeter
• Micrometer
• Nanometer
• Mile
• Yard
• Foot
• Inch
• Light Year
Convert 2.2 Centimeters to Inches (cm to in) with our conversion calculator.
2.2 Centimeters to Inches equals 0.8661 in.
To convert 2.2 cm to inches, you use the conversion factor that 1 inch is equivalent to 2.54 centimeters. The formula to convert centimeters to inches is the length in centimeters divided by 2.54. Therefore, for converting 2.2 cm to inches, you would perform the following calculation: ( \text{Inches} = \frac{2.2 \text{ cm}}{2.54} ).
The actual calculation would be ( \text{Inches} = \frac{2.2}{2.54} \approx 0.8661 ). This means that 2.2 centimeters is approximately equal to 0.8661 inches. This conversion is useful in many fields including but not limited to engineering, construction, and everyday uses where there is a need to convert measurements from the metric system (centimeters) to the imperial system (inches).
Here are 7 items that are approximately 2.2 cm in length:
1. Diameter of a US Nickel:
• Coins vary in sizes, but the diameter of a US Nickel is very close to 2.2 cm.
2. Standard USB Connector Width:
• The width of a standard USB connector (type A) is roughly 2.2 cm, making it a common item of this measurement.
3. A Standard Dice:
• A standard dice is about 1.6 cm on each side, but if it were irregular with one dimension longer, it could stretch to 2.2 cm.
4. Short Pencil Stub:
• A short pencil that's been used down to about 2.2 cm in length, typical for pencils near the end of their use.
5. A Large Paperclip:
• A large paperclip can be stretched or manipulated to measure approximately 2.2 cm in length.
6. SD Card Width:
• The width of an SD Card is exactly 2.2 cm, a standard size for many storage devices used in cameras and computers.
7. Part of a Standard Ruler:
• On any standard ruler, there is a segment that measures exactly 2.2 cm which is just a small portion of the ruler's total length.
These examples show common items that one might encounter or use daily, providing a tangible sense of how small or large an object that measures 2.2 cm in length is.
| 646 | 6,699,444 |
<MASK>
The problem is in the Cauchy Integral Formula section in Gamelin's "Complex Analysis". $$\oint_{|z-1|=3} \frac{dz}{z(z^2-4)e^z}$$ I have trouble with it because -2 is actually on the boundary.
-
It should be treated as principal value, i.e
$$\begin{eqnarray} \operatorname{P.V.} \oint_{|z-1|=3} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} &=& \frac{1}{2} \left( \oint_{|z-1|=3^+} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} + \oint_{|z-1|=3^-} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} \right) \\ &=& 2 \pi i \left( \left. \frac{\mathrm{e}^{-z}}{z^2-4} \right|_{z=0} + \left. \frac{\mathrm{e}^{-z}}{z(z+2)} \right|_{z=2} + \frac{1}{2} \left. \frac{\mathrm{e}^{-z}}{z(z-2)} \right|_{z=-2} \right) \\ &=& 2 \pi i \left( -\frac{1}{4} + \frac{\mathrm{e}^{-2}}{8} + \frac{\mathrm{e}^{2}}{16} \right) \end{eqnarray}$$
<MASK>
<UNMASK>
# A complex integral question
The problem is in the Cauchy Integral Formula section in Gamelin's "Complex Analysis". $$\oint_{|z-1|=3} \frac{dz}{z(z^2-4)e^z}$$ I have trouble with it because -2 is actually on the boundary.
-
It should be treated as principal value, i.e
$$\begin{eqnarray} \operatorname{P.V.} \oint_{|z-1|=3} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} &=& \frac{1}{2} \left( \oint_{|z-1|=3^+} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} + \oint_{|z-1|=3^-} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} \right) \\ &=& 2 \pi i \left( \left. \frac{\mathrm{e}^{-z}}{z^2-4} \right|_{z=0} + \left. \frac{\mathrm{e}^{-z}}{z(z+2)} \right|_{z=2} + \frac{1}{2} \left. \frac{\mathrm{e}^{-z}}{z(z-2)} \right|_{z=-2} \right) \\ &=& 2 \pi i \left( -\frac{1}{4} + \frac{\mathrm{e}^{-2}}{8} + \frac{\mathrm{e}^{2}}{16} \right) \end{eqnarray}$$
Added: Also can be done using Mathematica:
<MASK>
<UNMASK>
# A complex integral question
The problem is in the Cauchy Integral Formula section in Gamelin's "Complex Analysis". $$\oint_{|z-1|=3} \frac{dz}{z(z^2-4)e^z}$$ I have trouble with it because -2 is actually on the boundary.
-
It should be treated as principal value, i.e
$$\begin{eqnarray} \operatorname{P.V.} \oint_{|z-1|=3} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} &=& \frac{1}{2} \left( \oint_{|z-1|=3^+} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} + \oint_{|z-1|=3^-} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} \right) \\ &=& 2 \pi i \left( \left. \frac{\mathrm{e}^{-z}}{z^2-4} \right|_{z=0} + \left. \frac{\mathrm{e}^{-z}}{z(z+2)} \right|_{z=2} + \frac{1}{2} \left. \frac{\mathrm{e}^{-z}}{z(z-2)} \right|_{z=-2} \right) \\ &=& 2 \pi i \left( -\frac{1}{4} + \frac{\mathrm{e}^{-2}}{8} + \frac{\mathrm{e}^{2}}{16} \right) \end{eqnarray}$$
Added: Also can be done using Mathematica:
<MASK>
Out[217]= -I Pi/2 + I Pi/(4 E^2) + 1/8 I E^2 Pi
<MASK>
<UNMASK>
# A complex integral question
The problem is in the Cauchy Integral Formula section in Gamelin's "Complex Analysis". $$\oint_{|z-1|=3} \frac{dz}{z(z^2-4)e^z}$$ I have trouble with it because -2 is actually on the boundary.
-
It should be treated as principal value, i.e
$$\begin{eqnarray} \operatorname{P.V.} \oint_{|z-1|=3} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} &=& \frac{1}{2} \left( \oint_{|z-1|=3^+} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} + \oint_{|z-1|=3^-} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} \right) \\ &=& 2 \pi i \left( \left. \frac{\mathrm{e}^{-z}}{z^2-4} \right|_{z=0} + \left. \frac{\mathrm{e}^{-z}}{z(z+2)} \right|_{z=2} + \frac{1}{2} \left. \frac{\mathrm{e}^{-z}}{z(z-2)} \right|_{z=-2} \right) \\ &=& 2 \pi i \left( -\frac{1}{4} + \frac{\mathrm{e}^{-2}}{8} + \frac{\mathrm{e}^{2}}{16} \right) \end{eqnarray}$$
Added: Also can be done using Mathematica:
<MASK>
Out[217]= -I Pi/2 + I Pi/(4 E^2) + 1/8 I E^2 Pi
-
(Principal value, to be precise.) – Hans Lundmark Oct 25 '11 at 6:29
@HansLundmark Thank you, I have just corrected the spelling. – Sasha Oct 25 '11 at 10:26
I dont understand the solution, whats going on here, please teach me – Un Chien Andalou Apr 30 '13 at 14:09
<UNMASK>
# A complex integral question
The problem is in the Cauchy Integral Formula section in Gamelin's "Complex Analysis". $$\oint_{|z-1|=3} \frac{dz}{z(z^2-4)e^z}$$ I have trouble with it because -2 is actually on the boundary.
-
It should be treated as principal value, i.e
$$\begin{eqnarray} \operatorname{P.V.} \oint_{|z-1|=3} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} &=& \frac{1}{2} \left( \oint_{|z-1|=3^+} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} + \oint_{|z-1|=3^-} \frac{\mathrm{d}z}{z (z^2-4) \mathrm{e}^z} \right) \\ &=& 2 \pi i \left( \left. \frac{\mathrm{e}^{-z}}{z^2-4} \right|_{z=0} + \left. \frac{\mathrm{e}^{-z}}{z(z+2)} \right|_{z=2} + \frac{1}{2} \left. \frac{\mathrm{e}^{-z}}{z(z-2)} \right|_{z=-2} \right) \\ &=& 2 \pi i \left( -\frac{1}{4} + \frac{\mathrm{e}^{-2}}{8} + \frac{\mathrm{e}^{2}}{16} \right) \end{eqnarray}$$
Added: Also can be done using Mathematica:
In[217]:= TrigToExp[
Integrate @@ {1/(z (z^2 - 4) Exp[z]) Dt[z]/Dt[phi] /.
z -> 1 + (3) Exp[I phi], {phi, 0, 2 Pi}, PrincipalValue -> True}]
Out[217]= -I Pi/2 + I Pi/(4 E^2) + 1/8 I E^2 Pi
-
(Principal value, to be precise.) – Hans Lundmark Oct 25 '11 at 6:29
@HansLundmark Thank you, I have just corrected the spelling. – Sasha Oct 25 '11 at 10:26
I dont understand the solution, whats going on here, please teach me – Un Chien Andalou Apr 30 '13 at 14:09
| 2,291 | 6,699,445 |
Scalars And Vectors
<MASK>
1. Velocity vector of a stationary particle is a zero vector.
2. The acceleration vector of an object moving with a uniform velocity is a zero vector.
3. The position vector of the origin of coordinate axes is a zero vector.
<UNMASK>
Scalars And Vectors
<MASK>
## Properties of a Zero Vector
<MASK>
1. Velocity vector of a stationary particle is a zero vector.
2. The acceleration vector of an object moving with a uniform velocity is a zero vector.
3. The position vector of the origin of coordinate axes is a zero vector.
<UNMASK>
Scalars And Vectors
<MASK>
A vector whose magnitude is zero is called a zero vector or null vector. So, $(0,0)$ is the zero vector. In a zero vector, the origin and the terminal point coincide. The direction of a zero vector is indeterminate.
A zero vector is represented by $\overrightarrow{O}$. A vector when multiplied by $0$ gives a zero vector. $\therefore\overrightarrow{O}=\overrightarrow{0}$
## Properties of a Zero Vector
<MASK>
A zero vector has a lot of physical significance. It is useful in describing the physical situation involving vector quantities.
<MASK>
1. Velocity vector of a stationary particle is a zero vector.
2. The acceleration vector of an object moving with a uniform velocity is a zero vector.
3. The position vector of the origin of coordinate axes is a zero vector.
<UNMASK>
Scalars And Vectors
<MASK>
A vector whose magnitude is zero is called a zero vector or null vector. So, $(0,0)$ is the zero vector. In a zero vector, the origin and the terminal point coincide. The direction of a zero vector is indeterminate.
A zero vector is represented by $\overrightarrow{O}$. A vector when multiplied by $0$ gives a zero vector. $\therefore\overrightarrow{O}=\overrightarrow{0}$
## Properties of a Zero Vector
1. The addition of a zero vector to a given vector or subtraction of the zero vector from the given vector does not alter the given vector. Thus, $\overrightarrow{A}+\overrightarrow{O}=\overrightarrow{A}$ $\overrightarrow{A}-\overrightarrow{O}=\overrightarrow{A}$
2. If a zero vector is multiplied by a non zero scalar $n$, then again a zero vector is obtained. $\therefore n\overrightarrow{O}=\overrightarrow{O}$
3. If $m$ and $n$ are two different non zero scalars, then the relation $m\overrightarrow{A}=n\overrightarrow{B}$ can hold only if both $\overrightarrow{A}$ and $\overrightarrow{B}$ are zero vectors.
A zero vector has a lot of physical significance. It is useful in describing the physical situation involving vector quantities.
<MASK>
1. Velocity vector of a stationary particle is a zero vector.
2. The acceleration vector of an object moving with a uniform velocity is a zero vector.
3. The position vector of the origin of coordinate axes is a zero vector.
<UNMASK>
Scalars And Vectors
# Zero Vector
A vector whose magnitude is zero is called a zero vector or null vector. So, $(0,0)$ is the zero vector. In a zero vector, the origin and the terminal point coincide. The direction of a zero vector is indeterminate.
A zero vector is represented by $\overrightarrow{O}$. A vector when multiplied by $0$ gives a zero vector. $\therefore\overrightarrow{O}=\overrightarrow{0}$
## Properties of a Zero Vector
1. The addition of a zero vector to a given vector or subtraction of the zero vector from the given vector does not alter the given vector. Thus, $\overrightarrow{A}+\overrightarrow{O}=\overrightarrow{A}$ $\overrightarrow{A}-\overrightarrow{O}=\overrightarrow{A}$
2. If a zero vector is multiplied by a non zero scalar $n$, then again a zero vector is obtained. $\therefore n\overrightarrow{O}=\overrightarrow{O}$
3. If $m$ and $n$ are two different non zero scalars, then the relation $m\overrightarrow{A}=n\overrightarrow{B}$ can hold only if both $\overrightarrow{A}$ and $\overrightarrow{B}$ are zero vectors.
A zero vector has a lot of physical significance. It is useful in describing the physical situation involving vector quantities.
### Examples of Zero Vectors
1. Velocity vector of a stationary particle is a zero vector.
2. The acceleration vector of an object moving with a uniform velocity is a zero vector.
3. The position vector of the origin of coordinate axes is a zero vector.
| 1,000 | 6,699,446 |
<MASK>
<UNMASK>
Q7x
• Posts: 3
Joined: Apr 10, 2013
August 16th, 2013 at 8:36:08 AM permalink
Card removal and No Limit Hold'em ?
<MASK>
98s
combos 4, 3, 2, 1, 1, 0, 0.
<MASK>
<UNMASK>
Q7x
• Posts: 3
Joined: Apr 10, 2013
August 16th, 2013 at 8:36:08 AM permalink
Card removal and No Limit Hold'em ?
<MASK>
JTo
combos 12, 9, 7, 4, 3, 2, 0.
98s
combos 4, 3, 2, 1, 1, 0, 0.
<MASK>
<UNMASK>
Q7x
• Posts: 3
Joined: Apr 10, 2013
August 16th, 2013 at 8:36:08 AM permalink
Card removal and No Limit Hold'em ?
<MASK>
AKx
combos 16, 12, 9, 5, 4, 2, 0.
JTo
combos 12, 9, 7, 4, 3, 2, 0.
98s
combos 4, 3, 2, 1, 1, 0, 0.
<MASK>
<UNMASK>
Q7x
• Posts: 3
Joined: Apr 10, 2013
August 16th, 2013 at 8:36:08 AM permalink
Card removal and No Limit Hold'em ?
Hello, I'm still a relative beginner at counting combos and combinatorics, and I've been stumped lately on some math that I can't figure out. For some background on my situation, I know that: In No Limit Hold'em (NLHE), there are a total of 169 hands (1326 combos). Of those 169 hands; 13 hands are pairs (78 combos), and 78 hands are Non-Pairs (1248 combos). The Non-Pair hands break into the two groups of Off-suited hands and Suited hands. There are 78 Off-suited hands (936 combos) and 78 Suited hands (312 combos). As far as individual starting hands go; there are 6 combos per Pair hand, 16 combos per Non-Pair hand, 12 combos per Off-suited hand, and 4 combos per Suited hand. I am starting to try my hand at the math that shows how card removal (dead cards) affect the number of combos for a particular hand type. Pairs are easy; for 0, 1, 2, and 3 dead cards, there are 6, 3, 1, and 0 combos possible.
Main Question vvv
Where I'm starting to doubt my results is when calculating how the number of dead cards, from 2 - 6 decreases the amount of combos possible for Non-Pair, Off-suited, and Suited hands. So, I wonder if someone who knows about this topic can verify my results, or give correct answers so that I can go back and rethink how I'm going about calculating them? My results are shown below. Across the top, from left to right is the number of dead cards and then underneath there are three rows showing the # of possible combos for each of the 3 hand-types (Non-pairs AKx, Off-suited JTo, and Suited 98s):
cards > 0, 1, 2, 3, 4, 5, 6.
AKx
combos 16, 12, 9, 5, 4, 2, 0.
JTo
combos 12, 9, 7, 4, 3, 2, 0.
98s
combos 4, 3, 2, 1, 1, 0, 0.
The results are in whole numbers, so that they are more practical to use and apply in-game and when I'm seeing how ranges are affect by dead cards during a hand review. I'm wanting to know if they are accurate enough to use both during a live game, and also when I'm studying situations away from the table to get an idea of how I played a particular hand (good or bad)?
If the above "# combos" figures for the AKx hand are accurate enough, I may go back and use the more precise figures (to 3 decimal places) for the AKx hand. Then, I'd divide up the combos proportionally for the JTo and 98s hands (splitting the AKx hand into 3:1). I'd do this to see if it made a big enough difference to change the existing results for JTo and 98s. Once those two hands were sorted out into proper whole numbers, I could go back and round off the AKx figures to a whole number. Are the results above good enough to use for most situations that you know of? Thanks in advance for the help on this!
| 1,089 | 6,699,447 |
1. sarah_hendrix7
2. Hoa
do you want the whole steps or just a result? the final result is an = -7 *6^n
3. Hoa
a(n+1) -6 a(n) =0 characteristic equation is x -6 =0 x =6 the equation has the form an= C*6^n replace a1 = C *6^1 = -42 ---->C = -7 replace C into an form to get an= -7 *6^n Hope this help. are you in discrete math course?If so, you will understand what i mean
4. ParthKohli
LOL
5. Hoa
What's wrong? and what makes you laugh
6. Hoa
@ParthKohli : I am a student and want to study from others. Please tell me if I am wrong in my work, I appreciate when you correct my mistake or my flaw
7. ParthKohli
You have left a lot of space. :-)
8. Hoa
I don't know why the net post it up. I am not good at computer
9. ParthKohli
I think we can do this using the geometric sequence formula.
10. Hoa
yes, you are right. show me your work. now is your turn
11. ParthKohli
12. ParthKohli
$$a_1 = -42$$ and $$r = -6$$. Now use the following formula:$a_n = a_1 \times r^{n - 1}$
13. Hoa
how to get r?
14. ParthKohli
Strictly speaking, if you have:$a_{n + 1} = {\rm some ~number} \times a_n$For all $$n$$, then the "some number" is the common ratio denoted by $$r$$.
15. ParthKohli
I'd leave it to the asker to do the rest.
16. Hoa
I need help from my problem, may I have yours?
17. ParthKohli
18. Hoa
my problem is: find the coefficient of x^10 in the power series of (x^2 +x^4 +....) (x^3 +x^6 +x^9 +....)(x^4 +x^8 +x^12 +....)
19. ParthKohli
This is an interesting problem. Let's see.
20. ParthKohli
You can just message me here; I check this site much often... more than I check my email
21. Hoa
dealt. since I have to go. send me message when you've done. Nice to see.
22. ParthKohli
Cya!
Find more explanations on OpenStudy
| 591 | 6,699,448 |
<MASK>
<UNMASK>
largest possible values : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack
It is currently 23 Jan 2017, 23:54
<MASK>
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
<MASK>
Track
every week, we’ll send you an estimated GMAT score based on your performance
<MASK>
we will pick new questions that match your level based on your Timer History
# Events & Promotions
<MASK>
Intern
Joined: 21 Feb 2010
Posts: 33
Location: Ukraine
Followers: 1
<MASK>
### HideShow timer Statistics
<MASK>
3
4
5
6
7
<MASK>
Re: largest possible values [#permalink]
<MASK>
75^3=(25X3)^3=(5^2 X 3)^3 = 5^6 x3^3
which means that largest factor would be 5^6
<MASK>
Re: largest possible values [#permalink]
<MASK>
if $$27*5^6$$ is a multiple of $$5^m$$, then $$\frac{27*5^6}{5^m}$$ should result in an integer. Also, remember that m is an integer.
<MASK>
So, out of all the 4 choices that leave us with an integer result, the largest value of m is 6. Hence correct answer is D.
_________________
<MASK>
### Show Tags
<MASK>
Kudos [?]: 176 [0], given: 9
<MASK>
16 May 2010, 08:22
Hi,
I think the simplest method is...
Find out the highest factor of 5s in 75^3. i.e., 5^6 so, m=6. Because that will be the HCF.
Re: largest possible values [#permalink] 16 May 2010, 08:22
Similar topics Replies Last post
Similar
Topics:
3 If 20! × 20!/20^n is an integer, what is the largest possible value of 3 28 Apr 2016, 16:00
8 If 200!/10^n is an integer, what is the largest possible value of n? 7 28 Apr 2016, 02:36
3 What is the largest possible value of c if 5c + (d-12)^2 = 235? 6 24 Jul 2015, 04:11
10 What is the largest possible value of the following expression 4 16 Oct 2014, 22:21
largest possible value of actual area of rectangle ? 1 23 Jan 2011, 02:16
Display posts from previous: Sort by
# largest possible values
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
<MASK>
<UNMASK>
largest possible values : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack
It is currently 23 Jan 2017, 23:54
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
<MASK>
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
<MASK>
### Hide Tags
Intern
Joined: 21 Feb 2010
Posts: 33
Location: Ukraine
Followers: 1
Kudos [?]: 3 [1] , given: 9
<MASK>
(N/A)
<MASK>
### HideShow timer Statistics
<MASK>
3
4
5
6
7
Can somebody explain how to solve it?
Intern
Joined: 11 Jan 2007
Posts: 45
Location: United States
Concentration: Marketing, Healthcare
GMAT 1: 600 Q49 V25
GMAT 2: 650 Q49 V32
GPA: 3.5
WE: Pharmaceuticals (Consulting)
Followers: 0
<MASK>
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 06:17
IMO m=6. here is why
75^3=(25X3)^3=(5^2 X 3)^3 = 5^6 x3^3
which means that largest factor would be 5^6
<MASK>
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 06:54
2
KUDOS
$$75^3 = 75*75*75=(15*5)*(15*5)*(15*5)=(3*5*5)*(3*5*5)*(3*5*5)=3^3*5^6=27*5^6$$
if $$27*5^6$$ is a multiple of $$5^m$$, then $$\frac{27*5^6}{5^m}$$ should result in an integer. Also, remember that m is an integer.
Let us look at the answer options.
A. if m = 3, then the expression becomes $$\frac{27*5^6}{5^3}$$, which leaves $$27*5^3$$, an integer.
B. if m = 4, then the expression becomes $$\frac{27*5^6}{5^4}$$, which leaves $$27*5^2$$, an integer.
C. if m = 5, then the expression becomes $$\frac{27*5^6}{5^5}$$, which leaves $$27*5^1$$, an integer.
D. if m = 6, then the expression becomes $$\frac{27*5^6}{5^6}$$, which leaves $$27*5^0$$, an integer.
E. if m = 7, then the expression becomes $$\frac{27*5^6}{5^7}$$, which leaves $$\frac{27}{5}$$, NOT an integer.
So, out of all the 4 choices that leave us with an integer result, the largest value of m is 6. Hence correct answer is D.
_________________
Manager
Joined: 16 Feb 2010
Posts: 188
Followers: 2
<MASK>
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 08:30
great explanation agreed wid hideyoshi oa shud be d
Manager
Joined: 16 Mar 2010
Posts: 184
Followers: 3
Kudos [?]: 176 [0], given: 9
Re: largest possible values [#permalink]
<MASK>
16 May 2010, 08:22
Hi,
I think the simplest method is...
Find out the highest factor of 5s in 75^3. i.e., 5^6 so, m=6. Because that will be the HCF.
Re: largest possible values [#permalink] 16 May 2010, 08:22
Similar topics Replies Last post
Similar
Topics:
3 If 20! × 20!/20^n is an integer, what is the largest possible value of 3 28 Apr 2016, 16:00
8 If 200!/10^n is an integer, what is the largest possible value of n? 7 28 Apr 2016, 02:36
3 What is the largest possible value of c if 5c + (d-12)^2 = 235? 6 24 Jul 2015, 04:11
10 What is the largest possible value of the following expression 4 16 Oct 2014, 22:21
largest possible value of actual area of rectangle ? 1 23 Jan 2011, 02:16
Display posts from previous: Sort by
# largest possible values
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
<MASK>
<UNMASK>
largest possible values : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack
It is currently 23 Jan 2017, 23:54
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# largest possible values
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
### Hide Tags
Intern
Joined: 21 Feb 2010
Posts: 33
Location: Ukraine
Followers: 1
Kudos [?]: 3 [1] , given: 9
largest possible values [#permalink]
### Show Tags
<MASK>
(N/A)
Question Stats:
<MASK>
### HideShow timer Statistics
If m is a positive integer and 75^3 is a multiple of 5^m , what is the largest possible values for m?
3
4
5
6
7
Can somebody explain how to solve it?
Intern
Joined: 11 Jan 2007
Posts: 45
Location: United States
Concentration: Marketing, Healthcare
GMAT 1: 600 Q49 V25
GMAT 2: 650 Q49 V32
GPA: 3.5
WE: Pharmaceuticals (Consulting)
Followers: 0
Kudos [?]: 30 [0], given: 4
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 06:17
IMO m=6. here is why
75^3=(25X3)^3=(5^2 X 3)^3 = 5^6 x3^3
which means that largest factor would be 5^6
What is the OA ?
Affiliations: Scrum Alliance
Joined: 09 Feb 2010
Posts: 84
Location: United States (MI)
Concentration: Strategy, General Management
GMAT 1: 600 Q48 V25
GMAT 2: 710 Q48 V38
WE: Information Technology (Retail)
Followers: 1
Kudos [?]: 36 [2] , given: 18
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 06:54
2
KUDOS
$$75^3 = 75*75*75=(15*5)*(15*5)*(15*5)=(3*5*5)*(3*5*5)*(3*5*5)=3^3*5^6=27*5^6$$
if $$27*5^6$$ is a multiple of $$5^m$$, then $$\frac{27*5^6}{5^m}$$ should result in an integer. Also, remember that m is an integer.
Let us look at the answer options.
A. if m = 3, then the expression becomes $$\frac{27*5^6}{5^3}$$, which leaves $$27*5^3$$, an integer.
B. if m = 4, then the expression becomes $$\frac{27*5^6}{5^4}$$, which leaves $$27*5^2$$, an integer.
C. if m = 5, then the expression becomes $$\frac{27*5^6}{5^5}$$, which leaves $$27*5^1$$, an integer.
D. if m = 6, then the expression becomes $$\frac{27*5^6}{5^6}$$, which leaves $$27*5^0$$, an integer.
E. if m = 7, then the expression becomes $$\frac{27*5^6}{5^7}$$, which leaves $$\frac{27}{5}$$, NOT an integer.
So, out of all the 4 choices that leave us with an integer result, the largest value of m is 6. Hence correct answer is D.
_________________
Manager
Joined: 16 Feb 2010
Posts: 188
Followers: 2
Kudos [?]: 29 [0], given: 14
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 08:30
great explanation agreed wid hideyoshi oa shud be d
Manager
Joined: 16 Mar 2010
Posts: 184
Followers: 3
Kudos [?]: 176 [0], given: 9
Re: largest possible values [#permalink]
### Show Tags
16 May 2010, 08:22
Hi,
I think the simplest method is...
Find out the highest factor of 5s in 75^3. i.e., 5^6 so, m=6. Because that will be the HCF.
Re: largest possible values [#permalink] 16 May 2010, 08:22
Similar topics Replies Last post
Similar
Topics:
3 If 20! × 20!/20^n is an integer, what is the largest possible value of 3 28 Apr 2016, 16:00
8 If 200!/10^n is an integer, what is the largest possible value of n? 7 28 Apr 2016, 02:36
3 What is the largest possible value of c if 5c + (d-12)^2 = 235? 6 24 Jul 2015, 04:11
10 What is the largest possible value of the following expression 4 16 Oct 2014, 22:21
largest possible value of actual area of rectangle ? 1 23 Jan 2011, 02:16
Display posts from previous: Sort by
# largest possible values
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.
<UNMASK>
largest possible values : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack
It is currently 23 Jan 2017, 23:54
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# largest possible values
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
### Hide Tags
Intern
Joined: 21 Feb 2010
Posts: 33
Location: Ukraine
Followers: 1
Kudos [?]: 3 [1] , given: 9
largest possible values [#permalink]
### Show Tags
14 May 2010, 03:56
1
KUDOS
00:00
Difficulty:
(N/A)
Question Stats:
100% (01:38) correct 0% (00:00) wrong based on 5 sessions
### HideShow timer Statistics
If m is a positive integer and 75^3 is a multiple of 5^m , what is the largest possible values for m?
3
4
5
6
7
Can somebody explain how to solve it?
Intern
Joined: 11 Jan 2007
Posts: 45
Location: United States
Concentration: Marketing, Healthcare
GMAT 1: 600 Q49 V25
GMAT 2: 650 Q49 V32
GPA: 3.5
WE: Pharmaceuticals (Consulting)
Followers: 0
Kudos [?]: 30 [0], given: 4
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 06:17
IMO m=6. here is why
75^3=(25X3)^3=(5^2 X 3)^3 = 5^6 x3^3
which means that largest factor would be 5^6
What is the OA ?
Affiliations: Scrum Alliance
Joined: 09 Feb 2010
Posts: 84
Location: United States (MI)
Concentration: Strategy, General Management
GMAT 1: 600 Q48 V25
GMAT 2: 710 Q48 V38
WE: Information Technology (Retail)
Followers: 1
Kudos [?]: 36 [2] , given: 18
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 06:54
2
KUDOS
$$75^3 = 75*75*75=(15*5)*(15*5)*(15*5)=(3*5*5)*(3*5*5)*(3*5*5)=3^3*5^6=27*5^6$$
if $$27*5^6$$ is a multiple of $$5^m$$, then $$\frac{27*5^6}{5^m}$$ should result in an integer. Also, remember that m is an integer.
Let us look at the answer options.
A. if m = 3, then the expression becomes $$\frac{27*5^6}{5^3}$$, which leaves $$27*5^3$$, an integer.
B. if m = 4, then the expression becomes $$\frac{27*5^6}{5^4}$$, which leaves $$27*5^2$$, an integer.
C. if m = 5, then the expression becomes $$\frac{27*5^6}{5^5}$$, which leaves $$27*5^1$$, an integer.
D. if m = 6, then the expression becomes $$\frac{27*5^6}{5^6}$$, which leaves $$27*5^0$$, an integer.
E. if m = 7, then the expression becomes $$\frac{27*5^6}{5^7}$$, which leaves $$\frac{27}{5}$$, NOT an integer.
So, out of all the 4 choices that leave us with an integer result, the largest value of m is 6. Hence correct answer is D.
_________________
Manager
Joined: 16 Feb 2010
Posts: 188
Followers: 2
Kudos [?]: 29 [0], given: 14
Re: largest possible values [#permalink]
### Show Tags
14 May 2010, 08:30
great explanation agreed wid hideyoshi oa shud be d
Manager
Joined: 16 Mar 2010
Posts: 184
Followers: 3
Kudos [?]: 176 [0], given: 9
Re: largest possible values [#permalink]
### Show Tags
16 May 2010, 08:22
Hi,
I think the simplest method is...
Find out the highest factor of 5s in 75^3. i.e., 5^6 so, m=6. Because that will be the HCF.
Re: largest possible values [#permalink] 16 May 2010, 08:22
Similar topics Replies Last post
Similar
Topics:
3 If 20! × 20!/20^n is an integer, what is the largest possible value of 3 28 Apr 2016, 16:00
8 If 200!/10^n is an integer, what is the largest possible value of n? 7 28 Apr 2016, 02:36
3 What is the largest possible value of c if 5c + (d-12)^2 = 235? 6 24 Jul 2015, 04:11
10 What is the largest possible value of the following expression 4 16 Oct 2014, 22:21
largest possible value of actual area of rectangle ? 1 23 Jan 2011, 02:16
Display posts from previous: Sort by
# largest possible values
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.
| 4,625 | 6,699,449 |
1. ## differentiating implicitly....little help
i understand the main concept of differentiating implicitly, however i cant seem to get this problem,
sin(xy)=6x+7
mathaction
2. Originally Posted by mathaction
i understand the main concept of differentiating implicitly, however i cant seem to get this problem,
sin(xy)=6x+7
mathaction
you need the chain rule (coupled with the product rule) for the LHS. here goes:
$\displaystyle \sin (xy) = 6x + 7$
differentiating implicitly with respect to x, we get:
$\displaystyle \cos (xy) \cdot (y + x~y') = 6$
now solve for $\displaystyle y'$
3. Originally Posted by Jhevon
you need the chain rule (coupled with the product rule) for the LHS. here goes:
$\displaystyle \sin (xy) = 6x + 7$
differentiating implicitly with respect to x, we get:
$\displaystyle \cos (xy) \cdot (y + x~y') = 6$
now solve for $\displaystyle y'$
yeah when i was doing it for some reason i was putting dy/dx on the cos(xy), but this helped, i came up with...
(6-ycos(xy))/(xcos(xy))
4. Originally Posted by mathaction
yeah when i was doing it for some reason i was putting dy/dx on the cos(xy), but this helped, i came up with...
(6-ycos(xy))/(xcos(xy))
that's correct
5. i have just one more, about implicity differentiation...for this problem...
x^3 + 5x^2y + 2y^2 = 4y + 11
the last answer i came up with was...
(-3x^2-10xy)/(5x^2+4y-1)
then it wanted the slope of the tangent line at (1,2), but when i plugged these in the answer was wrong, and im sure thats from a faulty differentiation, can you please let me know where i messed up...thanks
6. Originally Posted by mathaction
i have just one more, about implicity differentiation...for this problem...
x^3 + 5x^2y + 2y^2 = 4y + 11
the last answer i came up with was...
(-3x^2-10xy)/(5x^2+4y-1)
then it wanted the slope of the tangent line at (1,2), but when i plugged these in the answer was wrong, and im sure thats from a faulty differentiation, can you please let me know where i messed up...thanks
the 1 should be a 4. it comes from the derivative of 4y
7. Originally Posted by Jhevon
the 1 should be a 4. it comes from the derivative of 4y
ok, so then the rest of it is good then...
8. Originally Posted by mathaction
ok, so then the rest of it is good then...
yes. change that 1 to a 4 and you should be fine
9. Originally Posted by Jhevon
yes. change that 1 to a 4 and you should be fine
yeah i had it that way before, and just didnt do the basic algebra right, just gotta be careful for the really easy stuff to overlook...thanks alot
| 727 | 6,699,450 |
## Conversion formula
The conversion factor from pounds to kilograms is 0.45359237, which means that 1 pound is equal to 0.45359237 kilograms:
1 lb = 0.45359237 kg
To convert 190.8 pounds into kilograms we have to multiply 190.8 by the conversion factor in order to get the mass amount from pounds to kilograms. We can also form a simple proportion to calculate the result:
1 lb → 0.45359237 kg
190.8 lb → M(kg)
Solve the above proportion to obtain the mass M in kilograms:
M(kg) = 190.8 lb × 0.45359237 kg
M(kg) = 86.545424196 kg
The final result is:
190.8 lb → 86.545424196 kg
We conclude that 190.8 pounds is equivalent to 86.545424196 kilograms:
190.8 pounds = 86.545424196 kilograms
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 kilogram is equal to 0.011554625900675 × 190.8 pounds.
Another way is saying that 190.8 pounds is equal to 1 ÷ 0.011554625900675 kilograms.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that one hundred ninety point eight pounds is approximately eighty-six point five four five kilograms:
190.8 lb ≅ 86.545 kg
An alternative is also that one kilogram is approximately zero point zero one two times one hundred ninety point eight pounds.
## Conversion table
### pounds to kilograms chart
For quick reference purposes, below is the conversion table you can use to convert from pounds to kilograms
pounds (lb) kilograms (kg)
191.8 pounds 86.999 kilograms
192.8 pounds 87.453 kilograms
193.8 pounds 87.906 kilograms
194.8 pounds 88.36 kilograms
195.8 pounds 88.813 kilograms
196.8 pounds 89.267 kilograms
197.8 pounds 89.721 kilograms
198.8 pounds 90.174 kilograms
199.8 pounds 90.628 kilograms
200.8 pounds 91.081 kilograms
| 491 | 6,699,451 |
# Every ordered field that has the least upper bound property is isomorphic to the real number system.
Okay, so here's a theorem from Rudin:
"Every ordered field that has the least upper bound property is isomorphic to the real number system."
Here's a definition:
"Ordered fields are isomorphic if there is a bijection between the underlying sets that preserves the field operations and the orders."
i.e.
Ordered fields (F, +, ·, ≤) and (K, ⊕, ⊙, ≼) are isomorphic if there exists a bijection h : F → K such that (a) for all x,y ∈ F, h(x+y)=h(x)⊕h(y), (b) for all x,y ∈ F, h(x·y)=h(x)⊙h(y), and (c) for all x,y ∈ F, if x≤y,then h(x)≼h(y).
Here are some things that I think are true:
1.) The field T with two elements {0, 1} is an ordered field that has the least upper bound property.
Because T is finite, every nonempty subset of T has a maximum. Max(T) = LUB(T) for all subsets of T that have a maximum element. Therefore all subsets of T have a least upper bound in T. Thus T has the least upper bound property.
2.) Since T has the LUB property, it is isomorphic to the real # system and therefore there exists a bijection between the real numbers and {0, 1}.
3.) ...but since {0, 1} is finite and the real numbers are infinite, there can be no surjection from {0,1} -> real numbers. Therefore there's no bijection between the reals and {0, 1}.
So... Which thing that I think is true is not actually true?
Thanks!
• The field with two elements is not an ordered field; $1 + 1$ is $0$ in that field, but it would have to be greater than $0$ in an ordered field. Every ordered field has characteristic $0$, actually. See en.wikipedia.org/wiki/Ordered_field – Carl Mummert Aug 30 '15 at 2:48
• Ah! Great! Thanks! – boxTurtle Aug 30 '15 at 2:51
• Ah hurray! And my next question was going to be "What about the integers then?" But that's not an ordered field either, because that's not a field! – boxTurtle Aug 30 '15 at 2:57
| 546 | 6,699,452 |
<MASK>
One way to think of a function is as a "black box" -- that is, a box that does some kind of operation on its input, and produces a number as a result. For example, suppose f is a function. When it accepts 2 as input, it spits out 7 as output. When 3 goes in, 12 comes out. When 4 goes in, 19 comes out. We could draw a picture that looks like this to illustrate f:
<MASK>
Do you recognize this function, f, as the same one we have used in previous examples? In each case, the function works by squaring its input value, and then adding three. The diagram, above, shows what happens to each of three input values. Algebraically, we would write it this way:
f(2) = 7
f(3) = 12
f(4) = 19
<MASK>
### Inverse
<MASK>
f(x) = x2+3
j(x) = x-2
<MASK>
Using set-builder notation {a : a has a certain property} means the set of all elements, a, such that a has a certain property,
<MASK>
Functions can be defined by closed formula, by cases, recursion, or general axiom(s).
The notation f: A —> B means domain(f) = A and range(f) is a subset of B
<MASK>
(f o g)(x) = f(g(x))
<MASK>
The following laws hold, at least where the left hand side of the equation is defined.
f~~ = f
IA~ = IA
f o IA = f
IA o f = f
(f o g)~ = g~ o f~
(f o g) o h = f o (g o h)
<MASK>
Mathworld: Periodic Function defines and illustrates periodic functions
<MASK>
<UNMASK>
<MASK>
### Function Definition
A function definition -- the definition of the function named "f" -- looks like this:
<MASK>
f(x) = x2+3
<MASK>
Perhaps the question looks meaningless to you, so bear with me for a minute. The point I'm making is that the definition of the function has nothing to do with any variable, x, that may have been discussed prior to (or even subsequent to) the definition of the function. The meaning of x exists only within the definition itself, and serves to define the function, f. The answer to the question is that the definition defines f.
<MASK>
The first one defines a function, f, and the second one defines a function, g. Are these functions different?
<MASK>
One way to think of a function is as a "black box" -- that is, a box that does some kind of operation on its input, and produces a number as a result. For example, suppose f is a function. When it accepts 2 as input, it spits out 7 as output. When 3 goes in, 12 comes out. When 4 goes in, 19 comes out. We could draw a picture that looks like this to illustrate f:
<MASK>
Do you recognize this function, f, as the same one we have used in previous examples? In each case, the function works by squaring its input value, and then adding three. The diagram, above, shows what happens to each of three input values. Algebraically, we would write it this way:
f(2) = 7
f(3) = 12
f(4) = 19
<MASK>
We've explored three different ways to depict a function: with a formal definition using a parameter, as a "black box" that accepts a number as input and produces a number as output, and as an algorithm. Yet another way to depict a function is by plotting its input and output on the x-axis and y-axis, respectively.
<MASK>
### Inverse
If we view a function as an algorithm, can we make it go backwards? That is, can we start with the result, and "undo" each of the steps, giving us back the original input? Let's define h is the inverse of f. Whereas f squares the input and then adds three, h subtracts three, and then finds the square root of the input. Notice that each step is undone, and the order is the reverse order. So if we start with 28 and run it through the function, h, we subtract 3 giving 25, then find the square root, 5. So h(28) = 5. Here it is in a chart:
Function h is the inverse of function f f squares its input, then adds 3 h subtracts 3, then finds the square root f(x) = x²2+3 h(x) = sqrt(x-3) f(2) = 7 h(7) = 2 f(3) = 12 h(12) = 3 f(4) = 19 h(19) = 4
<MASK>
h(x) = sqrt(x-3)
### Injective, Surjective, etc.
<MASK>
injective, or one-to-one, if each element of B is mapped from at most one element of A. If f is injective, then f(x)=f(y) implies x=y.
<MASK>
Can you write the black box diagrams of f(2), f(3), and f(4)? I'll repeat them here:
<MASK>
Now, can you write the black box diagrams of j(7), j(12), and j(19)? Here they are:
<MASK>
Now what if these black boxes were put end-to-end so the output of f were fed directly into j? It would look like this:
2 —> [ f ] —> 7 —> [ j ] —> 5
3 —> [ f ] —> 12 —> [ j ] —> 10
4 —> [ f ] —> 19 —> [ j ] —> 17
Now what if we bolted f and j together so we couldn't see anything in between them? It would look like this:
<MASK>
### Simplifying a composition of functions
<MASK>
f(x) = x2+3
j(x) = x-2
Putting it all together,
j(f(x)) = f(x)-2, from the definition of j
= (x2+3)-2, from the definition of f
= x2+1, by algebraic simplification
### Domain, Range, and Mapping
So far we've looked at functions in four ways -- by its formal definition, as a black box, as an algorithm, and as a graph. Yet another way to look at a function is as a mapping from one set to another. The set of possible input values is called the "domain" of the function. The set of possible output values is called the "range" of the function. The function maps each element of the domain to one element of the range.
Using set-builder notation {a : a has a certain property} means the set of all elements, a, such that a has a certain property,
The domain of the function, f, is domain(f) = {x : f(x) is defined}
<MASK>
Functions can be defined by closed formula, by cases, recursion, or general axiom(s).
The notation f: A —> B means domain(f) = A and range(f) is a subset of B
Composition of functions, f o g, is defined whenever domain(f) is a subset of range(g), by the equation
(f o g)(x) = f(g(x))
Composition is associative but not commutative.
Two functions, f and g, are inverses when
<MASK>
The inverse function to f is written f~ or f-1.
The identity function on a set A, IA, is defined
<MASK>
The following laws hold, at least where the left hand side of the equation is defined.
f~~ = f
IA~ = IA
f o IA = f
IA o f = f
(f o g)~ = g~ o f~
(f o g) o h = f o (g o h)
### Periodic Functions
<MASK>
a sin(b x)
Here, the period of the function (measured horizontally from crest to crest of the wave) is 2π/b, and the amplitude, measured vertically from the midline to the crest, is a.
Students sometimes have difficulty visualizing why the function containing bx (with x multiplied by b) results in a graph in which the labels of the x values are divided by b. The reason for this is that the argument (input) to the function is 2π when the x value is 2π/b. In other words, we have to divide the x value by b to counteract the fact that it's multiplied by b later on.
<MASK>
### Internet references
Visual Calculus: Composition of Functions, Domain and Range
<MASK>
Mathworld: Domain and Range defines the concepts as they are ordinarily used, and then proceeds to catalog all the other less common definitions -- just read the first paragraph or so of each of these pages.
Mathworld: Periodic Function defines and illustrates periodic functions
Mathwords: Periodic Function, Period
<MASK>
Set Description Notation
<MASK>
<UNMASK>
<MASK>
### Function Definition
A function definition -- the definition of the function named "f" -- looks like this:
<MASK>
f(x) = x2+3
<MASK>
Perhaps the question looks meaningless to you, so bear with me for a minute. The point I'm making is that the definition of the function has nothing to do with any variable, x, that may have been discussed prior to (or even subsequent to) the definition of the function. The meaning of x exists only within the definition itself, and serves to define the function, f. The answer to the question is that the definition defines f.
<MASK>
The first one defines a function, f, and the second one defines a function, g. Are these functions different?
<MASK>
One way to think of a function is as a "black box" -- that is, a box that does some kind of operation on its input, and produces a number as a result. For example, suppose f is a function. When it accepts 2 as input, it spits out 7 as output. When 3 goes in, 12 comes out. When 4 goes in, 19 comes out. We could draw a picture that looks like this to illustrate f:
<MASK>
Do you recognize this function, f, as the same one we have used in previous examples? In each case, the function works by squaring its input value, and then adding three. The diagram, above, shows what happens to each of three input values. Algebraically, we would write it this way:
f(2) = 7
f(3) = 12
f(4) = 19
<MASK>
We've explored three different ways to depict a function: with a formal definition using a parameter, as a "black box" that accepts a number as input and produces a number as output, and as an algorithm. Yet another way to depict a function is by plotting its input and output on the x-axis and y-axis, respectively.
<MASK>
### Inverse
If we view a function as an algorithm, can we make it go backwards? That is, can we start with the result, and "undo" each of the steps, giving us back the original input? Let's define h is the inverse of f. Whereas f squares the input and then adds three, h subtracts three, and then finds the square root of the input. Notice that each step is undone, and the order is the reverse order. So if we start with 28 and run it through the function, h, we subtract 3 giving 25, then find the square root, 5. So h(28) = 5. Here it is in a chart:
Function h is the inverse of function f f squares its input, then adds 3 h subtracts 3, then finds the square root f(x) = x²2+3 h(x) = sqrt(x-3) f(2) = 7 h(7) = 2 f(3) = 12 h(12) = 3 f(4) = 19 h(19) = 4
<MASK>
h(x) = sqrt(x-3)
### Injective, Surjective, etc.
<MASK>
injective, or one-to-one, if each element of B is mapped from at most one element of A. If f is injective, then f(x)=f(y) implies x=y.
<MASK>
Can you write the black box diagrams of f(2), f(3), and f(4)? I'll repeat them here:
<MASK>
Now, can you write the black box diagrams of j(7), j(12), and j(19)? Here they are:
<MASK>
Now what if these black boxes were put end-to-end so the output of f were fed directly into j? It would look like this:
2 —> [ f ] —> 7 —> [ j ] —> 5
3 —> [ f ] —> 12 —> [ j ] —> 10
4 —> [ f ] —> 19 —> [ j ] —> 17
Now what if we bolted f and j together so we couldn't see anything in between them? It would look like this:
<MASK>
### Simplifying a composition of functions
<MASK>
f(x) = x2+3
j(x) = x-2
Putting it all together,
j(f(x)) = f(x)-2, from the definition of j
= (x2+3)-2, from the definition of f
= x2+1, by algebraic simplification
### Domain, Range, and Mapping
So far we've looked at functions in four ways -- by its formal definition, as a black box, as an algorithm, and as a graph. Yet another way to look at a function is as a mapping from one set to another. The set of possible input values is called the "domain" of the function. The set of possible output values is called the "range" of the function. The function maps each element of the domain to one element of the range.
Using set-builder notation {a : a has a certain property} means the set of all elements, a, such that a has a certain property,
The domain of the function, f, is domain(f) = {x : f(x) is defined}
<MASK>
Functions can be defined by closed formula, by cases, recursion, or general axiom(s).
The notation f: A —> B means domain(f) = A and range(f) is a subset of B
Composition of functions, f o g, is defined whenever domain(f) is a subset of range(g), by the equation
(f o g)(x) = f(g(x))
Composition is associative but not commutative.
Two functions, f and g, are inverses when
<MASK>
The inverse function to f is written f~ or f-1.
The identity function on a set A, IA, is defined
<MASK>
The following laws hold, at least where the left hand side of the equation is defined.
f~~ = f
IA~ = IA
f o IA = f
IA o f = f
(f o g)~ = g~ o f~
(f o g) o h = f o (g o h)
### Periodic Functions
<MASK>
a sin(b x)
Here, the period of the function (measured horizontally from crest to crest of the wave) is 2π/b, and the amplitude, measured vertically from the midline to the crest, is a.
Students sometimes have difficulty visualizing why the function containing bx (with x multiplied by b) results in a graph in which the labels of the x values are divided by b. The reason for this is that the argument (input) to the function is 2π when the x value is 2π/b. In other words, we have to divide the x value by b to counteract the fact that it's multiplied by b later on.
<MASK>
### Internet references
Visual Calculus: Composition of Functions, Domain and Range
<MASK>
Mathworld: Domain and Range defines the concepts as they are ordinarily used, and then proceeds to catalog all the other less common definitions -- just read the first paragraph or so of each of these pages.
Mathworld: Periodic Function defines and illustrates periodic functions
Mathwords: Periodic Function, Period
<MASK>
Set Description Notation
The webmaster and author of this Math Help site is Graeme McRae.
<UNMASK>
<MASK>
### Function Definition
A function definition -- the definition of the function named "f" -- looks like this:
<MASK>
f(x) = x2+3
<MASK>
f(x) = x2+3
<MASK>
Perhaps the question looks meaningless to you, so bear with me for a minute. The point I'm making is that the definition of the function has nothing to do with any variable, x, that may have been discussed prior to (or even subsequent to) the definition of the function. The meaning of x exists only within the definition itself, and serves to define the function, f. The answer to the question is that the definition defines f.
<MASK>
The first one defines a function, f, and the second one defines a function, g. Are these functions different?
<MASK>
### A function as a black box
One way to think of a function is as a "black box" -- that is, a box that does some kind of operation on its input, and produces a number as a result. For example, suppose f is a function. When it accepts 2 as input, it spits out 7 as output. When 3 goes in, 12 comes out. When 4 goes in, 19 comes out. We could draw a picture that looks like this to illustrate f:
<MASK>
Do you recognize this function, f, as the same one we have used in previous examples? In each case, the function works by squaring its input value, and then adding three. The diagram, above, shows what happens to each of three input values. Algebraically, we would write it this way:
f(2) = 7
f(3) = 12
f(4) = 19
These are two equivalent ways to represent what the function, f, does to each of three input values.
<MASK>
Another way to think of a function is as an "algorithm" -- a series of steps that are performed in a particular order to get a result that depends on the input. If f is a function that squares its input and then adds three, this is an algorithm that tells you what f would do to any input. What would f(5) be? Well, start with 5, then square it. That gives you 25. Then add three. The result is 28. So f(5) = 28.
<MASK>
We've explored three different ways to depict a function: with a formal definition using a parameter, as a "black box" that accepts a number as input and produces a number as output, and as an algorithm. Yet another way to depict a function is by plotting its input and output on the x-axis and y-axis, respectively.
<MASK>
### Inverse
If we view a function as an algorithm, can we make it go backwards? That is, can we start with the result, and "undo" each of the steps, giving us back the original input? Let's define h is the inverse of f. Whereas f squares the input and then adds three, h subtracts three, and then finds the square root of the input. Notice that each step is undone, and the order is the reverse order. So if we start with 28 and run it through the function, h, we subtract 3 giving 25, then find the square root, 5. So h(28) = 5. Here it is in a chart:
Function h is the inverse of function f f squares its input, then adds 3 h subtracts 3, then finds the square root f(x) = x²2+3 h(x) = sqrt(x-3) f(2) = 7 h(7) = 2 f(3) = 12 h(12) = 3 f(4) = 19 h(19) = 4
<MASK>
h(x) = sqrt(x-3)
### Injective, Surjective, etc.
A function from set A (the domain) to set B (the range, or codomain) is...
injective, or one-to-one, if each element of B is mapped from at most one element of A. If f is injective, then f(x)=f(y) implies x=y.
<MASK>
Can you write the black box diagrams of f(2), f(3), and f(4)? I'll repeat them here:
<MASK>
Now, can you write the black box diagrams of j(7), j(12), and j(19)? Here they are:
<MASK>
Now what if these black boxes were put end-to-end so the output of f were fed directly into j? It would look like this:
2 —> [ f ] —> 7 —> [ j ] —> 5
3 —> [ f ] —> 12 —> [ j ] —> 10
4 —> [ f ] —> 19 —> [ j ] —> 17
Now what if we bolted f and j together so we couldn't see anything in between them? It would look like this:
<MASK>
This is what we mean by the "composition" of j and f -- the output of f goes directly into j as if f and j were one function. We write it this way:
<MASK>
### Simplifying a composition of functions
<MASK>
Recall our definitions of f and j:
f(x) = x2+3
j(x) = x-2
Putting it all together,
j(f(x)) = f(x)-2, from the definition of j
= (x2+3)-2, from the definition of f
= x2+1, by algebraic simplification
### Domain, Range, and Mapping
So far we've looked at functions in four ways -- by its formal definition, as a black box, as an algorithm, and as a graph. Yet another way to look at a function is as a mapping from one set to another. The set of possible input values is called the "domain" of the function. The set of possible output values is called the "range" of the function. The function maps each element of the domain to one element of the range.
Using set-builder notation {a : a has a certain property} means the set of all elements, a, such that a has a certain property,
The domain of the function, f, is domain(f) = {x : f(x) is defined}
<MASK>
Functions can be defined by closed formula, by cases, recursion, or general axiom(s).
The notation f: A —> B means domain(f) = A and range(f) is a subset of B
Composition of functions, f o g, is defined whenever domain(f) is a subset of range(g), by the equation
(f o g)(x) = f(g(x))
Composition is associative but not commutative.
Two functions, f and g, are inverses when
f(g(x)) = x for any x in domain(g), and
g(f(y)) = y for any y in domain(f)
<MASK>
The inverse function to f is written f~ or f-1.
The identity function on a set A, IA, is defined
<MASK>
The following laws hold, at least where the left hand side of the equation is defined.
f~~ = f
IA~ = IA
f o IA = f
IA o f = f
(f o g)~ = g~ o f~
(f o g) o h = f o (g o h)
### Periodic Functions
<MASK>
a sin(b x)
Here, the period of the function (measured horizontally from crest to crest of the wave) is 2π/b, and the amplitude, measured vertically from the midline to the crest, is a.
Students sometimes have difficulty visualizing why the function containing bx (with x multiplied by b) results in a graph in which the labels of the x values are divided by b. The reason for this is that the argument (input) to the function is 2π when the x value is 2π/b. In other words, we have to divide the x value by b to counteract the fact that it's multiplied by b later on.
<MASK>
### Internet references
Visual Calculus: Composition of Functions, Domain and Range
Translations of Sine and Cosine Curves explains: If the graphs of y = A Sin Bx and y = A Cos Bx are translated h units horizontally and k units vertically, then the resulting graphs have the equations: y-k=A Sin B(x-h) and y-k=A Cos B(x-h)
Mathworld: Domain and Range defines the concepts as they are ordinarily used, and then proceeds to catalog all the other less common definitions -- just read the first paragraph or so of each of these pages.
Mathworld: Periodic Function defines and illustrates periodic functions
Mathwords: Periodic Function, Period
The Math Page: Graphs of the Trig. Functions explains how to calculate the period of trig functions.
<MASK>
### Related pages in this website
Set Description Notation
The webmaster and author of this Math Help site is Graeme McRae.
<UNMASK>
# document.write (document.title)
Math Help > Basic Math > Functions
In the study of algebra, students are first exposed to the use of variables, such as x, to represent quantities that are unknown or that might change over time. This idea comes naturally to some students, but for others, a great deal of effort is expended to understand the concept.
The next major hurdle in algebra comes with the introduction of functions. Just as a variable represents a number, a function represents an expression or operation that is performed on a number. This page introduces the concept of the function, giving a variety of ways for the student to think about functions.
### Function Definition
A function definition -- the definition of the function named "f" -- looks like this:
f(x) = expression,
where expression is an algebraic expression that has some value, usually depending on x. Students sometimes question the meaning of "x" in the function definition, because they rightly discern that its use is different from the use of x they're familiar with. Here, the variable "x" is called a parameter of the function, and it has meaning only within the function definition itself. To shed light on the meaning of a function definition and the special role of the parameter, let's start with an example:
### Example
Here is a function definition:
f(x) = x2+3
This means that the function has the effect of squaring its parameter, and then adding three. Since x is a parameter, it is intended to show that f would yield the same result for anything used in place of x. For example, f(4) = 42+3, so f(4) is 19.
Refer to this definition:
f(x) = x2+3
Does this define f(x)? Or does it define f?
Perhaps the question looks meaningless to you, so bear with me for a minute. The point I'm making is that the definition of the function has nothing to do with any variable, x, that may have been discussed prior to (or even subsequent to) the definition of the function. The meaning of x exists only within the definition itself, and serves to define the function, f. The answer to the question is that the definition defines f.
Now, refer to these two definitions:
f(x) = x2+3
g(y) = y2+3
The first one defines a function, f, and the second one defines a function, g. Are these functions different?
At first, you might be tempted to say that f depends on x, and g depends on y, so they are different. But if you said that, you would be wrong, because f doesn't depend on x at all. Rather, the parameter, x, is used simply to define f. And a different parameter, y, is used to define g. The choice of parameter is arbitrary, and doesn't figure into the definition of the function at all. The answer: these two functions, f and g, are the same.
### A function as a black box
One way to think of a function is as a "black box" -- that is, a box that does some kind of operation on its input, and produces a number as a result. For example, suppose f is a function. When it accepts 2 as input, it spits out 7 as output. When 3 goes in, 12 comes out. When 4 goes in, 19 comes out. We could draw a picture that looks like this to illustrate f:
2 —> [ f ] —> 7
3 —> [ f ] —> 12
4 —> [ f ] —> 19
Do you recognize this function, f, as the same one we have used in previous examples? In each case, the function works by squaring its input value, and then adding three. The diagram, above, shows what happens to each of three input values. Algebraically, we would write it this way:
f(2) = 7
f(3) = 12
f(4) = 19
These are two equivalent ways to represent what the function, f, does to each of three input values.
### A function as an algorithm
Another way to think of a function is as an "algorithm" -- a series of steps that are performed in a particular order to get a result that depends on the input. If f is a function that squares its input and then adds three, this is an algorithm that tells you what f would do to any input. What would f(5) be? Well, start with 5, then square it. That gives you 25. Then add three. The result is 28. So f(5) = 28.
### Plotting a function
We've explored three different ways to depict a function: with a formal definition using a parameter, as a "black box" that accepts a number as input and produces a number as output, and as an algorithm. Yet another way to depict a function is by plotting its input and output on the x-axis and y-axis, respectively.
f(x) = x2+3
You may recognize this as the graph of y=x2+3. This is to be expected, because it is customary to plot the value (the output) of the function on the y axis, and the parameter (the input) of the function on the x axis. This is probably the reason that "x" is used so frequently as the parameter in formal definitions of functions, although, as we have seen, we don't have to use x for this purpose.
### Inverse
If we view a function as an algorithm, can we make it go backwards? That is, can we start with the result, and "undo" each of the steps, giving us back the original input? Let's define h is the inverse of f. Whereas f squares the input and then adds three, h subtracts three, and then finds the square root of the input. Notice that each step is undone, and the order is the reverse order. So if we start with 28 and run it through the function, h, we subtract 3 giving 25, then find the square root, 5. So h(28) = 5. Here it is in a chart:
Function h is the inverse of function f f squares its input, then adds 3 h subtracts 3, then finds the square root f(x) = x²2+3 h(x) = sqrt(x-3) f(2) = 7 h(7) = 2 f(3) = 12 h(12) = 3 f(4) = 19 h(19) = 4
One way to plot the inverse of a function is to simply relabel the axes. Look at the plot, above. What was called "x" can now be called "h(x)", and what was called "f(x)" can now be called "x". You see, by relabeling the axes we simply interchange the input and the output of the function. But then the axis labeled "x" is the vertical axis, which can be confusing, because we normally think of that as the "y" axis. So we complete the process of plotting the inverse by interchanging the axes, like this:
h(x) = sqrt(x-3)
### Injective, Surjective, etc.
A function from set A (the domain) to set B (the range, or codomain) is...
injective, or one-to-one, if each element of B is mapped from at most one element of A. If f is injective, then f(x)=f(y) implies x=y.
surjective, or onto, if each element of B is mapped from at least one element of A.
bijective (both one-to-one and onto) if each element of B is mapped from exactly one element of A.
### Composition of functions
Suppose we define functions f and j as follows:
f(x) = x2+3
j(x) = x-2
Can you write the black box diagrams of f(2), f(3), and f(4)? I'll repeat them here:
2 —> [ f ] —> 7
3 —> [ f ] —> 12
4 —> [ f ] —> 19
Now, can you write the black box diagrams of j(7), j(12), and j(19)? Here they are:
7 —> [ j ] —> 5
12 —> [ j ] —> 10
19 —> [ j ] —> 17
Now what if these black boxes were put end-to-end so the output of f were fed directly into j? It would look like this:
2 —> [ f ] —> 7 —> [ j ] —> 5
3 —> [ f ] —> 12 —> [ j ] —> 10
4 —> [ f ] —> 19 —> [ j ] —> 17
Now what if we bolted f and j together so we couldn't see anything in between them? It would look like this:
2 —> [ f ] [ j ] —> 5
3 —> [ f ] [ j ] —> 10
4 —> [ f ] [ j ] —> 17
This is what we mean by the "composition" of j and f -- the output of f goes directly into j as if f and j were one function. We write it this way:
(j o f) is the composition of j and f.
(j o f)(x) is the same as j(f(x))
### Simplifying a composition of functions
Here's another way to look at it: Remember that j(x) = x-2. This means j(3) = 3-2, and j(11) = 11-2. It also means that j(elephant) = elephant-2, and j(Empire State Building) = Empire State Building - 2. OK, I'm using extreme examples to make a point: no matter what you feed into j, you get a result that's smaller by 2. So f(x) is no exception. I'm treading lightly, because I know from experience this causes great consternation:
j(f(x)) = f(x)-2
Here we are feeding f(x) into j, and getting a result that is f(x) minus 2. It's no different from any of the other examples in that regard. Now, f(x) is defined as x2+3, so j(f(x)) = f(x)-2 = x2+3-2. This can be further simplified as x2+1.
Recall our definitions of f and j:
f(x) = x2+3
j(x) = x-2
Putting it all together,
j(f(x)) = f(x)-2, from the definition of j
= (x2+3)-2, from the definition of f
= x2+1, by algebraic simplification
### Domain, Range, and Mapping
So far we've looked at functions in four ways -- by its formal definition, as a black box, as an algorithm, and as a graph. Yet another way to look at a function is as a mapping from one set to another. The set of possible input values is called the "domain" of the function. The set of possible output values is called the "range" of the function. The function maps each element of the domain to one element of the range.
Using set-builder notation {a : a has a certain property} means the set of all elements, a, such that a has a certain property,
The domain of the function, f, is domain(f) = {x : f(x) is defined}
The range of f, range(f) = {f(x) : f(x) is defined}
Functions can be defined by closed formula, by cases, recursion, or general axiom(s).
The notation f: A —> B means domain(f) = A and range(f) is a subset of B
Composition of functions, f o g, is defined whenever domain(f) is a subset of range(g), by the equation
(f o g)(x) = f(g(x))
Composition is associative but not commutative.
Two functions, f and g, are inverses when
f(g(x)) = x for any x in domain(g), and
g(f(y)) = y for any y in domain(f)
In these circumstances domain(f) = range(g) and domain(g) = range(f). A function has an inverse if and only if it is one to one.
The inverse function to f is written f~ or f-1.
The identity function on a set A, IA, is defined
IA : A —> A
IA(x) = x
The following laws hold, at least where the left hand side of the equation is defined.
f~~ = f
IA~ = IA
f o IA = f
IA o f = f
(f o g)~ = g~ o f~
(f o g) o h = f o (g o h)
### Periodic Functions
Functions that take on values that repeat themselves again and again at regular intervals are called "periodic" functions. An example of a periodic function is
a sin(b x)
Here, the period of the function (measured horizontally from crest to crest of the wave) is 2π/b, and the amplitude, measured vertically from the midline to the crest, is a.
Students sometimes have difficulty visualizing why the function containing bx (with x multiplied by b) results in a graph in which the labels of the x values are divided by b. The reason for this is that the argument (input) to the function is 2π when the x value is 2π/b. In other words, we have to divide the x value by b to counteract the fact that it's multiplied by b later on.
### Graphing the Cosine Function
Cosine is a "sinusoidal" function, meaning it has the shape of a "sine wave". The general form of a sinusoidal function is:
f(t) = a cos(bt+c) + d, where
|a| is the amplitude of the function, a<0 to "flip" the function upside down,
b is 2π/period
c is the phase shift (in radians), i.e. how far to the left (right if c<0) the graph is shifted
d is the vertical shift, i.e. how far up (down if d<0) the graph is shifted.
For a detailed explanation of this, see . . . . . . (this page needs to be updated to provide a step-by-step procedure for beginning students to graph the following: sin, cos, tan, csc, sec, cot)
### Internet references
Visual Calculus: Composition of Functions, Domain and Range
Translations of Sine and Cosine Curves explains: If the graphs of y = A Sin Bx and y = A Cos Bx are translated h units horizontally and k units vertically, then the resulting graphs have the equations: y-k=A Sin B(x-h) and y-k=A Cos B(x-h)
Mathworld: Domain and Range defines the concepts as they are ordinarily used, and then proceeds to catalog all the other less common definitions -- just read the first paragraph or so of each of these pages.
Mathworld: Periodic Function defines and illustrates periodic functions
Mathwords: Periodic Function, Period
The Math Page: Graphs of the Trig. Functions explains how to calculate the period of trig functions.
Wikipedia: Function (mathematics)
The following seem to be unavailable, which is a shame...
Graphing the Sine and Cosine Functions is an excellent demonstration of the effect of each of the constants in f(t) = a cos(bt+c) + d.
### Related pages in this website
Set Description Notation
The webmaster and author of this Math Help site is Graeme McRae.
| 8,942 | 6,699,453 |
# Difference of 2 number is 11 and their product is 60, find the sum of numbers ?
677 views
posted Jul 16, 2015
## 3 Solutions
+1 vote
The ans is 19
Numbers are 15 & 4
solution Jul 16, 2015
can you expand please
+1 vote
19
The two numbers are:
4 & 15
solution Jul 16, 2015 by anonymous
x-y=11
x*y=60{ x=60/y}
(60/y)-y=11
y^2+11y-60=0
(y-4)(y-15)=0
y=4,y=15
therefore the two numbers are 4 and 15
then the sum of two numbers are 19.
solution Mar 31, 2016
Similar Puzzles
| 185 | 6,699,454 |
Question
# Find out the following in the electric circuit given in figure alongside:(a) Effective resistance of two $$8\Omega$$ resistors in the combination(b) Current flowing through $$4\Omega$$ resistor(c) Potential difference across $$4\Omega$$ resistance(d) Power dissipated in $$42$$ resistor(e) Difference in ammeter readings, if any.
Solution
## (a) $$R = \dfrac {R_{1}R_{2}}{R_{1}R_{2}} = \left (\dfrac {8\times 8}{8 + 8}\right ) = 4\Omega$$(b) $$I = \dfrac {V}{R} = \dfrac {8}{4 + \left (\dfrac {8\times 8}{8 + 8}\right )} = \dfrac {8}{8} = 1A$$(c) $$V = IR = 1\times 4 = 4V$$(d) $$P = I^{2} R = 1^{2}\times 4 = 4W$$(e) No difference, same current flows through each element in a series circuit.Physics
Suggest Corrections
0
Similar questions
View More
People also searched for
View More
| 270 | 6,699,455 |
Math pattern solver
Math pattern solver can be a useful tool for these scholars. Our website can solve math problems for you.
The Best Math pattern solver
This Math pattern solver helps to quickly and easily solve any math problems. Differential equations describe the change in one quantity as a function of time. They are used to solve problems that involve both change and time. Differential equations can be solved using the method of undetermined coefficients, but they can also be solved using integration or difference quotients. Integration is useful when you want to determine the area under a curve, while difference quotients can help you determine the magnitude of an unknown quantity's change (usually expressed as a percentage). As with all math problems, it's important to make sure your solution makes sense. If it doesn't, there's a good chance it's wrong! Solving differential equations is a helpful skill to have in any field, so keep practicing!
From there, you can continue to build your programming skills by learning more complex languages and frameworks. As you progress through learning basic programming concepts, it’s important to keep in mind that learning a language is different than learning how to write code. A programming language is simply a set of instructions written in a specific syntax that tell the computer what to do. A code snippet is just a short piece of code that demonstrates how to implement a specific logic function. Writing code is more about practicing and honing your programming skills. As you practice writing code, it’s important to keep the end goal in mind and make sure you are learning only what you need to know at the moment.
You can solve the problem of water retention by eliminating the main things that can cause it: dehydration, lack of salt, and iron deficiency. You could also try an herbal supplement like milk thistle (a great liver cleanser and antioxidant), which can help to get rid of excess fluid. One of the most effective ways to eliminate excess water is by taking diuretics. There are many different types of diuretics, but they all work basically the same way: they increase your urine output, which leads you to eliminate more fluid from your body. Some examples of diuretics include coffee, beer, cranberry juice and even tea. If you’re having a lot of problems with water retention, you may want to start taking a diuretic as soon as possible.
A trig factoring calculator can take care of this for you by quickly calculating the amount of money you would receive if you took out a loan. With a trig factoring calculator, you simply input an amount that you would like to borrow and it will tell you how much money you would receive if you took out the loan. It’s not always easy to understand how to factor a trignometry equation because it requires some math skills. But with a trig factoring calculator, it’s simple to see how much money you would receive if you took out a loan. The first thing that needs to be done is input the principal amount that you want to borrow. Next, input the interest rate and the term of your loan. Finally, press calculate and your result will be displayed.
As a math teacher, this app is very useful for me as it helps me generate homework answers faster. Also helps me when I mark homework. This app singlehandedly taught me more about math than 10 years of school ever did, I love it and I cannot recommend it enough
Coraline Cooper
No ads and no nothing. App has a ton of features for free. Works great. But sometimes the camera lags, nothing serious though. Would recommend. Very good for studying and can teach better than some teachers. Highly recommend to any students in need of math help.
Serenity Murphy
App to solve algebra problems Web math Finite math tutoring 3 variable system of equation solver Math answerer Math help college algebra
| 780 | 6,699,456 |
A businessman imported Laptops, worth Rs 210000, Mobile phones worth Rs 100000 and Television sets worth Rs 150000. He had to pay 10% duty on laptops, 8% on Phones and 5% on Television sets as a special case. How much total duty (in Rupees) he had to pay on all items as per above details?
Free Practice With Testbook Mock Tests
Options:
1. 36500
2. 37000
3. 37250
4. 37500
Correct Answer: Option 1 (Solution Below)
This question was previously asked in
SSC CGL Tier 2 Quant Previous Paper 13 (Held On: 12 Jan 2017)
Solution:
Worth of Laptops = Rs. 210000
Duty on laptops = 210000 × (10/100) = 21000
Worth of Mobile phones = Rs. 100000
Duty on Mobile phones = 100000 × 8/100 = 8000
Worth of Television sets = Rs. 150000
Duty on Television sets = 150000 × (5/100) = 7500
Total duty on all three items = 21000 + 8000 + 7500 = 36500
| 271 | 6,699,457 |
# How Do You Prove Lines Are Parallel?
## What are five ways to prove two lines are parallel?
Ways to Prove Two Lines ParallelShow that corresponding angles are equal.Show that alternative interior angles are equal.Show that consecutive interior angles are supplementary.Show that consecutive exterior angles are supplementary.In a plane, show that the lines are perpendicular to the same line..
## What is the instrument used to draw the parallel lines?
ClinographClinograph. Clinograph is an instrument used to draw parallel lines to the inclined lines. It contains one adjustable wing or strip which can be adjusted to required angle. So, it can be termed as adjustable set square.
## Can lines be congruent?
Line segments are congruent if they have the same length. However, they need not be parallel. They can be at any angle or orientation on the plane. … Rays and lines cannot be congruent because they do not have both end points defined, and so have no definite length.
## What angles are always congruent?
Vertical angles are always congruent, which means that they are equal. Adjacent angles are angles that come out of the same vertex.
## Are there common values between parallel lines?
Always the same distance apart and never touching. Parallel lines also point in the same direction. Parallel lines have so much in common. It’s a shame they will never meet!
## Which theorem correctly justifies why the lines m and n are parallel when cut by transversal k?
Let m and n are two lines and the lines are cut by transversal k. Then, if we show that the alternate interior angles are equal, then m and n become parallel to each other. So, the converse of the alternate interior angles theorem correctly justifies that the lines are parallel when cut by transversal.
Which lines are parallel? Justify your answer. Lines e and f are parallel because their alternate exterior angles are congruent. Lines c and d are parallel lines cut by transversal p.
## Do parallel lines intersect?
Parallel lines are lines in a plane that are always the same distance apart. Parallel lines never intersect.
## What are the corresponding angles?
: any pair of angles each of which is on the same side of one of two lines cut by a transversal and on the same side of the transversal.
## Are parallel lines congruent in a triangle?
Theorem 6.1: If two parallel lines are transected by a third, the alternate interior angles are the same size. … Theorem 6.2: If a line intersects two other lines then the following conditions are equivalent. a) The alternate interior angles are the same size.
## How do you prove parallel lines in congruent triangles?
To really understand this problem you have to remember the ways to prove lines parallel: the converse of the corresponding angles postulate, the converse of the alternate interior angles theorem and the converse of the same-side interior angles theorem. So, to prove that segment AB is congruent to Segment CD.
## Can you prove that lines P and Q are parallel?
is it possible to prove that lines p and q are parallel? … If the lines are cut by a transversal so that (alternate interior, alternate exterior, corresponding) angles are congruent, then the lines are parallel. 10. Complete the two-column proof.
## How do you prove two lines are parallel on a graph?
Lines that are parallel have the same gradient . The graphs above, y = 2 x + 1 and y = 2 x − 2 have the same gradient of 2. The lines are parallel. Two lines will be parallel if they have the same gradient.
## How do you prove lines are parallel proof?
If two parallel lines are cut by a transversal, then corresponding angles are congruent. If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
| 797 | 6,699,458 |
<MASK>
## What is the content of a polynomial?
<MASK>
## What polynomials Cannot be factored?
<MASK>
## Is a polynomial irreducible?
<MASK>
Now, from this carachterization it is obvious that x−1 is an irreducible polynomial (over any field indeed). About g:=x2+x+1, suppose it is not irreducible; then there must be a polynomial ax+b∈R[x] (a≠0) such that ax+b∣g, i.e −b/a must be a root for g.
<MASK>
Units are certainly not reducible. For example, in the integers, is neither prime nor composite — indeed, it is a unit. Moreover, it is neither irreducible nor reducible, as the concept of irreducible and prime coincides in (being a unique factorization domain).
<MASK>
## What is a minimal polynomial?
<MASK>
## What does it mean when a polynomial is irreducible?
<MASK>
## What is the difference between reducible and irreducible representation?
In a given representation (reducible or irreducible), the characters of all matrices belonging to symmetry operations in the same class are identical. The number of irreducible representations of a group is equal to the number of classes in the group.
<MASK>
## What is the meaning of irreducible?
<MASK>
adjective. not redeemable; incapable of being bought back or paid off. irremediable; irreparable; hopeless. beyond redemption; irreclaimable. (of paper money) not convertible into gold or silver.
## What is irreparable?
<MASK>
## What does deplorable mean?
<MASK>
Begin typing your search term above and press enter to search. Press ESC to cancel.
<UNMASK>
<MASK>
## What is the content of a polynomial?
<MASK>
## What polynomials Cannot be factored?
<MASK>
## Is a polynomial irreducible?
Over the complex field, and, more generally, over an algebraically closed field, a univariate polynomial is irreducible if and only if its degree is one. This fact is known as the fundamental theorem of algebra in the case of the complex numbers and, in general, as the condition of being algebraically closed.
<MASK>
Now, from this carachterization it is obvious that x−1 is an irreducible polynomial (over any field indeed). About g:=x2+x+1, suppose it is not irreducible; then there must be a polynomial ax+b∈R[x] (a≠0) such that ax+b∣g, i.e −b/a must be a root for g.
## How do you know if a quadratic is irreducible?
<MASK>
Units are certainly not reducible. For example, in the integers, is neither prime nor composite — indeed, it is a unit. Moreover, it is neither irreducible nor reducible, as the concept of irreducible and prime coincides in (being a unique factorization domain).
<MASK>
## What is a minimal polynomial?
<MASK>
## What does it mean when a polynomial is irreducible?
<MASK>
## What does it mean for a polynomial to be reducible?
<MASK>
## What is reducible and irreducible?
<MASK>
## What is the difference between reducible and irreducible representation?
In a given representation (reducible or irreducible), the characters of all matrices belonging to symmetry operations in the same class are identical. The number of irreducible representations of a group is equal to the number of classes in the group.
## What is an irreducible matrix?
<MASK>
## What is the meaning of irreducible?
<MASK>
adjective. not redeemable; incapable of being bought back or paid off. irremediable; irreparable; hopeless. beyond redemption; irreclaimable. (of paper money) not convertible into gold or silver.
## What is irreparable?
<MASK>
## What does deplorable mean?
<MASK>
Begin typing your search term above and press enter to search. Press ESC to cancel.
<UNMASK>
<MASK>
## What is the content of a polynomial?
<MASK>
## What polynomials Cannot be factored?
<MASK>
## Is a polynomial irreducible?
Over the complex field, and, more generally, over an algebraically closed field, a univariate polynomial is irreducible if and only if its degree is one. This fact is known as the fundamental theorem of algebra in the case of the complex numbers and, in general, as the condition of being algebraically closed.
<MASK>
Now, from this carachterization it is obvious that x−1 is an irreducible polynomial (over any field indeed). About g:=x2+x+1, suppose it is not irreducible; then there must be a polynomial ax+b∈R[x] (a≠0) such that ax+b∣g, i.e −b/a must be a root for g.
## How do you know if a quadratic is irreducible?
When it comes to irreducible quadratic factors, there can’t be any x-intercepts corresponding to this factor, since there are no real zeros. In other words, if we have an irreducible quadratic factor, f(x), then the graph will have no x-intercepts if we graph y = f(x).
<MASK>
Units are certainly not reducible. For example, in the integers, is neither prime nor composite — indeed, it is a unit. Moreover, it is neither irreducible nor reducible, as the concept of irreducible and prime coincides in (being a unique factorization domain).
<MASK>
## What is a minimal polynomial?
<MASK>
## What does it mean when a polynomial is irreducible?
A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field.
## What does it mean for a polynomial to be reducible?
<MASK>
## What is reducible and irreducible?
<MASK>
## What is the difference between reducible and irreducible representation?
In a given representation (reducible or irreducible), the characters of all matrices belonging to symmetry operations in the same class are identical. The number of irreducible representations of a group is equal to the number of classes in the group.
## What is an irreducible matrix?
<MASK>
## What is the meaning of irreducible?
<MASK>
adjective. not redeemable; incapable of being bought back or paid off. irremediable; irreparable; hopeless. beyond redemption; irreclaimable. (of paper money) not convertible into gold or silver.
## What is irreparable?
<MASK>
## What does deplorable mean?
<MASK>
Begin typing your search term above and press enter to search. Press ESC to cancel.
<UNMASK>
# What is the content of a polynomial?
## What is the content of a polynomial?
<MASK>
## What polynomials Cannot be factored?
A polynomial with integer coefficients that cannot be factored into polynomials of lower degree , also with integer coefficients, is called an irreducible or prime polynomial .
## Is a polynomial irreducible?
Over the complex field, and, more generally, over an algebraically closed field, a univariate polynomial is irreducible if and only if its degree is one. This fact is known as the fundamental theorem of algebra in the case of the complex numbers and, in general, as the condition of being algebraically closed.
<MASK>
Now, from this carachterization it is obvious that x−1 is an irreducible polynomial (over any field indeed). About g:=x2+x+1, suppose it is not irreducible; then there must be a polynomial ax+b∈R[x] (a≠0) such that ax+b∣g, i.e −b/a must be a root for g.
## How do you know if a quadratic is irreducible?
When it comes to irreducible quadratic factors, there can’t be any x-intercepts corresponding to this factor, since there are no real zeros. In other words, if we have an irreducible quadratic factor, f(x), then the graph will have no x-intercepts if we graph y = f(x).
<MASK>
Units are certainly not reducible. For example, in the integers, is neither prime nor composite — indeed, it is a unit. Moreover, it is neither irreducible nor reducible, as the concept of irreducible and prime coincides in (being a unique factorization domain).
## Is the zero polynomial irreducible?
You’re correct, if working over a field; that is, if the polynomial ring you’re talking about is F[x] where F is a field, then any non-zero constant polynomial is a unit of the ring F[x], and the constant zero polynomial is the zero element of the ring F[x], and therefore none of them are irreducible elements of F[x].
## What is a minimal polynomial?
<MASK>
## What does it mean when a polynomial is irreducible?
A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field.
## What does it mean for a polynomial to be reducible?
<MASK>
If is reducible, it has a factor of degree 1 or a factor of degree 2. Use long division or other arguments to show that none of these is actually a factor. If a polynomial with degree 2 or higher is irreducible in , then it has no roots in .
## What is reducible and irreducible?
<MASK>
## What is the difference between reducible and irreducible representation?
In a given representation (reducible or irreducible), the characters of all matrices belonging to symmetry operations in the same class are identical. The number of irreducible representations of a group is equal to the number of classes in the group.
## What is an irreducible matrix?
<MASK>
## What is the meaning of irreducible?
<MASK>
adjective. not redeemable; incapable of being bought back or paid off. irremediable; irreparable; hopeless. beyond redemption; irreclaimable. (of paper money) not convertible into gold or silver.
## What is irreparable?
<MASK>
## What does deplorable mean?
1 : deserving censure or contempt deplorable behavior : wretched deplorable living conditions. 2 : lamentable a deplorable death.
Begin typing your search term above and press enter to search. Press ESC to cancel.
<UNMASK>
# What is the content of a polynomial?
## What is the content of a polynomial?
In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content.
## How do you find irreducible polynomials?
p2(x)=p(x+1)=(x+1)5+(x+1)2+1=x5+x4+x2+x+1 is irreducible. Fun fact. If q(x) is irreducible with non-zero constant term(over any field), then so is its reciprocal polynomial ˜q(x)=xdegqq(1x). are also irreducible.
## What polynomials Cannot be factored?
A polynomial with integer coefficients that cannot be factored into polynomials of lower degree , also with integer coefficients, is called an irreducible or prime polynomial .
## Is a polynomial irreducible?
Over the complex field, and, more generally, over an algebraically closed field, a univariate polynomial is irreducible if and only if its degree is one. This fact is known as the fundamental theorem of algebra in the case of the complex numbers and, in general, as the condition of being algebraically closed.
## Is x1 irreducible?
Now, from this carachterization it is obvious that x−1 is an irreducible polynomial (over any field indeed). About g:=x2+x+1, suppose it is not irreducible; then there must be a polynomial ax+b∈R[x] (a≠0) such that ax+b∣g, i.e −b/a must be a root for g.
## How do you know if a quadratic is irreducible?
When it comes to irreducible quadratic factors, there can’t be any x-intercepts corresponding to this factor, since there are no real zeros. In other words, if we have an irreducible quadratic factor, f(x), then the graph will have no x-intercepts if we graph y = f(x).
## Are units irreducible?
Units are certainly not reducible. For example, in the integers, is neither prime nor composite — indeed, it is a unit. Moreover, it is neither irreducible nor reducible, as the concept of irreducible and prime coincides in (being a unique factorization domain).
## Is the zero polynomial irreducible?
You’re correct, if working over a field; that is, if the polynomial ring you’re talking about is F[x] where F is a field, then any non-zero constant polynomial is a unit of the ring F[x], and the constant zero polynomial is the zero element of the ring F[x], and therefore none of them are irreducible elements of F[x].
## What is a minimal polynomial?
The minimal polynomial of a matrix is the monic polynomial in of smallest degree such that. (1) The minimal polynomial divides any polynomial with. and, in particular, it divides the characteristic polynomial.
## What does it mean when a polynomial is irreducible?
A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field.
## What does it mean for a polynomial to be reducible?
: a polynomial expressible as the product of two or more polynomials of lower degree.
## How do you know if a polynomial is reducible?
If is reducible, it has a factor of degree 1 or a factor of degree 2. Use long division or other arguments to show that none of these is actually a factor. If a polynomial with degree 2 or higher is irreducible in , then it has no roots in .
## What is reducible and irreducible?
Definition: Let be a field and let . Then is said to be Irreducible over if cannot be factored into a product of polynomials all of which having lower degree than . If is not irreducible over then we say that is Reducible over .
## What is the difference between reducible and irreducible representation?
In a given representation (reducible or irreducible), the characters of all matrices belonging to symmetry operations in the same class are identical. The number of irreducible representations of a group is equal to the number of classes in the group.
## What is an irreducible matrix?
A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size). Also, a Markov chain is irreducible if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state.
## What is the meaning of irreducible?
1 : impossible to transform into or restore to a desired or simpler condition an irreducible matrix specifically : incapable of being factored into polynomials of lower degree with coefficients in some given field (such as the rational numbers) or integral domain (such as the integers) an irreducible equation.
## What is the irreducible minimum meaning?
not reducible; incapable of being reduced or of being diminished or simplified further: the irreducible minimum. incapable of being brought into a different condition or form. Mathematics.
## What does irremediably mean?
: impossible to remedy, correct, or redress irremediable harm irremediable conduct. Other Words from irremediable.
## What is the meaning of irredeemable?
adjective. not redeemable; incapable of being bought back or paid off. irremediable; irreparable; hopeless. beyond redemption; irreclaimable. (of paper money) not convertible into gold or silver.
## What is irreparable?
: impossible to repair, remedy, or undo. Other Words from irreparable. irreparably adverb.
## What does deplorable mean?
1 : deserving censure or contempt deplorable behavior : wretched deplorable living conditions. 2 : lamentable a deplorable death.
Begin typing your search term above and press enter to search. Press ESC to cancel.
| 3,385 | 6,699,459 |
<MASK>
Tags: Olive Garden Pasta Tales Essay Writing 2014Writing Essays For CollegeMein Beruf EssayNursing Assessment Process EssayReasons For WritingThesis And Falun Gong
After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect.
Example 1: If 58 out of 100 students in a school are boys, then write a decimal for the part of the school that consists of boys.
<MASK>
Ask the children to perform the division to find the quotient by applying long division method. Decimals: Multiplication and Division These decimal worksheets emphasize decimal multiplication and division.
<MASK>
Our word problems worksheets cover addition, subtraction, multiplication, division, fractions, decimals, measurement (volume, mass and length), GCF / LCM and variables and expressions.
<MASK>
Decimal Word Problem Worksheets. Extensive decimal word problems are presented in these sets of worksheets, which require the learner to perform addition, subtraction, multiplication, and division operations. Our decimal word problems are replete with engaging scenarios. Free samples are included.…
<MASK>
<UNMASK>
<MASK>
Tags: Olive Garden Pasta Tales Essay Writing 2014Writing Essays For CollegeMein Beruf EssayNursing Assessment Process EssayReasons For WritingThesis And Falun Gong
After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect.
Example 1: If 58 out of 100 students in a school are boys, then write a decimal for the part of the school that consists of boys.
<MASK>
Ask the children to perform the division to find the quotient by applying long division method. Decimals: Multiplication and Division These decimal worksheets emphasize decimal multiplication and division.
The perfect blend of word problems makes the children stronger in performing the multiplication and division operation.
<MASK>
Our word problems worksheets cover addition, subtraction, multiplication, division, fractions, decimals, measurement (volume, mass and length), GCF / LCM and variables and expressions.
<MASK>
Solving More Decimal Word Problems. Answer Paul will make 36 monthly payments of \$529.47 each. Example 7 What is the average speed in miles per hour of a car that travels 956.4 miles in 15.9 hours? Round your answer to the nearest tenth. Analysis We will divide 956.4 by 15.9, then round the quotient to the nearest tenth.…
<MASK>
• ###### Decimal Word Problem Worksheets - Math Worksheets 4 Kids
Decimal Word Problem Worksheets. Extensive decimal word problems are presented in these sets of worksheets, which require the learner to perform addition, subtraction, multiplication, and division operations. Our decimal word problems are replete with engaging scenarios. Free samples are included.…
<MASK>
Free decimal worksheets grades 3-7 This versatile generator produces worksheets for addition, subtraction, multiplication, and division of decimals for grades 3-7. You can create easy decimal problems to be solved with mental math, worksheets for multiplying by 10, 100, or 1000, decimal long division problems, missing number problems, and more.…
<MASK>
<UNMASK>
# Decimal Problem Solving Worksheet
Tags: Olive Garden Pasta Tales Essay Writing 2014Writing Essays For CollegeMein Beruf EssayNursing Assessment Process EssayReasons For WritingThesis And Falun Gong
After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect.
Example 1: If 58 out of 100 students in a school are boys, then write a decimal for the part of the school that consists of boys.
Then we must determine how the least compares with the winning score. Analysis: We must compare and order these decimals to help us solve this problem.
<MASK>
Ask the children to perform the division to find the quotient by applying long division method. Decimals: Multiplication and Division These decimal worksheets emphasize decimal multiplication and division.
The perfect blend of word problems makes the children stronger in performing the multiplication and division operation.
To do this, we will round one factor up and one factor down.
Analysis: We need to estimate the product of .50 and 15.5.
<MASK>
Our word problems worksheets cover addition, subtraction, multiplication, division, fractions, decimals, measurement (volume, mass and length), GCF / LCM and variables and expressions.
## Comments Decimal Problem Solving Worksheet
<MASK>
Print Solving Problems Using Decimal Numbers Worksheet 1. A model house is built to 1/8 or 0.125 of real-world size, so everything in the model house is exactly 0.125 times the size of its real.…
<MASK>
Solving More Decimal Word Problems. Answer Paul will make 36 monthly payments of \$529.47 each. Example 7 What is the average speed in miles per hour of a car that travels 956.4 miles in 15.9 hours? Round your answer to the nearest tenth. Analysis We will divide 956.4 by 15.9, then round the quotient to the nearest tenth.…
• ###### Decimals Worksheets Dynamically Created Decimal Worksheets
You may select the number of decimals in the dividend for the problems. These decimal worksheets produce 9 problems per worksheet. 3 Digit Decimal Division Worksheets Horizontal Format These decimal worksheets produces problems in which you must divide a 3 digit decimal number by a single digit number. You may select between 12, 15, 18, 21, 24 or 30 problems for these decimal worksheets.…
Decimal addition and subtraction worksheets Our grade 5 addition and subtraction of decimals worksheets provide practice exercises in adding and subtracting numbers with up to 3 decimal digits. These math worksheets complement our online math program.…
• ###### Decimal Word Problem Worksheets - Math Worksheets 4 Kids
Decimal Word Problem Worksheets. Extensive decimal word problems are presented in these sets of worksheets, which require the learner to perform addition, subtraction, multiplication, and division operations. Our decimal word problems are replete with engaging scenarios. Free samples are included.…
• ###### Worksheet on Decimal Word Problems - math-only-
Worksheet on Decimal Word Problems Solve the questions given in the worksheet on decimal word problems at your own space. This worksheet provides a mixture of questions on decimals involving order of operations i.e. addition, subtraction, multiplication and division.…
<MASK>
Free decimal worksheets grades 3-7 This versatile generator produces worksheets for addition, subtraction, multiplication, and division of decimals for grades 3-7. You can create easy decimal problems to be solved with mental math, worksheets for multiplying by 10, 100, or 1000, decimal long division problems, missing number problems, and more.…
<MASK>
<UNMASK>
# Decimal Problem Solving Worksheet
Tags: Olive Garden Pasta Tales Essay Writing 2014Writing Essays For CollegeMein Beruf EssayNursing Assessment Process EssayReasons For WritingThesis And Falun Gong
After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect.
Example 1: If 58 out of 100 students in a school are boys, then write a decimal for the part of the school that consists of boys.
Then we must determine how the least compares with the winning score. Analysis: We must compare and order these decimals to help us solve this problem.
Step 1: Example 4: To make a miniature ice cream truck, you need tires with a diameter between 1.465 cm and 1.472 cm. Specifically, we need to determine if the third decimal is between the first two.
Ask the children to perform the division to find the quotient by applying long division method. Decimals: Multiplication and Division These decimal worksheets emphasize decimal multiplication and division.
The perfect blend of word problems makes the children stronger in performing the multiplication and division operation.
To do this, we will round one factor up and one factor down.
Analysis: We need to estimate the product of .50 and 15.5.
<MASK>
Our word problems worksheets cover addition, subtraction, multiplication, division, fractions, decimals, measurement (volume, mass and length), GCF / LCM and variables and expressions.
## Comments Decimal Problem Solving Worksheet
• ###### Quiz & Worksheet - Problem Solving with Decimals
Print Solving Problems Using Decimal Numbers Worksheet 1. A model house is built to 1/8 or 0.125 of real-world size, so everything in the model house is exactly 0.125 times the size of its real.…
• ###### Solving More Decimal Word Problems Math Goodies
Solving More Decimal Word Problems. Answer Paul will make 36 monthly payments of \$529.47 each. Example 7 What is the average speed in miles per hour of a car that travels 956.4 miles in 15.9 hours? Round your answer to the nearest tenth. Analysis We will divide 956.4 by 15.9, then round the quotient to the nearest tenth.…
• ###### Decimals Worksheets Dynamically Created Decimal Worksheets
You may select the number of decimals in the dividend for the problems. These decimal worksheets produce 9 problems per worksheet. 3 Digit Decimal Division Worksheets Horizontal Format These decimal worksheets produces problems in which you must divide a 3 digit decimal number by a single digit number. You may select between 12, 15, 18, 21, 24 or 30 problems for these decimal worksheets.…
Decimal addition and subtraction worksheets Our grade 5 addition and subtraction of decimals worksheets provide practice exercises in adding and subtracting numbers with up to 3 decimal digits. These math worksheets complement our online math program.…
• ###### Decimal Word Problem Worksheets - Math Worksheets 4 Kids
Decimal Word Problem Worksheets. Extensive decimal word problems are presented in these sets of worksheets, which require the learner to perform addition, subtraction, multiplication, and division operations. Our decimal word problems are replete with engaging scenarios. Free samples are included.…
• ###### Worksheet on Decimal Word Problems - math-only-
Worksheet on Decimal Word Problems Solve the questions given in the worksheet on decimal word problems at your own space. This worksheet provides a mixture of questions on decimals involving order of operations i.e. addition, subtraction, multiplication and division.…
<MASK>
Free decimal worksheets grades 3-7 This versatile generator produces worksheets for addition, subtraction, multiplication, and division of decimals for grades 3-7. You can create easy decimal problems to be solved with mental math, worksheets for multiplying by 10, 100, or 1000, decimal long division problems, missing number problems, and more.…
• ###### Solving Decimal Word Problems Math Goodies
Solving Decimal Word Problems. Step 2 The least decimal is 9.75. Now we must determine how 9.75 compares with the winning score. Answer The last swimmer must get a score less than 9.75 s in order to win. Example 4 To make a miniature ice cream truck, you need tires with a diameter between 1.465 cm and 1.472 cm.…
<UNMASK>
# Decimal Problem Solving Worksheet
Tags: Olive Garden Pasta Tales Essay Writing 2014Writing Essays For CollegeMein Beruf EssayNursing Assessment Process EssayReasons For WritingThesis And Falun Gong
After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect.
Example 1: If 58 out of 100 students in a school are boys, then write a decimal for the part of the school that consists of boys.
Then we must determine how the least compares with the winning score. Analysis: We must compare and order these decimals to help us solve this problem.
Step 1: Example 4: To make a miniature ice cream truck, you need tires with a diameter between 1.465 cm and 1.472 cm. Specifically, we need to determine if the third decimal is between the first two.
Ask the children to perform the division to find the quotient by applying long division method. Decimals: Multiplication and Division These decimal worksheets emphasize decimal multiplication and division.
The perfect blend of word problems makes the children stronger in performing the multiplication and division operation.
To do this, we will round one factor up and one factor down.
Analysis: We need to estimate the product of .50 and 15.5.
provide students with real world word problems that students can solve with grade 5 math concepts.
Our word problems worksheets cover addition, subtraction, multiplication, division, fractions, decimals, measurement (volume, mass and length), GCF / LCM and variables and expressions.
## Comments Decimal Problem Solving Worksheet
• ###### Quiz & Worksheet - Problem Solving with Decimals
Print Solving Problems Using Decimal Numbers Worksheet 1. A model house is built to 1/8 or 0.125 of real-world size, so everything in the model house is exactly 0.125 times the size of its real.…
• ###### Solving More Decimal Word Problems Math Goodies
Solving More Decimal Word Problems. Answer Paul will make 36 monthly payments of \$529.47 each. Example 7 What is the average speed in miles per hour of a car that travels 956.4 miles in 15.9 hours? Round your answer to the nearest tenth. Analysis We will divide 956.4 by 15.9, then round the quotient to the nearest tenth.…
• ###### Decimals Worksheets Dynamically Created Decimal Worksheets
You may select the number of decimals in the dividend for the problems. These decimal worksheets produce 9 problems per worksheet. 3 Digit Decimal Division Worksheets Horizontal Format These decimal worksheets produces problems in which you must divide a 3 digit decimal number by a single digit number. You may select between 12, 15, 18, 21, 24 or 30 problems for these decimal worksheets.…
Decimal addition and subtraction worksheets Our grade 5 addition and subtraction of decimals worksheets provide practice exercises in adding and subtracting numbers with up to 3 decimal digits. These math worksheets complement our online math program.…
• ###### Decimal Word Problem Worksheets - Math Worksheets 4 Kids
Decimal Word Problem Worksheets. Extensive decimal word problems are presented in these sets of worksheets, which require the learner to perform addition, subtraction, multiplication, and division operations. Our decimal word problems are replete with engaging scenarios. Free samples are included.…
• ###### Worksheet on Decimal Word Problems - math-only-
Worksheet on Decimal Word Problems Solve the questions given in the worksheet on decimal word problems at your own space. This worksheet provides a mixture of questions on decimals involving order of operations i.e. addition, subtraction, multiplication and division.…
• ###### Free decimal worksheets for grades 3-7 -
Free decimal worksheets grades 3-7 This versatile generator produces worksheets for addition, subtraction, multiplication, and division of decimals for grades 3-7. You can create easy decimal problems to be solved with mental math, worksheets for multiplying by 10, 100, or 1000, decimal long division problems, missing number problems, and more.…
• ###### Solving Decimal Word Problems Math Goodies
Solving Decimal Word Problems. Step 2 The least decimal is 9.75. Now we must determine how 9.75 compares with the winning score. Answer The last swimmer must get a score less than 9.75 s in order to win. Example 4 To make a miniature ice cream truck, you need tires with a diameter between 1.465 cm and 1.472 cm.…
| 3,174 | 6,699,460 |
## sasogeek 3 years ago If the US population was 200,000,000 and the cost of gasoline was \$2.00 per gallon and gas mileage is 20 mi/gal and you buy every single person in the US a \$2,000 car and enough gas to run it nonstop at 20 mph for a year: How much money would you spend.
1. apoorvk
some trillions... basically that would bust Uncle Sam. :D
2. sasogeek
lol
3. apoorvk
On a more serious note. So every person is guzzling up a gallon an hour. that is \$2 an hour. so totally the whole of USA is spending \$400m per hour. Now multiply that the no. of hours in an year (24x365), and you have your BILL. Now pay! :P
4. apoorvk
@Diyadiya Uncle Sam just guzzled up all your oil^. What you gonna do now? :P
5. apoorvk
oh yes add to that 2000 times 200million, cost of the cars.
6. apoorvk
@sasogeek don't tell me you made this up yourself. please. I have a gun to my head.
7. ZhangYan
That looks very hard, but it is actually quite easy. First off, find out how much you spend on the cars alone. 200,000,000 people and each get a \$2,000 car so 200,000,000*2,000 = \$400 billion. Next find out how many gallons of gas you need. So we know that the car travels 20 miles in 1 hr. So 20*24 gives us 480 miles in one day. And 480*365 gives us 175,200 miles in one year. If the mileage is 20 mil/gal then divided 175,200 by 20 and we get 8,760 gallons of gas for each car. At \$2.00 per gallon we get 2*8760 = 17.520 for each car to travel 20 mi/hr nonstop for a year We then multiply this amount by 200,000,000 and get \$3,504,000,000,000. Add this amount to \$400,000,000,000 and you get a grand total of \$3,904,000,000,000 or if you want it in words - 3 trillion, nine-hundred and four billion dollars.
8. sasogeek
http://answers.yahoo.com/question/index?qid=20100424164819AAfF7Sn there << i'm searching for questions everywhere i can to solve. i only post the interesting ones here :)
9. apoorvk
Wait. WAIT. the answer is \$0. You asked how much would 'I' spend. nothing. I am not financing them, i don't even live in the States. :p
10. sasogeek
LOL well that's quite an extensive solution Zoe :) thanks!
11. sasogeek
it's always good to know how to copy and paste @ZhangYan :) I know what u did :)
12. ZhangYan
hahaha
| 679 | 6,699,461 |
<MASK>
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
<MASK>
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
<MASK>
“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
<MASK>
1) Number of movie theaters that sold at least 100 tickets were more than half of the total number of movie theaters.
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
<MASK>
For con 1), even if you assume the number of theaters that sold at least 100 tickets as 5, since the rest of the 5 movie theaters sold at least 80 tickets, so the total number of tickets sold becomes 100(5)+80(5)=900. Since it said that it is half greater than the total number of movie theaters, which is 10, so it is always greater than 900, hence yes, it is sufficient.
<MASK>
"Each stage of the journey is crucial to attaining new heights of knowledge."
<MASK>
This Fact tells us that MORE than half of the theaters sold at least 100 tickets. That would be at least 6 theaters (at the minimum)...
(6 theaters)(100 tickets each) = 600 tickets
<MASK>
<UNMASK>
<MASK>
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
<MASK>
Track
<MASK>
# There are a total 10 movie theaters. Minimum number of tickets that ea
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
<MASK>
31 Mar 2017, 14:58
3
00:00
<MASK>
71% (01:41) correct 29% (01:56) wrong based on 138 sessions
<MASK>
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
<MASK>
_________________
"Be challenged at EVERY MOMENT."
“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
<MASK>
01 Apr 2017, 01:31
1
ziyuen wrote:
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
1) Number of movie theaters that sold at least 100 tickets were more than half of the total number of movie theaters.
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
OFFICIAL EXPLANATION
<MASK>
For con 1), even if you assume the number of theaters that sold at least 100 tickets as 5, since the rest of the 5 movie theaters sold at least 80 tickets, so the total number of tickets sold becomes 100(5)+80(5)=900. Since it said that it is half greater than the total number of movie theaters, which is 10, so it is always greater than 900, hence yes, it is sufficient.
<MASK>
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
<MASK>
### Show Tags
21 Dec 2017, 11:07
Hi All,
While this question is awkwardly-worded, it is essentially a 'limit' question. From the prompt, we know that each of the 10 theaters sold AT LEAST 80 tickets. We're asked if the average number of tickets sold for each theater was GREATER than 90? We can 'rewrite' this prompt - it's asking "Is the TOTAL number of tickets sold greater than 900?" This is a YES/NO question.
<MASK>
This Fact tells us that MORE than half of the theaters sold at least 100 tickets. That would be at least 6 theaters (at the minimum)...
(6 theaters)(100 tickets each) = 600 tickets
<MASK>
Total MINIMUM number of tickets sold = 600 + 320 = 920 tickets. Increasing the number of theaters that sold at least 100 tickets would increase the total number of tickets sold. Thus, the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
We can deal with Fact 2 in the same general way that we dealt with Fact 1...
(4 theaters)(more than 105 tickets each) = more than 420 tickets
<MASK>
(6 theaters)(80 tickets each) = 480 tickets
<MASK>
The Course Used By GMAT Club Moderators To Earn 750+
<MASK>
# There are a total 10 movie theaters. Minimum number of tickets that ea
<MASK>
<UNMASK>
<MASK>
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
<MASK>
Track
<MASK>
# There are a total 10 movie theaters. Minimum number of tickets that ea
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
<MASK>
31 Mar 2017, 14:58
3
00:00
Difficulty:
<MASK>
71% (01:41) correct 29% (01:56) wrong based on 138 sessions
### HideShow timer Statistics
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
<MASK>
_________________
"Be challenged at EVERY MOMENT."
“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
<MASK>
01 Apr 2017, 01:31
1
ziyuen wrote:
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
1) Number of movie theaters that sold at least 100 tickets were more than half of the total number of movie theaters.
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
OFFICIAL EXPLANATION
<MASK>
For con 1), even if you assume the number of theaters that sold at least 100 tickets as 5, since the rest of the 5 movie theaters sold at least 80 tickets, so the total number of tickets sold becomes 100(5)+80(5)=900. Since it said that it is half greater than the total number of movie theaters, which is 10, so it is always greater than 900, hence yes, it is sufficient.
<MASK>
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
<MASK>
### Show Tags
21 Dec 2017, 11:07
Hi All,
While this question is awkwardly-worded, it is essentially a 'limit' question. From the prompt, we know that each of the 10 theaters sold AT LEAST 80 tickets. We're asked if the average number of tickets sold for each theater was GREATER than 90? We can 'rewrite' this prompt - it's asking "Is the TOTAL number of tickets sold greater than 900?" This is a YES/NO question.
<MASK>
This Fact tells us that MORE than half of the theaters sold at least 100 tickets. That would be at least 6 theaters (at the minimum)...
(6 theaters)(100 tickets each) = 600 tickets
<MASK>
Total MINIMUM number of tickets sold = 600 + 320 = 920 tickets. Increasing the number of theaters that sold at least 100 tickets would increase the total number of tickets sold. Thus, the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
We can deal with Fact 2 in the same general way that we dealt with Fact 1...
(4 theaters)(more than 105 tickets each) = more than 420 tickets
<MASK>
(6 theaters)(80 tickets each) = 480 tickets
<MASK>
The Course Used By GMAT Club Moderators To Earn 750+
<MASK>
# There are a total 10 movie theaters. Minimum number of tickets that ea
<MASK>
<UNMASK>
GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
It is currently 31 May 2020, 13:41
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
<MASK>
Track
<MASK>
Practice
Pays
<MASK>
# There are a total 10 movie theaters. Minimum number of tickets that ea
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
### Hide Tags
Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1343
Location: Malaysia
There are a total 10 movie theaters. Minimum number of tickets that ea [#permalink]
### Show Tags
31 Mar 2017, 14:58
3
00:00
Difficulty:
<MASK>
71% (01:41) correct 29% (01:56) wrong based on 138 sessions
### HideShow timer Statistics
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
<MASK>
_________________
"Be challenged at EVERY MOMENT."
“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
<MASK>
01 Apr 2017, 01:31
1
ziyuen wrote:
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
1) Number of movie theaters that sold at least 100 tickets were more than half of the total number of movie theaters.
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
OFFICIAL EXPLANATION
<MASK>
For con 1), even if you assume the number of theaters that sold at least 100 tickets as 5, since the rest of the 5 movie theaters sold at least 80 tickets, so the total number of tickets sold becomes 100(5)+80(5)=900. Since it said that it is half greater than the total number of movie theaters, which is 10, so it is always greater than 900, hence yes, it is sufficient.
For con 2), if you assume that from the total of 10 movie theaters, 4 movie theaters sold 105 tickets and the 6 movie theaters sold 80 tickets, then the total number of tickets sold is 105(4)+80(6)=900, and since it said that the 4 movie theaters sold more than 105 tickets each, so it is always greater than 900, hence it is yes and sufficient. Therefore, the answer is D.
_________________
"Be challenged at EVERY MOMENT."
<MASK>
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 16759
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: There are a total 10 movie theaters. Minimum number of tickets that ea [#permalink]
### Show Tags
21 Dec 2017, 11:07
Hi All,
While this question is awkwardly-worded, it is essentially a 'limit' question. From the prompt, we know that each of the 10 theaters sold AT LEAST 80 tickets. We're asked if the average number of tickets sold for each theater was GREATER than 90? We can 'rewrite' this prompt - it's asking "Is the TOTAL number of tickets sold greater than 900?" This is a YES/NO question.
<MASK>
This Fact tells us that MORE than half of the theaters sold at least 100 tickets. That would be at least 6 theaters (at the minimum)...
(6 theaters)(100 tickets each) = 600 tickets
From the prompt, we know that each theater sold AT LEAST 80 tickets, so we can find the minimum total that the other 4 theaters sold...
(4 theaters)(80 tickets each) = 320 tickets
Total MINIMUM number of tickets sold = 600 + 320 = 920 tickets. Increasing the number of theaters that sold at least 100 tickets would increase the total number of tickets sold. Thus, the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
We can deal with Fact 2 in the same general way that we dealt with Fact 1...
(4 theaters)(more than 105 tickets each) = more than 420 tickets
Each of the other theaters sold AT LEAST 80 tickets, so we can find the minimum total that the other 6 theaters sold...
(6 theaters)(80 tickets each) = 480 tickets
<MASK>
The Course Used By GMAT Club Moderators To Earn 750+
souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
Re: There are a total 10 movie theaters. Minimum number of tickets that ea [#permalink] 21 Dec 2017, 11:07
# There are a total 10 movie theaters. Minimum number of tickets that ea
<MASK>
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
<UNMASK>
GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
It is currently 31 May 2020, 13:41
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# There are a total 10 movie theaters. Minimum number of tickets that ea
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
### Hide Tags
Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1343
Location: Malaysia
There are a total 10 movie theaters. Minimum number of tickets that ea [#permalink]
### Show Tags
31 Mar 2017, 14:58
3
00:00
Difficulty:
45% (medium)
Question Stats:
71% (01:41) correct 29% (01:56) wrong based on 138 sessions
### HideShow timer Statistics
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
1) Number of movie theaters that sold at least 100 tickets were more than half of the total number of movie theaters.
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
_________________
"Be challenged at EVERY MOMENT."
“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1343
Location: Malaysia
There are a total 10 movie theaters. Minimum number of tickets that ea [#permalink]
### Show Tags
01 Apr 2017, 01:31
1
ziyuen wrote:
There are a total 10 movie theaters. Minimum number of tickets that each movie theaters sold was 80 tickets. Is total number of average tickets sold more than 90 tickets?
1) Number of movie theaters that sold at least 100 tickets were more than half of the total number of movie theaters.
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
OFFICIAL EXPLANATION
FROM MathRevolution
If you modify the original condition and the question, the average number of tickets sold>90? And this becomes the total number of tickets sold>90(10)=900? Then, there are 10 variables (10 movie theaters) and 1 equation (minimum of 80 tickets were sold), and in order to match the number of variables to the number of equations, there must be 9 more equations. Therefore, E is most likely to be the answer. By solving con 1) and con 2) together,
If you assume that the number of movie theaters that sold at least 100 tickets as 6 and from the 6 movie theaters, 4 of them sold 106 tickets, from the total number of tickets sold=100(6)+106(4)=1,024>900, it is always yes, hence it is sufficient. The answer is C. However, this is an integer question, one of the key questions, so you apply CMT 4 (A: if you get C too easily, consider A and B) and CMT 4 (B: if you get A or B too easily, consider D).
For con 1), even if you assume the number of theaters that sold at least 100 tickets as 5, since the rest of the 5 movie theaters sold at least 80 tickets, so the total number of tickets sold becomes 100(5)+80(5)=900. Since it said that it is half greater than the total number of movie theaters, which is 10, so it is always greater than 900, hence yes, it is sufficient.
For con 2), if you assume that from the total of 10 movie theaters, 4 movie theaters sold 105 tickets and the 6 movie theaters sold 80 tickets, then the total number of tickets sold is 105(4)+80(6)=900, and since it said that the 4 movie theaters sold more than 105 tickets each, so it is always greater than 900, hence it is yes and sufficient. Therefore, the answer is D.
_________________
"Be challenged at EVERY MOMENT."
“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum | Please DO NOT post short answer in your post!
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 16759
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: There are a total 10 movie theaters. Minimum number of tickets that ea [#permalink]
### Show Tags
21 Dec 2017, 11:07
Hi All,
While this question is awkwardly-worded, it is essentially a 'limit' question. From the prompt, we know that each of the 10 theaters sold AT LEAST 80 tickets. We're asked if the average number of tickets sold for each theater was GREATER than 90? We can 'rewrite' this prompt - it's asking "Is the TOTAL number of tickets sold greater than 900?" This is a YES/NO question.
1) Number of movie theaters that sold at least 100 tickets were more than half of the total number of movie theaters.
This Fact tells us that MORE than half of the theaters sold at least 100 tickets. That would be at least 6 theaters (at the minimum)...
(6 theaters)(100 tickets each) = 600 tickets
From the prompt, we know that each theater sold AT LEAST 80 tickets, so we can find the minimum total that the other 4 theaters sold...
(4 theaters)(80 tickets each) = 320 tickets
Total MINIMUM number of tickets sold = 600 + 320 = 920 tickets. Increasing the number of theaters that sold at least 100 tickets would increase the total number of tickets sold. Thus, the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
2) In 4 movie theaters of the total number of movie theaters, number of each tickets sold were more than 105.
We can deal with Fact 2 in the same general way that we dealt with Fact 1...
(4 theaters)(more than 105 tickets each) = more than 420 tickets
Each of the other theaters sold AT LEAST 80 tickets, so we can find the minimum total that the other 6 theaters sold...
(6 theaters)(80 tickets each) = 480 tickets
Total MINIMUM number of tickets sold = (more than 420) + 480 = more than 900 tickets. Thus, the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT
GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: [email protected]
The Course Used By GMAT Club Moderators To Earn 750+
souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
Re: There are a total 10 movie theaters. Minimum number of tickets that ea [#permalink] 21 Dec 2017, 11:07
# There are a total 10 movie theaters. Minimum number of tickets that ea
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
| 4,999 | 6,699,462 |
<MASK>
<UNMASK>
<MASK>
Let $A$ be recursive and $B=\{(\langle\,M\,\rangle,\langle\,N\,\rangle)\mid M,N\text{ are TMs and }L(M)=L(N)\}$. This language (sometimes called $\text{EQ}_{\text{TM}}$) is known to be neither r.e nor co-r.e. However, let
<MASK>
<UNMASK>
# If $B$, $\overline{B}\neq \varnothing$ , then for every recursive set $A$, $A \leq_m B$
<MASK>
Since we have $A \leq_m B$ and $A$ is recursive, it means checking membership (characteristic function) in $A$ is recursive, so it means checking $f(x) \in B$ is recursive, right?
I guess it means $B$ must be recursive too and it means every recursive set is mapping reducible in to every other recursive set. Am I right?
<MASK>
This should be a comment on the original post by @Drupalist and on Ds D's answer, but it's too long. It's not enough to say that $A\le_M B$ requires $B$ be recursive or that one of $B, \overline B$ be r.e.
Let $A$ be recursive and $B=\{(\langle\,M\,\rangle,\langle\,N\,\rangle)\mid M,N\text{ are TMs and }L(M)=L(N)\}$. This language (sometimes called $\text{EQ}_{\text{TM}}$) is known to be neither r.e nor co-r.e. However, let
M(x) = N(x) =
return accept if x = 0
return reject
else
return accept
<MASK>
f(x) =
run D on x
if D(x) = accept
return (<M>, <M>)
else
return (<M>, <N>)
<MASK>
As I know the definition you gave is definition of many one reduction.
If at least one of $B$ and $B^c$ be r.e. (we can assume that B is r.e.) then we have a Turing machine $T_B$ which accept $B$ (I mean $\forall x \in B$ Turing Machine $T_B$ answer YES and halt) and another Turing machine $T_A$ that decides $A$. The computable function $f$ which we need is a function that for all input $x \in A$ which $T_A$ answers YES, $T_B$ on input $f(x)$ answer YES. it's possible to make such function so $A \leq_m B$.
<MASK>
• What is the problem with my reasoning in the update? Commented Mar 2, 2015 at 14:28
• I guess not right! I think it is true that :'every recursive set is mapping reducible in to every other recursive." but the thing that you are going to prove is something like to prove recursive is subset of r.e. Commented Mar 2, 2015 at 14:37
<UNMASK>
# If $B$, $\overline{B}\neq \varnothing$ , then for every recursive set $A$, $A \leq_m B$
<MASK>
UPDATE
$A,B$ are sets, $A \leq_m B$ if there is a computable function $f$ such that
<MASK>
Since we have $A \leq_m B$ and $A$ is recursive, it means checking membership (characteristic function) in $A$ is recursive, so it means checking $f(x) \in B$ is recursive, right?
I guess it means $B$ must be recursive too and it means every recursive set is mapping reducible in to every other recursive set. Am I right?
• What have you tried and where did you get stuck? Hint: what kind of reduction is $\leq_m$? Unwrap the definition. Commented Mar 2, 2015 at 11:55
• Like I said I really have no idea to prove this I guess $\leq_m$ is clear! it is mapping reduction Commented Mar 2, 2015 at 12:45
• I think Raphael wants you to spell out the definition of $\leq_m$. Then you can see how a computable set might be $\leq_m$ any other set (non-empty, nor full). Commented Mar 2, 2015 at 12:49
• You're correct in saying that every recursive set is mapping reducible to every non-trivial recursive set, but as I show below, there are other $B$s for which $A\le_M B$. Commented Apr 1, 2015 at 17:31
This should be a comment on the original post by @Drupalist and on Ds D's answer, but it's too long. It's not enough to say that $A\le_M B$ requires $B$ be recursive or that one of $B, \overline B$ be r.e.
Let $A$ be recursive and $B=\{(\langle\,M\,\rangle,\langle\,N\,\rangle)\mid M,N\text{ are TMs and }L(M)=L(N)\}$. This language (sometimes called $\text{EQ}_{\text{TM}}$) is known to be neither r.e nor co-r.e. However, let
M(x) = N(x) =
return accept if x = 0
return reject
else
return accept
<MASK>
f(x) =
run D on x
if D(x) = accept
return (<M>, <M>)
else
return (<M>, <N>)
<MASK>
As I know the definition you gave is definition of many one reduction.
If at least one of $B$ and $B^c$ be r.e. (we can assume that B is r.e.) then we have a Turing machine $T_B$ which accept $B$ (I mean $\forall x \in B$ Turing Machine $T_B$ answer YES and halt) and another Turing machine $T_A$ that decides $A$. The computable function $f$ which we need is a function that for all input $x \in A$ which $T_A$ answers YES, $T_B$ on input $f(x)$ answer YES. it's possible to make such function so $A \leq_m B$.
<MASK>
• What is the problem with my reasoning in the update? Commented Mar 2, 2015 at 14:28
• I guess not right! I think it is true that :'every recursive set is mapping reducible in to every other recursive." but the thing that you are going to prove is something like to prove recursive is subset of r.e. Commented Mar 2, 2015 at 14:37
<UNMASK>
# If $B$, $\overline{B}\neq \varnothing$ , then for every recursive set $A$, $A \leq_m B$
How to prove if $B$, $\overline{B}\neq \varnothing$ , then for every recursive set $A$, $A \leq_m B$ ?
it means every recursive set is mapping reducible to set $B\neq \aleph$. I really have no idea to prove this.
UPDATE
$A,B$ are sets, $A \leq_m B$ if there is a computable function $f$ such that
$A = \{ x \in N | f(x) \in B\}$
Since we have $A \leq_m B$ and $A$ is recursive, it means checking membership (characteristic function) in $A$ is recursive, so it means checking $f(x) \in B$ is recursive, right?
I guess it means $B$ must be recursive too and it means every recursive set is mapping reducible in to every other recursive set. Am I right?
• What have you tried and where did you get stuck? Hint: what kind of reduction is $\leq_m$? Unwrap the definition. Commented Mar 2, 2015 at 11:55
• Like I said I really have no idea to prove this I guess $\leq_m$ is clear! it is mapping reduction Commented Mar 2, 2015 at 12:45
• I think Raphael wants you to spell out the definition of $\leq_m$. Then you can see how a computable set might be $\leq_m$ any other set (non-empty, nor full). Commented Mar 2, 2015 at 12:49
• You're correct in saying that every recursive set is mapping reducible to every non-trivial recursive set, but as I show below, there are other $B$s for which $A\le_M B$. Commented Apr 1, 2015 at 17:31
This should be a comment on the original post by @Drupalist and on Ds D's answer, but it's too long. It's not enough to say that $A\le_M B$ requires $B$ be recursive or that one of $B, \overline B$ be r.e.
Let $A$ be recursive and $B=\{(\langle\,M\,\rangle,\langle\,N\,\rangle)\mid M,N\text{ are TMs and }L(M)=L(N)\}$. This language (sometimes called $\text{EQ}_{\text{TM}}$) is known to be neither r.e nor co-r.e. However, let
M(x) = N(x) =
return accept if x = 0
return reject
else
return accept
We have $(\langle\,M\,\rangle,\langle\,M\,\rangle)\in B$ and $(\langle\,M\,\rangle,\langle\,N\,\rangle)\in \overline{B}$. Then if $A$ is recursive, there is a decider TM, $D$, for A and if we define $f$ by
f(x) =
run D on x
if D(x) = accept
return (<M>, <M>)
else
return (<M>, <N>)
Then $f$ is Turing-computable and $x\in A\Longleftrightarrow f(x)\in B$, which is the definition of $A\le_M B$. So we can have such a reduction in every case where it's possible to find (not necessarily by a TM) two instances $y\in B$ and $n\in\overline{B}$.
Note that I'm not claiming that this technique will work for all possible $B$ nor am I claiming that this technique is the only possible one. All I've shown is that the problem isn't as simple as it seems at first glance.
As I know the definition you gave is definition of many one reduction.
If at least one of $B$ and $B^c$ be r.e. (we can assume that B is r.e.) then we have a Turing machine $T_B$ which accept $B$ (I mean $\forall x \in B$ Turing Machine $T_B$ answer YES and halt) and another Turing machine $T_A$ that decides $A$. The computable function $f$ which we need is a function that for all input $x \in A$ which $T_A$ answers YES, $T_B$ on input $f(x)$ answer YES. it's possible to make such function so $A \leq_m B$.
If both of $B$ and $B^c$ be not r.e. then there is no Turing machine that accepts them or computes them. since we can construct a TM for all functions it's not possible to have many one reduction between $A$ and $B$.
• What is the problem with my reasoning in the update? Commented Mar 2, 2015 at 14:28
• I guess not right! I think it is true that :'every recursive set is mapping reducible in to every other recursive." but the thing that you are going to prove is something like to prove recursive is subset of r.e. Commented Mar 2, 2015 at 14:37
| 2,622 | 6,699,463 |
<MASK>
<UNMASK>
<MASK>
Bookmark it
<MASK>
# final-1004-2008-revised - MATH 1004 Final Examination...
<MASK>
• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.
<MASK>
• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.
<MASK>
<UNMASK>
{[ promptMessage ]}
Bookmark it
<MASK>
final-1004-2008-revised
# final-1004-2008-revised - MATH 1004 Final Examination...
This preview shows pages 1–3. Sign up to view the full content.
<MASK>
This preview has intentionally blurred sections. Sign up to view the full version.
<MASK>
{[ snackBarMessage ]}
<MASK>
Kiran Temple University Fox School of Business ‘17, Course Hero Intern
• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.
Dana University of Pennsylvania ‘17, Course Hero Intern
• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.
Jill Tulane University ‘16, Course Hero Intern
<UNMASK>
{[ promptMessage ]}
Bookmark it
{[ promptMessage ]}
final-1004-2008-revised
# final-1004-2008-revised - MATH 1004 Final Examination...
This preview shows pages 1–3. Sign up to view the full content.
MATH 1004 Final Examination December 2008 1 Multiple-Choice Questions Please choose only one answer and insert in PENCIL in your Scantron sheet. 1. [3 marks] Evaluate lim x 0 sin 8 x 16 x . (a) 1 (b) 1 / 2 (c) 3 / 2 (d) 0 2. [4 marks] Let f ( x ) = Arctan x 2 + 1. Evaluate f (1). (Note that Arctan x and tan 1 x represent the same function). (a) f (1) = 1 3 (b) f (1) = 1 2 (c) f (1) = 1 3 2 (d) f (1) = 1 3. [3 marks] Let f ( x ) = 2 | x + 1 | + 1. Calculate L = lim h 0 f ( 1 + h ) f ( 1) h . 4. [4 marks] Find the derivative of the function f defined by f ( x ) = x 3 x . (a) 3 x 3 x (1 + ln x ) (b) 3 x 3 x 1 (c) 3 x 3 x (d) x 3 x (1 + 2 ln x )
This preview has intentionally blurred sections. Sign up to view the full version.
<MASK>
{[ snackBarMessage ]}
<MASK>
Kiran Temple University Fox School of Business ‘17, Course Hero Intern
• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.
Dana University of Pennsylvania ‘17, Course Hero Intern
• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.
Jill Tulane University ‘16, Course Hero Intern
<UNMASK>
{[ promptMessage ]}
Bookmark it
{[ promptMessage ]}
final-1004-2008-revised
# final-1004-2008-revised - MATH 1004 Final Examination...
This preview shows pages 1–3. Sign up to view the full content.
MATH 1004 Final Examination December 2008 1 Multiple-Choice Questions Please choose only one answer and insert in PENCIL in your Scantron sheet. 1. [3 marks] Evaluate lim x 0 sin 8 x 16 x . (a) 1 (b) 1 / 2 (c) 3 / 2 (d) 0 2. [4 marks] Let f ( x ) = Arctan x 2 + 1. Evaluate f (1). (Note that Arctan x and tan 1 x represent the same function). (a) f (1) = 1 3 (b) f (1) = 1 2 (c) f (1) = 1 3 2 (d) f (1) = 1 3. [3 marks] Let f ( x ) = 2 | x + 1 | + 1. Calculate L = lim h 0 f ( 1 + h ) f ( 1) h . 4. [4 marks] Find the derivative of the function f defined by f ( x ) = x 3 x . (a) 3 x 3 x (1 + ln x ) (b) 3 x 3 x 1 (c) 3 x 3 x (d) x 3 x (1 + 2 ln x )
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2 MATH 1004 Final Examination December 2008 5. [3 marks] A differentiable function f with a differentiable inverse, F , has the property that f (1) = 1 / 2 and f (1) = 1 / 2. What is the value of the derivative of the inverse of f at x = 1 / 2? That is, calculate F (1 / 2). 6. [3 marks] Let f ( x ) = 5 x +1 . Evaluate f (8). In other words, find the derivative of f at x = 8.
This is the end of the preview. Sign up to access the rest of the document.
{[ snackBarMessage ]}
### What students are saying
• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.
Kiran Temple University Fox School of Business ‘17, Course Hero Intern
• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.
Dana University of Pennsylvania ‘17, Course Hero Intern
• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.
Jill Tulane University ‘16, Course Hero Intern
| 1,599 | 6,699,464 |
# How do you find h(4) given h(n)=n^3+6n?
May 27, 2017
$h \left(4\right) = 88$
#### Explanation:
$\text{to evaluate " h(4)" substitute " n=4" into } h \left(n\right)$
$h \left(\textcolor{red}{4}\right) = {\left(\textcolor{red}{4}\right)}^{3} + \left(6 \times \textcolor{red}{4}\right)$
$\textcolor{w h i t e}{h \left(4\right)} = 64 + 24$
$\textcolor{w h i t e}{h \left(4\right)} = 88$
| 164 | 6,699,465 |
Pinexam——There is royal road to learning.
## What is the inequality that corresponds to the graph
(A) y > 3x + 2
(B) y ≤ 3x + 2
(C) y ≥ 3x + 2
(D) y < 3x + 2
(E) y < 3x + 2
Explanation
If the solutions to the equation (x + a)(x + b) = 0 are x = 8 and x = -, then a + b = ?
（A）-13 （B） -12 （C） -6.5 （D） 6.5 （E） 12
| 141 | 6,699,466 |
# How to fully factor a polynomial of 4th degree?
How to fully factor this polynomial? $$2x^4+3x^3-32x^2-48x$$
Can anyone describe the full steps to factor it? Thanks for the help.
• There is a general formula for a 4th order polynomial, but it is way too long to remember. Just try to guesstimate two solutions and then quadratic formula the rest. I'm sorry I can't be more specific. Commented Dec 13, 2014 at 22:38
• haha +1 for guesstimate:D
– Marc
Commented Dec 13, 2014 at 22:41
• As Edward has shown, there is not need for the quartic formula or guestimation. Cubic polynomials $ax^3+bx^2+cx+d$ (which this becomes when an $x$ is factored out) are easily factorable when $a/c=b/d$ or $a/b=c/d$. Commented Dec 13, 2014 at 23:20
$$2x^4+3x^3-32x^2-48x$$ $$=x(2x^3+3x^2-32x-48)$$ $$=x(x^2(2x+3)-16(2x+3))$$ $$=x(2x+3)(x^2-16)$$ $$=x(2x+3)(x-4)(x+4)$$
| 325 | 6,699,467 |
A rectangular prism is a three-dimensional shape, having six faces, where all the faces (top, bottom, and lateral faces) of the prism are rectangles such that every two opposite side faces are identical. … A rectangular prism is also known as a cuboid.
Also, What is the height of the prism?
Prisms are polyhedrons, three-dimensional solids with two identical and parallel polygonal bases or ends. The prism’s height is the distance between its two bases and is an important measurement in the calculation of the prism’s volume and surface area.
Hereof, What is a rectangular prism?
A rectangular prism is a 3D figure with 6 rectangular faces. To find the volume of a rectangular prism, multiply its 3 dimensions: length x width x height. The volume is expressed in cubic units. Created by Sal Khan.
Also to know What is the height of rectangular prism? Finding the Height of a Rectangular Prism With a Known Volume. equals the height of the prism. The base of a prism is one of its congruent sides. Since all opposite sides of a rectangular prism are congruent, any side can be used as the base, as long as you are consistent in your calculations.
Which figure is a rectangular prism?
A rectangular prism is a polyhedron with two congruent and parallel bases. It is also a cuboid. It has six faces, and all the faces are in a rectangle shape and have twelve edges. Because of its cross-section along the length, it is said to be a prism.
23 Related Questions Answers Found
## What is base in prism?
In its basic form, a prism is simply a triangular wedge of optical material (glass, plastic, etc). The base of the prism (or hypotenuse for those who think in geometry terms) is the thickest part, and the apex is the point, opposite the base on an equilateral triangle or prism.
## What is the height of a hexagonal prism?
Answer: The height of the hexagonal prism is 9 inches.
## What is an example of a prism?
Prism-shaped objects you’ll see in everyday life include ice cubes, barns and candy bars. The regular geometry of the prism makes it useful for designing buildings and simple products. You’ll also find prisms in the natural world, such as mineral crystals.
## What type of shape is a prism?
A prism is a 3D shape which has a constant cross section – both ends of the solid are the same shape and anywhere you cut parallel to these ends will give you the same shape. For example, in the prism below, the cross section is a hexagon. This is called a hexagonal prism.
## What objects are rectangular prisms?
Right rectangular prisms or cuboids are all around us. A few of the examples are books, boxes, buildings, bricks, boards, doors, containers, cabinets, mobiles, and laptops. Non-examples of right rectangular prism: This shape is a prism but its top and base do not have right angles in the shape.
## What is the dimension of a rectangular prism?
A rectangular prism is a 3-dimensional object with six rectangular faces.
## How do you calculate the height of a prism?
Height of Prism
So to calculate height, divide the volume of a prism by its base area. For this example, the volume of the prism is 500 and its base area is 50. Dividing 500 by 50 results in 10. The height of the prism is 10.
## What is a real life example of a rectangular prism?
Rectangular Prisms: Boxes and Tanks
Some examples in everyday life include: rectangular tissue boxes, juice boxes, laptop computers, school notebooks and binders, standard birthday presents — such as shirt boxes — cereal boxes and aquariums.
## Which of these is an example of a rectangular prism?
Right rectangular prisms or cuboids are all around us. A few of the examples are books, boxes, buildings, bricks, boards, doors, containers, cabinets, mobiles, and laptops. Non-examples of right rectangular prism: This shape is a prism but its top and base do not have right angles in the shape.
## How many sides are in a rectangular prism?
A rectangular prism has 8 vertices, 12 sides and 6 rectangular faces. All the opposite faces of a rectangular prism are equal.
## What is base out prism?
Base –out prism bar is placed in front of one eye in increasing steps until the patient reports diplopia or inability to fuse. Base-out determines convergence and base-in determines divergence amplitudes.
## How much is a prism?
Most temporary (Fresnel) prism glasses cost about \$250 to \$500 and permanent (ground) prism glasses cost about \$600 to \$1500. There are instances in which prism glasses cost several thousand dollars. Speak to your doctor about how much prism glasses cost and to discuss ways to reduce the out-of-pocket expenses.
## What is the formula of hexagonal?
A Regular hexagon has six sides and angles that are congruent and is made up of six equilateral triangles. The formula to find out the area of a regular hexagon is as given; Area = (3√3 s2)/ 2 where, ‘s’ represent the length of a side of the regular hexagon.
## What does a triangular prism look like?
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. … All cross-sections parallel to the base faces are the same triangle.
## How does a hexagonal prism look like?
A hexagonal prism is a 3D-shaped figure with the top and bottom shaped like a hexagon. It is a polyhedron with 8 faces, 18 edges, and 12 vertices where out of the 8 faces, 6 faces are in the shape of rectangles and 2 faces are in the shape of hexagons.
## What is a prism in math?
In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases.
## What is Angle of prism?
angle of prism is the angle between the two surface of the prism from which the light enters into the prism and from the light goes out after refraction.
## How are prisms classified?
Prisms are classified and named by the number of sides and flat planes in the solid figure. … A pyramid has one polygon as a base and sides that are triangular faces and connect in one vertex at the top. A cone has one circular base and the sides of the cone meet in one vertex at the top.
## What is the NSA prism?
The Ism Prism is not meant to “fix” personal bias, but to assist participants to understand their biases and to learn an alternative way of thinking about and addressing them. Being conscious of personal attitudes and behaviours helps make meaning of everyday experiences.
## What are 3 things pyramids and prisms have in common?
Similarities: Common Features
Prisms and pyramids are three-dimensional solid shapes that contain sides and faces that are polygons — two-dimensional shapes with straight sides. Both shapes fall under the large category — polyhedrons — because the sides and bases are polygons.
## What is a prism Year 1?
A prism is 3D shape that has two identical shapes facing one another. The shapes that make up the bases can be any polygons.
Tagged in:
| 1,581 | 6,699,468 |
# 329fall09hw1 - ECE 329 Consider the 3D vectors Homework 1...
This preview shows pages 1–2. Sign up to view the full content.
ECE 329 Homework 1 Due: Sept 1, 2009, 5PM 1. Review exercises on vectors: Consider the 3D vectors A = 3ˆ x y - z, B x y - ˆ C x - y + 3ˆ where ˆ x (1 , 0 , 0) , ˆ y (0 , 1 , 0) , and ˆ z (0 , 0 , 1) constitute an orthogonal set of unit vectors directed along the principal axes of a right-handed Cartesian coordinate system. Vectors can also be represented in component form — e.g., A = (3 , 1 , - 2) , which is equivalent to x y - z . Determine: a) The vector D A + B , b) The vector A + B - 4 C , c) The vector magnitude | A + B - 4 C | . d) The unit vector ˆ u along vector A +2 B - C . e) The dot product A · B . f) The cross product B × C . 2. A particle with charge q =1 C passing through the origin r =( x, y, z ) = 0 of the lab frame is observed to accelerate with forces F 1 = 3ˆ F 2 F 3 = 3ˆ z + 4ˆ y N when the velocity of the particle is v 1 =0 , v 2 = 1ˆ y, v 3 = 2ˆ z m s , in turns. Use the Lorentz force equation F = q ( E + v × B ) to determine the Felds E and B at the origin.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
## This note was uploaded on 02/21/2010 for the course ECE 329 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.
### Page1 / 2
329fall09hw1 - ECE 329 Consider the 3D vectors Homework 1...
This preview shows document pages 1 - 2. Sign up to view the full document.
View Full Document
Ask a homework question - tutors are online
| 526 | 6,699,469 |
<MASK>
<UNMASK>
<MASK>
square root of each side.
2x-3= +- 5
<UNMASK>
<MASK>
posted by on .
<MASK>
A) 2x - 3 = 5
B) 2x - 3 = -5
C) 2x - 3 = 5 and 2x - 3 = -5
<MASK>
square root of each side.
2x-3= +- 5
<UNMASK>
# Math
posted by on .
Which of the following is equivalent to (2x-3)^2=25 ?
A) 2x - 3 = 5
B) 2x - 3 = -5
C) 2x - 3 = 5 and 2x - 3 = -5
<MASK>
square root of each side.
2x-3= +- 5
<UNMASK>
# Math
posted by on .
Which of the following is equivalent to (2x-3)^2=25 ?
A) 2x - 3 = 5
B) 2x - 3 = -5
C) 2x - 3 = 5 and 2x - 3 = -5
• Math - ,
square root of each side.
2x-3= +- 5
| 274 | 6,699,470 |
Home > Math > Cardinal Splines Part 1
## Cardinal Splines Part 1
September 24, 2009
When anyone hears the term ‘cardinal spline’ for the first time, the most common question is why the name ‘cardinal?’ Yes, that was my first question way back in the day 🙂 As it happens, there is a subtle relation between the spline and the cardinal series [1]. I’ll leave it to interested readers to pursue the history and mathematics of the series as an aside.
I believe it was Schoenberg who said that the cardinal spline bridges the gap between linear splines and the cardinal series [2]. Again, I’ll leave the details to those who want to either read Schoenberg’s book or pursue the matter to whatever degree it is documented online. In this series, we will be interested in how to construct cardinal splines and get them into Degrafa.
First, let’s back up and look at cubic Hermite interpolation. Quadratic Hermite interpolation was discussed in the Quadratic Hermite Curve series (see the Degrafa page for links to the entire series). This case involved interpolating two points with a quadratic polynomial. The three degrees of freedom were resolved by forcing the curve to interpolate the two points and assigning a start tangent. This selection inferred an end tangent as shown below.
Start and End Tangents for a polynomial curve
Instead of selecting the start tangent, T0 and forcing the end tangent, T1, what if we allowed both tangents to be variable? This introduces an extra degree of freedom, allowing a cubic curve to be constructed. The cubic curve has more flexibility, but requires both T0 and T1 to be initially specified.
The natural question is why not develop a cubic Hermite spline? We could, although it requires two initial parameter selections. Most designers prefer to have a single element of control or have everything automatically done for them. An alternative approach is to examine specialized cases of Hermite interpolation where the tangents are automatically chosen.
In part 2, we will look at one such specification and how this method produces a computationally efficient interpolant with an adjustable tension parameter for shape control.
References:
[1] Higgins, J.R., “Five Short Stories About the Cardianal Series”, Bulletin of the American Mathematical Society, Vol. 12, No. 1, 1985.
[2] Schoenberg, I.J., “Cardinal Spline Interpolation”, The Mathematics Research Center, Univ. of Wisconsin-Madison, Regional Conference Series in Applied Mathematics, Capital City Press, 1993.
| 544 | 6,699,471 |
<MASK>
```Input: nums = [3,2,1,6,0,5]
Output: [6,3,5,null,2,0,null,null,1]
Explanation: The recursive calls are as follow:
- The largest value in [3,2,1,6,0,5] is 6. Left prefix is [3,2,1] and right suffix is [0,5].
- The largest value in [3,2,1] is 3. Left prefix is [] and right suffix is [2,1].
- Empty array, so no child.
- The largest value in [2,1] is 2. Left prefix is [] and right suffix is [1].
- Empty array, so no child.
- Only one element, so child is a node with value 1.
- The largest value in [0,5] is 5. Left prefix is [0] and right suffix is [].
- Only one element, so child is a node with value 0.
- Empty array, so no child.
```
<MASK>
``````
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
return DFS(nums, 0, nums.size());
}
<MASK>
### #4 Code Example with C# Programming
<MASK>
private int GetMaxIndex(int[] nums, int left, int right)
{
var result = left;
for (int i = left; i < right; i++)
if (nums[result] < nums[i]) result = i;
<MASK>
<UNMASK>
<MASK>
You are given an integer array `nums` with no duplicates. A maximum binary tree can be built recursively from `nums` using the following algorithm:
<MASK>
Return the maximum binary tree built from `nums`.
<MASK>
```Input: nums = [3,2,1,6,0,5]
Output: [6,3,5,null,2,0,null,null,1]
Explanation: The recursive calls are as follow:
- The largest value in [3,2,1,6,0,5] is 6. Left prefix is [3,2,1] and right suffix is [0,5].
- The largest value in [3,2,1] is 3. Left prefix is [] and right suffix is [2,1].
- Empty array, so no child.
- The largest value in [2,1] is 2. Left prefix is [] and right suffix is [1].
- Empty array, so no child.
- Only one element, so child is a node with value 1.
- The largest value in [0,5] is 5. Left prefix is [0] and right suffix is [].
- Only one element, so child is a node with value 0.
- Empty array, so no child.
```
<MASK>
```Input: nums = [3,2,1]
Output: [3,null,2,null,1]
```
Constraints:
<MASK>
``````
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
return DFS(nums, 0, nums.size());
}
TreeNode* DFS(vector& nums, int l, int r){
if(l == r) return NULL;
int maxPos = l;
for(int i = l; i < r; i++) if(nums[i] > nums[maxPos]) maxPos = i;
TreeNode* root = new TreeNode(nums[maxPos]);
root->left = DFS(nums, l, maxPos);
root->right = DFS(nums, maxPos + 1, r);
return root;
}
};
// O(n)
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
dequeq;
for(auto x: nums){
TreeNode* p = new TreeNode(x);
while(!q.empty() && q.back()->val < x) p->left = q.back(), q.pop_back();
if(!q.empty() && q.back()->val > x) q.back()->right = p;
q.push_back(p);
}
return q.front();
}
};
``````
Copy The Code &
<MASK>
cmd
nums = [3,2,1,6,0,5]
Output
<MASK>
### #2 Code Example with Java Programming
<MASK>
``````
class Solution {
public TreeNode constructMaximumBinaryTree(int[] nums) {
if (nums.length == 0) {
return null;
}
TreeNode root = helper(nums, 0, nums.length - 1);
return root;
}
private TreeNode helper(int[] nums, int start, int end) {
if (start > end) {
return null;
}
int maxIdx = getMaxIdx(nums, start, end);
TreeNode root = new TreeNode(nums[maxIdx]);
root.left = helper(nums, start, maxIdx - 1);
root.right = helper(nums, maxIdx + 1, end);
return root;
}
<MASK>
Input
<MASK>
Input
<MASK>
### #4 Code Example with C# Programming
<MASK>
private TreeNode ConstructMaximumBinaryTree(int[] nums, int left, int right)
{
if (left == right) return null;
var maxIndex = GetMaxIndex(nums, left, right);
<MASK>
return node;
}
private int GetMaxIndex(int[] nums, int left, int right)
{
var result = left;
for (int i = left; i < right; i++)
if (nums[result] < nums[i]) result = i;
return result;
}
}
}
``````
Copy The Code &
<MASK>
Output
<MASK>
<UNMASK>
## Algorithm
<MASK>
You are given an integer array `nums` with no duplicates. A maximum binary tree can be built recursively from `nums` using the following algorithm:
1. Create a root node whose value is the maximum value in `nums`.
2. Recursively build the left subtree on the subarray prefix to the left of the maximum value.
3. Recursively build the right subtree on the subarray suffix to the right of the maximum value.
Return the maximum binary tree built from `nums`.
Example 1:
```Input: nums = [3,2,1,6,0,5]
Output: [6,3,5,null,2,0,null,null,1]
Explanation: The recursive calls are as follow:
- The largest value in [3,2,1,6,0,5] is 6. Left prefix is [3,2,1] and right suffix is [0,5].
- The largest value in [3,2,1] is 3. Left prefix is [] and right suffix is [2,1].
- Empty array, so no child.
- The largest value in [2,1] is 2. Left prefix is [] and right suffix is [1].
- Empty array, so no child.
- Only one element, so child is a node with value 1.
- The largest value in [0,5] is 5. Left prefix is [0] and right suffix is [].
- Only one element, so child is a node with value 0.
- Empty array, so no child.
```
<MASK>
```Input: nums = [3,2,1]
Output: [3,null,2,null,1]
```
Constraints:
• `1 <= nums.length <= 1000`
• `0 <= nums[i] <= 1000`
• All integers in `nums` are unique.
## Code Examples
<MASK>
``````
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
return DFS(nums, 0, nums.size());
}
TreeNode* DFS(vector& nums, int l, int r){
if(l == r) return NULL;
int maxPos = l;
for(int i = l; i < r; i++) if(nums[i] > nums[maxPos]) maxPos = i;
TreeNode* root = new TreeNode(nums[maxPos]);
root->left = DFS(nums, l, maxPos);
root->right = DFS(nums, maxPos + 1, r);
return root;
}
};
// O(n)
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
dequeq;
for(auto x: nums){
TreeNode* p = new TreeNode(x);
while(!q.empty() && q.back()->val < x) p->left = q.back(), q.pop_back();
if(!q.empty() && q.back()->val > x) q.back()->right = p;
q.push_back(p);
}
return q.front();
}
};
``````
Copy The Code &
<MASK>
cmd
nums = [3,2,1,6,0,5]
Output
<MASK>
### #2 Code Example with Java Programming
```Code - Java Programming```
``````
class Solution {
public TreeNode constructMaximumBinaryTree(int[] nums) {
if (nums.length == 0) {
return null;
}
TreeNode root = helper(nums, 0, nums.length - 1);
return root;
}
private TreeNode helper(int[] nums, int start, int end) {
if (start > end) {
return null;
}
int maxIdx = getMaxIdx(nums, start, end);
TreeNode root = new TreeNode(nums[maxIdx]);
root.left = helper(nums, start, maxIdx - 1);
root.right = helper(nums, maxIdx + 1, end);
return root;
}
<MASK>
Input
<MASK>
cmd
[6,3,5,null,2,0,null,null,1]
### #3 Code Example with Python Programming
<MASK>
Input
cmd
nums = [3,2,1]
<MASK>
### #4 Code Example with C# Programming
<MASK>
``````
namespace LeetCode
{
public class _0654_MaximumBinaryTree
{
public TreeNode ConstructMaximumBinaryTree(int[] nums)
{
return ConstructMaximumBinaryTree(nums, 0, nums.Length);
}
private TreeNode ConstructMaximumBinaryTree(int[] nums, int left, int right)
{
if (left == right) return null;
var maxIndex = GetMaxIndex(nums, left, right);
<MASK>
return node;
}
private int GetMaxIndex(int[] nums, int left, int right)
{
var result = left;
for (int i = left; i < right; i++)
if (nums[result] < nums[i]) result = i;
return result;
}
}
}
``````
Copy The Code &
<MASK>
Output
<MASK>
<UNMASK>
## Algorithm
<MASK>
You are given an integer array `nums` with no duplicates. A maximum binary tree can be built recursively from `nums` using the following algorithm:
1. Create a root node whose value is the maximum value in `nums`.
2. Recursively build the left subtree on the subarray prefix to the left of the maximum value.
3. Recursively build the right subtree on the subarray suffix to the right of the maximum value.
Return the maximum binary tree built from `nums`.
Example 1:
```Input: nums = [3,2,1,6,0,5]
Output: [6,3,5,null,2,0,null,null,1]
Explanation: The recursive calls are as follow:
- The largest value in [3,2,1,6,0,5] is 6. Left prefix is [3,2,1] and right suffix is [0,5].
- The largest value in [3,2,1] is 3. Left prefix is [] and right suffix is [2,1].
- Empty array, so no child.
- The largest value in [2,1] is 2. Left prefix is [] and right suffix is [1].
- Empty array, so no child.
- Only one element, so child is a node with value 1.
- The largest value in [0,5] is 5. Left prefix is [0] and right suffix is [].
- Only one element, so child is a node with value 0.
- Empty array, so no child.
```
<MASK>
```Input: nums = [3,2,1]
Output: [3,null,2,null,1]
```
Constraints:
• `1 <= nums.length <= 1000`
• `0 <= nums[i] <= 1000`
• All integers in `nums` are unique.
## Code Examples
### #1 Code Example with C++ Programming
```Code - C++ Programming```
``````
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
return DFS(nums, 0, nums.size());
}
TreeNode* DFS(vector& nums, int l, int r){
if(l == r) return NULL;
int maxPos = l;
for(int i = l; i < r; i++) if(nums[i] > nums[maxPos]) maxPos = i;
TreeNode* root = new TreeNode(nums[maxPos]);
root->left = DFS(nums, l, maxPos);
root->right = DFS(nums, maxPos + 1, r);
return root;
}
};
// O(n)
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
dequeq;
for(auto x: nums){
TreeNode* p = new TreeNode(x);
while(!q.empty() && q.back()->val < x) p->left = q.back(), q.pop_back();
if(!q.empty() && q.back()->val > x) q.back()->right = p;
q.push_back(p);
}
return q.front();
}
};
``````
Copy The Code &
<MASK>
cmd
nums = [3,2,1,6,0,5]
Output
<MASK>
### #2 Code Example with Java Programming
```Code - Java Programming```
``````
class Solution {
public TreeNode constructMaximumBinaryTree(int[] nums) {
if (nums.length == 0) {
return null;
}
TreeNode root = helper(nums, 0, nums.length - 1);
return root;
}
private TreeNode helper(int[] nums, int start, int end) {
if (start > end) {
return null;
}
int maxIdx = getMaxIdx(nums, start, end);
TreeNode root = new TreeNode(nums[maxIdx]);
root.left = helper(nums, start, maxIdx - 1);
root.right = helper(nums, maxIdx + 1, end);
return root;
}
<MASK>
Input
<MASK>
cmd
[6,3,5,null,2,0,null,null,1]
### #3 Code Example with Python Programming
```Code - Python Programming```
``````
class Solution:
def constructMaximumBinaryTree(self, nums):
"""
:type nums: List[int]
:rtype: TreeNode
"""
if nums:
pos = nums.index(max(nums))
root = TreeNode(nums[pos])
root.left = self.constructMaximumBinaryTree(nums[:pos])
root.right = self.constructMaximumBinaryTree(nums[pos+1:])
return root
``````
Copy The Code &
Input
cmd
nums = [3,2,1]
Output
<MASK>
### #4 Code Example with C# Programming
```Code - C# Programming```
``````
namespace LeetCode
{
public class _0654_MaximumBinaryTree
{
public TreeNode ConstructMaximumBinaryTree(int[] nums)
{
return ConstructMaximumBinaryTree(nums, 0, nums.Length);
}
private TreeNode ConstructMaximumBinaryTree(int[] nums, int left, int right)
{
if (left == right) return null;
var maxIndex = GetMaxIndex(nums, left, right);
<MASK>
return node;
}
private int GetMaxIndex(int[] nums, int left, int right)
{
var result = left;
for (int i = left; i < right; i++)
if (nums[result] < nums[i]) result = i;
return result;
}
}
}
``````
Copy The Code &
<MASK>
Output
<MASK>
<UNMASK>
## Algorithm
Problem Name: 654. Maximum Binary Tree
You are given an integer array `nums` with no duplicates. A maximum binary tree can be built recursively from `nums` using the following algorithm:
1. Create a root node whose value is the maximum value in `nums`.
2. Recursively build the left subtree on the subarray prefix to the left of the maximum value.
3. Recursively build the right subtree on the subarray suffix to the right of the maximum value.
Return the maximum binary tree built from `nums`.
Example 1:
```Input: nums = [3,2,1,6,0,5]
Output: [6,3,5,null,2,0,null,null,1]
Explanation: The recursive calls are as follow:
- The largest value in [3,2,1,6,0,5] is 6. Left prefix is [3,2,1] and right suffix is [0,5].
- The largest value in [3,2,1] is 3. Left prefix is [] and right suffix is [2,1].
- Empty array, so no child.
- The largest value in [2,1] is 2. Left prefix is [] and right suffix is [1].
- Empty array, so no child.
- Only one element, so child is a node with value 1.
- The largest value in [0,5] is 5. Left prefix is [0] and right suffix is [].
- Only one element, so child is a node with value 0.
- Empty array, so no child.
```
Example 2:
```Input: nums = [3,2,1]
Output: [3,null,2,null,1]
```
Constraints:
• `1 <= nums.length <= 1000`
• `0 <= nums[i] <= 1000`
• All integers in `nums` are unique.
## Code Examples
### #1 Code Example with C++ Programming
```Code - C++ Programming```
``````
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
return DFS(nums, 0, nums.size());
}
TreeNode* DFS(vector& nums, int l, int r){
if(l == r) return NULL;
int maxPos = l;
for(int i = l; i < r; i++) if(nums[i] > nums[maxPos]) maxPos = i;
TreeNode* root = new TreeNode(nums[maxPos]);
root->left = DFS(nums, l, maxPos);
root->right = DFS(nums, maxPos + 1, r);
return root;
}
};
// O(n)
class Solution {
public:
TreeNode* constructMaximumBinaryTree(vector& nums) {
dequeq;
for(auto x: nums){
TreeNode* p = new TreeNode(x);
while(!q.empty() && q.back()->val < x) p->left = q.back(), q.pop_back();
if(!q.empty() && q.back()->val > x) q.back()->right = p;
q.push_back(p);
}
return q.front();
}
};
``````
Copy The Code &
Input
cmd
nums = [3,2,1,6,0,5]
Output
cmd
[6,3,5,null,2,0,null,null,1]
### #2 Code Example with Java Programming
```Code - Java Programming```
``````
class Solution {
public TreeNode constructMaximumBinaryTree(int[] nums) {
if (nums.length == 0) {
return null;
}
TreeNode root = helper(nums, 0, nums.length - 1);
return root;
}
private TreeNode helper(int[] nums, int start, int end) {
if (start > end) {
return null;
}
int maxIdx = getMaxIdx(nums, start, end);
TreeNode root = new TreeNode(nums[maxIdx]);
root.left = helper(nums, start, maxIdx - 1);
root.right = helper(nums, maxIdx + 1, end);
return root;
}
private int getMaxIdx(int[] nums, int start, int end) {
int maxVal = Integer.MIN_VALUE;
int maxIdx = -1;
while (start <= end) {
if (nums[start] > maxVal) {
maxVal = nums[start];
maxIdx = start;
}
start++;
}
return maxIdx;
}
}
``````
Copy The Code &
Input
cmd
nums = [3,2,1,6,0,5]
Output
cmd
[6,3,5,null,2,0,null,null,1]
### #3 Code Example with Python Programming
```Code - Python Programming```
``````
class Solution:
def constructMaximumBinaryTree(self, nums):
"""
:type nums: List[int]
:rtype: TreeNode
"""
if nums:
pos = nums.index(max(nums))
root = TreeNode(nums[pos])
root.left = self.constructMaximumBinaryTree(nums[:pos])
root.right = self.constructMaximumBinaryTree(nums[pos+1:])
return root
``````
Copy The Code &
Input
cmd
nums = [3,2,1]
Output
cmd
[3,null,2,null,1]
### #4 Code Example with C# Programming
```Code - C# Programming```
``````
namespace LeetCode
{
public class _0654_MaximumBinaryTree
{
public TreeNode ConstructMaximumBinaryTree(int[] nums)
{
return ConstructMaximumBinaryTree(nums, 0, nums.Length);
}
private TreeNode ConstructMaximumBinaryTree(int[] nums, int left, int right)
{
if (left == right) return null;
var maxIndex = GetMaxIndex(nums, left, right);
var node = new TreeNode(nums[maxIndex]);
node.left = ConstructMaximumBinaryTree(nums, left, maxIndex);
node.right = ConstructMaximumBinaryTree(nums, maxIndex + 1, right);
return node;
}
private int GetMaxIndex(int[] nums, int left, int right)
{
var result = left;
for (int i = left; i < right; i++)
if (nums[result] < nums[i]) result = i;
return result;
}
}
}
``````
Copy The Code &
Input
cmd
nums = [3,2,1]
Output
cmd
[3,null,2,null,1]
| 4,571 | 6,699,472 |
# 2.1 Moving Average Models (MA models)
2.1 Moving Average Models (MA models)
Time series models known as ARIMA models may include autoregressive terms and/or moving average terms. In Week 1, we learned an autoregressive term in a time series model for the variable $$x_t$$ is a lagged value of $$x_t$$. For instance, a lag 1 autoregressive term is $$x_{t-1}$$(multiplied by a coefficient). This lesson defines moving average terms.
A moving average term in a time series model is a past error (multiplied by a coefficient).
Let $$w_t \overset{iid}{\sim} N(0, \sigma^2_w)$$, meaning that the wt are identically, independently distributed, each with a normal distribution having mean 0 and the same variance.
The 1st order moving average model, denoted by MA(1) is:
$$x_t = \mu + w_t +\theta_1w_{t-1}$$
The 2nd order moving average model, denoted by MA(2) is:
$$x_t = \mu + w_t +\theta_1w_{t-1}+\theta_2w_{t-2}$$
The qth order moving average model, denoted by MA(q) is:
$$x_t = \mu + w_t +\theta_1w_{t-1}+\theta_2w_{t-2}+\dots + \theta_qw_{t-q}$$
Note!
Many textbooks and software programs define the model with negative signs before the $$\theta$$ terms. This doesn’t change the general theoretical properties of the model, although it does flip the algebraic signs of estimated coefficient values and (unsquared) $$\theta$$ terms in formulas for ACFs and variances. You need to check your software to verify whether negative or positive signs have been used in order to correctly write the estimated model. R uses positive signs in its underlying model, as we do here.
### Theoretical Properties of a Time Series with an MA(1) Model
• Mean is $$E(x_t)=\mu$$
• Variance is $$Var(x_t)= \sigma^2_w(1+\theta^2_1)$$
• Autocorrelation function (ACF) is:
$$\rho_1 = \dfrac{\theta_1}{1+\theta^2_1}, \text{ and } \rho_h = 0 \text{ for } h \ge 2$$
Note!
That the only nonzero value in the theoretical ACF is for lag 1. All other autocorrelations are 0. Thus a sample ACF with a significant autocorrelation only at lag 1 is an indicator of a possible MA(1) model.
For interested students, proofs of these properties are in the appendix.
## Example 2-1
Suppose that an MA(1) model is $$x_t=10+w_t+.7w_{t-1}$$, where $$w_t \overset{iid}{\sim} N(0,1)$$. Thus the coefficient $$\theta_1=0.7$$. The theoretical ACF is given by:
$$\rho_1 = \dfrac{0.7}{1+0.7^2} = 0.4698, \text{ and } \rho_h = 0 \text{ for all lags } h \ge 2$$
A plot of this ACF follows:
The plot just shown is the theoretical ACF for an MA(1) with $$\theta_1=0.7$$. In practice, a sample won’t usually provide such a clear pattern. Using R, we simulated n = 100 sample values using the model $$x_t=10+w_t+.7w_{t-1}$$ where $$w_t \overset{iid}{\sim} N(0,1)$$. For this simulation, a time series plot of the sample data follows. We can’t tell much from this plot.
The sample ACF for the simulated data follows. We see a “spike” at lag 1 followed by generally non-significant values for lags past 1. Note that the sample ACF does not exactly match the theoretical pattern of the underlying MA(1), which is that all autocorrelations for lags above lag 1 equal zero. A different sample would have a slightly different sample ACF shown below, but would likely have the same broad features.
## Theoretical Properties of a Time Series with an MA(2) Model
For the MA(2) model, theoretical properties are the following:
• Mean is $$E(x_t)=\mu$$
• Variance is $$Var(x_t)=\sigma^2_w(1+\theta^2_1+\theta^2_2)$$
• Autocorrelation function (ACF) is:
$$\rho_1 = \dfrac{\theta_1+\theta_1\theta_2}{1+\theta^2_1 +\theta^2_2}, \text{ } \rho_2 = \dfrac{\theta_2}{1+\theta^2_1 +\theta^2_2}, \text{ and } \rho_h = 0 \text{ for } h \ge 3$$
Note!
The only nonzero values in the theoretical ACF are for lags 1 and 2. Autocorrelations for higher lags are 0. So, a sample ACF with significant autocorrelations at lags 1 and 2, but non-significant autocorrelations for higher lags indicates a possible MA(2) model.
## Example 2-2
Consider the MA(2) model $$x_t=10+w_t+.5w_{t-1}+.3w_{t-2}$$, where $$w_t \overset{iid}{\sim} N(0,1)$$. The coefficients are $$\theta_1=0.5$$ and $$\theta_2= 0.3$$. Because this is an MA(2), the theoretical ACF will have nonzero values only at lags 1 and 2.
Values of the two nonzero autocorrelations are:
$$\rho_1 = \dfrac{0.5+0.5 \times 0.3}{1+0.5^2 +0.3^2} = 0.4851 \text{ and } \rho_2 = \dfrac{0.3}{1+0.5^2 +0.3^2}= 0.2239$$
A plot of the theoretical ACF follows:
As nearly always is the case, sample data won’t behave quite so perfectly as theory. We simulated n = 150 sample values for the model $$x_t=10+w_t+.5w_{t-1}+.3w_{t-2}$$, where $$w_t \overset{iid}{\sim} N(0,1)$$. The time series plot of the data follows. As with the time series plot for the MA(1) sample data, you can’t tell much from it.
The sample ACF for the simulated data follows. The pattern is typical for situations where an MA(2) model may be useful. There are two statistically significant “spikes” at lags 1 and 2 followed by non-significant values for other lags. Note that due to sampling error, the sample ACF did not match the theoretical pattern exactly.
## ACF for General MA(q) Models
A property of MA(q) models in general is that there are nonzero autocorrelations for the first q lags and autocorrelations = 0 for all lags > q.
Non-uniqueness of connection between values of $$\theta_1$$ and $$\rho_1$$ in MA(1) Model.
In the MA(1) model, for any value of $$\theta_1$$, the reciprocal $$1/\theta_1$$ gives the same value for:
$$\rho_1 = \dfrac{\theta_1}{1+\theta^2_1}$$
As an example, use +0.5 for $$\theta_1$$, and then use 1/(0.5) = 2 for $$\theta_1$$. You’ll get $$\rho_1 = 0.4$$ in both instances.
To satisfy a theoretical restriction called invertibility, we restrict MA(1) models to have values with absolute value less than 1. In the example just given, $$\theta_1 = 0.5$$ will be an allowable parameter value, whereas $$\theta_1 = 1/0.5 = 2$$ will not.
### Invertibility of MA models
An MA model is said to be invertible if it is algebraically equivalent to a converging infinite order AR model. By converging, we mean that the AR coefficients decrease to 0 as we move back in time.
Invertibility is a restriction programmed into time series software used to estimate the coefficients of models with MA terms. It’s not something that we check for in the data analysis. Additional information about the invertibility restriction for MA(1) models is given in the appendix.
For a MA(q) model with a specified ACF, there is only one invertible model. The necessary condition for invertibility is that the $$\theta$$ coefficients have values such that the equation $$1-\theta_1y-...-\theta_qy^q=0$$ has solutions for $$y$$ that fall outside the unit circle.
## R Code for the Examples
In Example 1, we plotted the theoretical ACF of the model $$x_t=10+w_t+.7w_{t-1}$$, and then simulated n = 150 values from this model and plotted the sample time series and the sample ACF for the simulated data. The R commands used to plot the theoretical ACF were:
acfma1=ARMAacf(ma=c(0.7), lag.max=10) # 10 lags of ACF for MA(1) with theta1 = 0.7
lags=0:10 #creates a variable named lags that ranges from 0 to 10.
plot(lags,acfma1,xlim=c(1,10), ylab="r",type="h", main = "ACF for MA(1) with theta1 = 0.7")
abline(h=0) #adds a horizontal axis to the plot
The first command determines the ACF and stores it in an object named acfma1 (our choice of name).
The plot command (the 3rd command) plots lags versus the ACF values for lags 1 to 10. The ylab parameter labels the y-axis and the "main" parameter puts a title on the plot.
To see the numerical values of the ACF simply use the command acfma1.
The simulation and plots were done with the following commands:
xc=arima.sim(n=150, list(ma=c(0.7))) #Simulates n = 150 values from MA(1)
x=xc+10 # adds 10 to make mean = 10. Simulation defaults to mean = 0.
plot(x,type="b", main="Simulated MA(1) data")
acf(x, xlim=c(1,10), main="ACF for simulated sample data")
In Example 2, we plotted the theoretical ACF of the model $$x_t=10+w_t+.5w_{t-1}+.3w_{t-2}$$, and then simulated n = 150 values from this model and plotted the sample time series and the sample ACF for the simulated data. The R commands used were:
acfma2=ARMAacf(ma=c(0.5,0.3), lag.max=10)
acfma2
lags=0:10
plot(lags,acfma2,xlim=c(1,10), ylab="r",type="h", main = "ACF for MA(2) with theta1 = 0.5,theta2=0.3")
abline(h=0)
xc=arima.sim(n=150, list(ma=c(0.5, 0.3)))
x=xc+10
plot(x, type="b", main = "Simulated MA(2) Series")
acf(x, xlim=c(1,10), main="ACF for simulated MA(2) Data")
## Appendix: Proof of Properties of MA(1)
For interested students, here are proofs for theoretical properties of the MA(1) model.
The 1st order moving average model , denoted by MA(1) is $$x_t=\mu+w_t+\theta_1w_{t-1}$$, where $$w_t \overset{iid}{\sim} N(0,\sigma^2_w)$$.
Mean: $$E(x_t)=E(\mu + w_t + \theta_1 w_{t-1} ) = \mu + 0 + (\theta_1)(0) = \mu$$
Variance: $$\text{Var}(x_t) = \text{Var} (\mu + w_t + \theta_1 w_{t-1}) = 0 + \text{Var}(w_t) + \text{Var}(\theta_1w_{t-1}) = \sigma^2_w + \theta^2_1\sigma^2_w = (1+\theta^2_1)\sigma^2_w$$
ACF: Consider the covariance between $$x_t$$ and $$x_{t-h}$$. This is $$E(x_t-\mu)(x_{t-h}-\mu)$$, which equals
$$E[(w_t + \theta_1w_{t-1})(w_{t-h}+\theta_1w_{t-h-1})] = E[w_tw_{t-h} + \theta_1w_{t-1}w_{t-h} +\theta_1w_tw_{t-h-1}+\theta^2_1 w_{t-1}w_{t-h-1}]$$
When $$h=1$$, the previous expression = $$\theta_1 \sigma_w^2$$. For any $$h \ge 2$$, the previous expression = 0. The reason is that, by definition of independence of the $$w_t$$, $$E(w_k w_k)=0$$ for any $$k \ne j$$. Further, because the $$w_t$$ have mean 0, $$E(w_k w_k)=E(w_j^2)=\sigma_w^2$$.
For a time series,
$$\rho_h = \dfrac{\text{Covariance for lag h}}{\text{Variance}}$$
Apply this result to get the ACF given above.
### Invertibility Restriction:
An invertible MA model is one that can be written as an infinite order AR model that converges so that the AR coefficients converge to 0 as we move infinitely back in time. We’ll demonstrate invertibility for the MA(1) model.
The MA(1) model can be written as $$x_t-\mu=w_t+\theta_1 w_{t-1}$$.
If we let $$z_t=x_t-\mu$$, then the MA(1) model is
(1) $$z_t = w_t +\theta_1w_{t-1}$$.
At time $$t-1$$, the model is $$z_{t-1}=w_{t-1}+\theta_1 w_{t-2}$$ which can be reshuffled to
(2) $$w_{t-1} = z_{t-1}-\theta_1w_{t-2}$$.
We then substitute relationship (2) for $$w_{t-1}$$ in equation (1)
(3) $$z_t = w_t +\theta_1(z_{t-1}-\theta_1w_{t-2}) = w_t +\theta_1z_{t-1} -\theta^2w_{t-2}$$
At time $$t-2$$, equation (2) becomes
(4) $$w_{t-2} = z_{t-2}-\theta_1w_{t-3}$$.
We then substitute relationship (4) for $$w_{t-2}$$ in equation (3)
$$z_t = w_t +\theta_1 z_{t-1}-\theta^2_1w_{t-2} = w_t + \theta_1z_{t-1} -\theta^2_1(z_{t-2}-\theta_1w_{t-3}) = w_t +\theta_1z_{t-1} -\theta_1^2z_{t-2}+\theta^3_1w_{t-3}$$
If we were to continue (infinitely), we would get the infinite order AR model
$$z_t = w_t +\theta_1 z_{t-1} - \theta^2_1z_{t-2} +\theta^3_1z_{t-3}-\theta^4_1z_{t-4}+\dots$$
Note!
However, that if $$\lvert\theta_1\rvert \ge 1$$, the coefficients multiplying the lags of $$z$$ will increase (infinitely) in size as we move back in time. To prevent this, we need $$\lvert\theta_1\rvert < 1$$. This is the condition for an invertible MA(1) model.
### Infinite Order MA model
In week 3, we’ll see that an AR(1) model can be converted to an infinite order MA model:
$$x_t -\mu = w_t +\phi_1w_{t-1}+\phi^2_1w_{t-2} + \dots + \phi^k_1 w_{t-k} +\dots = \sum_{j=0}^{\infty} \phi^j_1w_{t-j}$$
This summation of past white noise terms is known as the causal representation of an AR(1). In other words, $$x_t$$ is a special type of MA with an infinite number of terms going back in time. This is called an infinite order MA or MA($$\infty$$). A finite order MA is an infinite order AR and any finite order AR is an infinite order MA.
Recall in Week 1, we noted that a requirement for a stationary AR(1) is that $$\lvert\phi_1\rvert< 1$$. Let’s calculate the $$\text{Var}(x_t)$$ using the causal representation.
$$\text{Var}(x_t) = \text{Var} \left(\sum_{j=0}^{\infty} \phi^j_1w_{t-j} = \sum_{j=0}^{\infty}\text{Var}(\phi^j_1w_{t-j}) = \sum_{j=0}^{\infty}\phi^{2j}_1\sigma^2_w = \sigma^2_w \sum_{j=0}^{\infty}\phi^{2j}_1 = \frac{\sigma^2_w}{1-\phi^2_1} \right)$$
This last step uses a basic fact about geometric series that requires $$\lvert\phi_1\rvert <1$$; otherwise the series diverges.
[1] Link ↥ Has Tooltip/Popover Toggleable Visibility
| 4,235 | 6,699,473 |
## What does an index number mean
Differences in how index values are calculated can occur depending on the the number of shares included in the index, and each stock's weight in the index is An index, in your example, refers to a position within an ordered list. Python strings can be thought of as lists of characters; each character is
The formulas below show how INDEX can be used to get a value: The area_num is argument is supplied as a number that acts like a numeric index. The first What is a person's ideal weight, and how do height, age, and other factors affect However, there is not one ideal healthy weight for each person, because a number of Body mass index (BMI) is a common tool for deciding whether a person has they have more muscle mass, but this does not mean they are overweight. 21 May 2019 The UV Index scale used in the United States conforms with international by the World Health OrganizationExitLearn how to read the UV index Scale to help you You can safely stay outside using minimal sun protection. As your database tables grow, an increasing number of rows need to be inspected each time which can slow the overall performance of the database and In weighted average form, the index would combine series of the volume of work What is necessary is that the great bulk of a country's industrial production in. An index number is the measure of change in a variable (or group of variables) over time. It is typically used in economics to measure trends in a wide variety of areas including: stock market prices, cost of living, industrial or agricultural production, and imports.
## In weighted average form, the index would combine series of the volume of work What is necessary is that the great bulk of a country's industrial production in.
Index numbers possess much practical importance in measuring changes in the cost We can know from such as index number the actual condition of different Therefore, it is useful to understand how index numbers are constructed and and Financial Ratios and Price Indices · The Geometric, and the Harmonic Means. 11 Dec 2014 Learn how useful this statistical number is in the real world. You will also see unique items. What exactly does this index number mean then? That is, BLS sets the average index level (representing the average price level)— for the 36-month period covering How to use it Movements of the index from one date to another can be expressed as changes in index points (simply, the
### Index numbers possess much practical importance in measuring changes in the cost We can know from such as index number the actual condition of different
18 Feb 2020 Index number definition is - a number used to indicate change in magnitude Views expressed in the examples do not represent the opinion of Indicator of average percentage change in a series of figures where one figure ( called the base) is assigned an arbitrary value of 100, and other figures are Index numbers are a useful way of expressing economic data time series and comparing / contrasting What are some frequently used examples of index numbers? Average motor insurance premiums paid in the United Kingdom The base value does not necessarily have to be a specific year - it can also be a country. Index numbers are used to measure changes and simplify comparisons. are interested in how changes in the monetary value of economic transactions can be Mathematically, an economic index number is an average of the many different 25 Sep 2001 A measure reflecting the average of the proportionate changes in the or the number of services; the quantity index has no meaning from an
### The noun INDEX NUMBER has 1 sense: 1. a number or ratio (a value on a scale of measurement) derived from a series of observed facts; can reveal relative changes as a function of time Familiarity information: INDEX NUMBER used as a noun is very rare.
As your database tables grow, an increasing number of rows need to be inspected each time which can slow the overall performance of the database and In weighted average form, the index would combine series of the volume of work What is necessary is that the great bulk of a country's industrial production in. An index number is the measure of change in a variable (or group of variables) over time. It is typically used in economics to measure trends in a wide variety of areas including: stock market prices, cost of living, industrial or agricultural production, and imports.
## As your database tables grow, an increasing number of rows need to be inspected each time which can slow the overall performance of the database and
The noun INDEX NUMBER has 1 sense: 1. a number or ratio (a value on a scale of measurement) derived from a series of observed facts; can reveal relative changes as a function of time Familiarity information: INDEX NUMBER used as a noun is very rare. An index number is a measure that summarizes changes in prices, levels of activity, quantities, lengths, and other measurements. Index numbers are created to study the change in effects of factors that cannot be measured directly. index number. n. A number indicating change in magnitude, as of price, wage, employment, or production shifts, relative to the magnitude at a specified point usually taken as 100.
Index numbers are used to measure changes and simplify comparisons. are interested in how changes in the monetary value of economic transactions can be Mathematically, an economic index number is an average of the many different 25 Sep 2001 A measure reflecting the average of the proportionate changes in the or the number of services; the quantity index has no meaning from an 13 Oct 2016 The composite index number is a weighted mean of the elementary index numbers in which the weighting represents the "mass" of the
| 1,164 | 6,699,474 |
<MASK>
Customized
for You
<MASK>
Everyone of us have understood that without him helping us
<MASK>
Question Stats:
<MASK>
Everyone of us have understood that without him helping us we would not have succeeded in our program over the past six months
<MASK>
a. A thoroughly frightened child was seen by her cowering in the corner of the room.
b. Cowering in the corner of the room a thoroughly frightened child was seen by her.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her.
e. She saw a thoroughly frightened child who was cowering in the corner of the room.
Intern
Joined: 20 Jan 2006
Posts: 14
<MASK>
16 Mar 2006, 02:41
1) ^ D ^
<MASK>
It sounds like She was cowering .... of the room. But it was not intended from original sentence.
Director
Joined: 27 Feb 2006
Posts: 622
<MASK>
C)She saw, cowering in the corner of the room, a thoroughly frightened child.
<MASK>
<UNMASK>
It is currently 23 Jun 2017, 14:22
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
Events & Promotions
Events & Promotions in June
Open Detailed Calendar
Everyone of us have understood that without him helping us
Author Message
TAGS:
Hide Tags
Director
Joined: 02 Mar 2006
Posts: 575
Location: France
Everyone of us have understood that without him helping us [#permalink]
Show Tags
15 Mar 2006, 16:25
1
This post was
BOOKMARKED
00:00
Difficulty:
(N/A)
Question Stats:
100% (01:05) correct 0% (00:00) wrong based on 12 sessions
HideShow timer Statistics
Everyone of us have understood that without him helping us we would not have succeeded in our program over the past six months
a. Everyone of us have understood that without him helping us
b. Everyone of us has understood that without his helping us
c. Everyone of us have understood that without his help
d. Everyone of us has understood that without him helping us
e. Every single one of us have understood that without him helping us
A thoroughly frightened child was seen by her cowering in the corner of the room.
a. A thoroughly frightened child was seen by her cowering in the corner of the room.
b. Cowering in the corner of the room a thoroughly frightened child was seen by her.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her.
e. She saw a thoroughly frightened child who was cowering in the corner of the room.
Intern
Joined: 20 Jan 2006
Posts: 14
Show Tags
15 Mar 2006, 17:11
1. 'Everyone' is singular , so 'has' is required.
A & C are out.
E is wordy
Between B & D.
usage of 'his helping us' is wrong.
Will opt for D.
<MASK>
Show Tags
15 Mar 2006, 17:18
(1) D for Subject-Verb agreement and
(2) E - in active voice and uses modifier correctly .
Intern
Joined: 20 Jan 2006
Posts: 14
Show Tags
<MASK>
Thanks chuckle.
GMAT Club Legend
Joined: 07 Jul 2004
Posts: 5043
Location: Singapore
Show Tags
<MASK>
Show Tags
15 Mar 2006, 20:00
Q2)
a. awkward
b. awkward
d. Reject -> passive voice
Between c and e, e looks a better choice though c isn't all that bad.
Director
Joined: 17 Sep 2005
Posts: 903
<MASK>
15 Mar 2006, 23:14
SC1: IMO it's D. 'has' is required and 'without him helping us' is correct.
SC2: It has to be E. Simple and concise.
Regards,
Brajesh
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
16 Mar 2006, 02:41
1) ^ D ^
We need "has" after everyone.
a. Everyone of us have understood that without him helping us
b. Everyone of us has understood that without his helping us ---> "his" is wrong here
c. Everyone of us have understood that without his help
e. Every single one of us have understood that without him helping us
2) ^ E ^
a. A thoroughly frightened child was seen by her cowering in the corner of the room. ---> wrong modifier
b. Cowering in the corner of the room a thoroughly frightened child was seen by her. ---> we need a comma after "room"
c. She saw, cowering in the corner of the room, a thoroughly frightened child. ---> wrong modifier
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her. ---> passive
Manager
Joined: 06 Aug 2005
Posts: 90
Location: Minneapolis
Show Tags
<MASK>
Show Tags
16 Mar 2006, 10:44
in the second question if "cowering in the corner of the room" is the modifier for "she", then is nothing wrong with it and is correct one.
Intern
Joined: 24 Jan 2006
Posts: 10
Show Tags
<MASK>
what's wrong with C?
it's nice and short, and makes sense. i don't see a modifier problem. please elaborate on the incorrectness of C.
_________________
dyin inside but outside you lookin fearless...
www.djcham.com
= )
Senior Manager
Joined: 22 Nov 2005
Posts: 474
Show Tags
16 Mar 2006, 13:53
djcham wrote:
i choose C for question 2.
<MASK>
it's nice and short, and makes sense. i don't see a modifier problem. please elaborate on the incorrectness of C.
C has modifier problem.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
It sounds like She was cowering .... of the room. But it was not intended from original sentence.
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
16 Mar 2006, 14:13
To Ketrin1 and djcham
C)She saw, cowering in the corner of the room, a thoroughly frightened child.
This choice has a modifier problem. It sounds as if the cowering person is "she". However, the person cowering should be "the throughly frightened child"
Manager
Joined: 13 Dec 2005
Posts: 224
Location: Milwaukee,WI
Show Tags
16 Mar 2006, 14:19
1 D everyone requires has ..... and him is correct usage
2. E active,concise and clear
Director
Joined: 02 Mar 2006
Posts: 575
Location: France
Show Tags
18 Mar 2006, 05:56
No one has answered like OA, but Ketrin1 and djcham for Q2.
So the OA is:
Q1. B. Everyone is singular and requires the singular has. The preposition without requires the gerund helping preceded by the possessive his.
Q2. C. This is suspenseful sentence since what "she saw" is held off to the very last word in the sentence. Also, an active verb, saw, is preferable to the passive was seen.
Manager
Joined: 20 Feb 2006
Posts: 213
Show Tags
18 Mar 2006, 10:20
Vow!!! Both of my answers are wrong
I can at least attempt to understand the answer for Question 1 to be B.
But, I still do not understand the answer for Question 2.
Won't the "Cowering" modify "she" and not the "child" in C???
SVP
Joined: 14 Dec 2004
Posts: 1689
Show Tags
18 Mar 2006, 11:01
I disagree on SC2 OA, "E" is the best.
<MASK>
What is the source of these questions, karlfurt?
Manager
Joined: 04 Sep 2005
Posts: 141
Location: Fringes of the Boreal, Canada
Show Tags
18 Mar 2006, 12:58
Q2: E doesn't seem right to me. The "who was" is redundant because the past tense is already used in "She saw". The sentence would sound better if "who was" was omitted: eg. "She saw a thoroughly frightened child cowering in the corner of the room."
_________________
"To hell with circumstances; I create opportunities." - Bruce Lee
Senior Manager
Joined: 08 Sep 2004
Posts: 257
Location: New York City, USA
Show Tags
18 Mar 2006, 13:10
Both of my answers were wrong too. But I agree with the OAs. First one is definitely B (how could I pick 'him helping' ).
The second question is even more interesting. The important thing to note is that 'cowering in the corner of the room' modifies the 'frightened child'. D is definitely wrong because of passive voice. E is wordy. The modifier 'who was' immidiately follows 'frightened child' without comma. It can be removed without changing the meaning of the sentence in any way. But ofcourse all this is after getting to know the official answer. As they say, hindsight is 20-20.
- Vipin
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
18 Mar 2006, 16:17
vipin7um wrote:
Both of my answers were wrong too. But I agree with the OAs. First one is definitely B (how could I pick 'him helping' ).
The second question is even more interesting. The important thing to note is that 'cowering in the corner of the room' modifies the 'frightened child'. D is definitely wrong because of passive voice. E is wordy. The modifier 'who was' immidiately follows 'frightened child' without comma. It can be removed without changing the meaning of the sentence in any way. But ofcourse all this is after getting to know the official answer. As they say, hindsight is 20-20.
- Vipin
I agree with you for the first question.
However, for the second one I still support ^ E ^. In C ,even if "who was" is omitted, we cannot think that it modifes the child because modifier is in wrong place.
18 Mar 2006, 16:17
<MASK>
Similar topics Replies Last post
Similar
Topics:
2 I want to know when to use 'Having been'? How it is used? It 2 20 Jun 2016, 05:21
26 Researchers have questioned the use of costly and 16 08 Mar 2017, 09:22
In recent years cattle breeders have increasingly used 0 13 Dec 2016, 02:35
In recent years cattle breeders have increasingly used 0 29 Dec 2015, 07:45
230 In recent years cattle breeders have increasingly used 57 22 Jun 2017, 10:38
Display posts from previous: Sort by
<UNMASK>
It is currently 23 Jun 2017, 14:22
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
Events & Promotions
Events & Promotions in June
Open Detailed Calendar
Everyone of us have understood that without him helping us
Author Message
TAGS:
Hide Tags
Director
Joined: 02 Mar 2006
Posts: 575
Location: France
Everyone of us have understood that without him helping us [#permalink]
Show Tags
15 Mar 2006, 16:25
1
This post was
BOOKMARKED
00:00
Difficulty:
(N/A)
Question Stats:
100% (01:05) correct 0% (00:00) wrong based on 12 sessions
HideShow timer Statistics
Everyone of us have understood that without him helping us we would not have succeeded in our program over the past six months
a. Everyone of us have understood that without him helping us
b. Everyone of us has understood that without his helping us
c. Everyone of us have understood that without his help
d. Everyone of us has understood that without him helping us
e. Every single one of us have understood that without him helping us
A thoroughly frightened child was seen by her cowering in the corner of the room.
a. A thoroughly frightened child was seen by her cowering in the corner of the room.
b. Cowering in the corner of the room a thoroughly frightened child was seen by her.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her.
e. She saw a thoroughly frightened child who was cowering in the corner of the room.
Intern
Joined: 20 Jan 2006
Posts: 14
Show Tags
15 Mar 2006, 17:11
1. 'Everyone' is singular , so 'has' is required.
A & C are out.
E is wordy
Between B & D.
usage of 'his helping us' is wrong.
Will opt for D.
2. Modifier quesion.
D is my choice
Manager
Joined: 20 Feb 2006
Posts: 213
Show Tags
15 Mar 2006, 17:18
(1) D for Subject-Verb agreement and
(2) E - in active voice and uses modifier correctly .
Intern
Joined: 20 Jan 2006
Posts: 14
Show Tags
15 Mar 2006, 17:48
Yes, E is the choice for Question #2.
Thanks chuckle.
GMAT Club Legend
Joined: 07 Jul 2004
Posts: 5043
Location: Singapore
Show Tags
<MASK>
Show Tags
15 Mar 2006, 20:00
Q2)
a. awkward
b. awkward
d. Reject -> passive voice
Between c and e, e looks a better choice though c isn't all that bad.
Director
Joined: 17 Sep 2005
Posts: 903
Show Tags
15 Mar 2006, 23:14
SC1: IMO it's D. 'has' is required and 'without him helping us' is correct.
SC2: It has to be E. Simple and concise.
Regards,
Brajesh
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
16 Mar 2006, 02:41
1) ^ D ^
We need "has" after everyone.
a. Everyone of us have understood that without him helping us
b. Everyone of us has understood that without his helping us ---> "his" is wrong here
c. Everyone of us have understood that without his help
e. Every single one of us have understood that without him helping us
2) ^ E ^
a. A thoroughly frightened child was seen by her cowering in the corner of the room. ---> wrong modifier
b. Cowering in the corner of the room a thoroughly frightened child was seen by her. ---> we need a comma after "room"
c. She saw, cowering in the corner of the room, a thoroughly frightened child. ---> wrong modifier
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her. ---> passive
Manager
Joined: 06 Aug 2005
Posts: 90
Location: Minneapolis
Show Tags
<MASK>
Show Tags
16 Mar 2006, 10:44
in the second question if "cowering in the corner of the room" is the modifier for "she", then is nothing wrong with it and is correct one.
Intern
Joined: 24 Jan 2006
Posts: 10
Show Tags
16 Mar 2006, 12:24
i choose C for question 2.
what's wrong with C?
it's nice and short, and makes sense. i don't see a modifier problem. please elaborate on the incorrectness of C.
_________________
dyin inside but outside you lookin fearless...
www.djcham.com
= )
Senior Manager
Joined: 22 Nov 2005
Posts: 474
Show Tags
16 Mar 2006, 13:53
djcham wrote:
i choose C for question 2.
<MASK>
it's nice and short, and makes sense. i don't see a modifier problem. please elaborate on the incorrectness of C.
C has modifier problem.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
It sounds like She was cowering .... of the room. But it was not intended from original sentence.
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
16 Mar 2006, 14:13
To Ketrin1 and djcham
C)She saw, cowering in the corner of the room, a thoroughly frightened child.
This choice has a modifier problem. It sounds as if the cowering person is "she". However, the person cowering should be "the throughly frightened child"
Manager
Joined: 13 Dec 2005
Posts: 224
Location: Milwaukee,WI
Show Tags
16 Mar 2006, 14:19
1 D everyone requires has ..... and him is correct usage
2. E active,concise and clear
Director
Joined: 02 Mar 2006
Posts: 575
Location: France
Show Tags
18 Mar 2006, 05:56
No one has answered like OA, but Ketrin1 and djcham for Q2.
So the OA is:
Q1. B. Everyone is singular and requires the singular has. The preposition without requires the gerund helping preceded by the possessive his.
Q2. C. This is suspenseful sentence since what "she saw" is held off to the very last word in the sentence. Also, an active verb, saw, is preferable to the passive was seen.
Manager
Joined: 20 Feb 2006
Posts: 213
Show Tags
18 Mar 2006, 10:20
Vow!!! Both of my answers are wrong
I can at least attempt to understand the answer for Question 1 to be B.
But, I still do not understand the answer for Question 2.
Won't the "Cowering" modify "she" and not the "child" in C???
SVP
Joined: 14 Dec 2004
Posts: 1689
Show Tags
18 Mar 2006, 11:01
I disagree on SC2 OA, "E" is the best.
"C" clearly has modifier issue.
What is the source of these questions, karlfurt?
Manager
Joined: 04 Sep 2005
Posts: 141
Location: Fringes of the Boreal, Canada
Show Tags
18 Mar 2006, 12:58
Q2: E doesn't seem right to me. The "who was" is redundant because the past tense is already used in "She saw". The sentence would sound better if "who was" was omitted: eg. "She saw a thoroughly frightened child cowering in the corner of the room."
_________________
"To hell with circumstances; I create opportunities." - Bruce Lee
Senior Manager
Joined: 08 Sep 2004
Posts: 257
Location: New York City, USA
Show Tags
18 Mar 2006, 13:10
Both of my answers were wrong too. But I agree with the OAs. First one is definitely B (how could I pick 'him helping' ).
The second question is even more interesting. The important thing to note is that 'cowering in the corner of the room' modifies the 'frightened child'. D is definitely wrong because of passive voice. E is wordy. The modifier 'who was' immidiately follows 'frightened child' without comma. It can be removed without changing the meaning of the sentence in any way. But ofcourse all this is after getting to know the official answer. As they say, hindsight is 20-20.
- Vipin
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
18 Mar 2006, 16:17
vipin7um wrote:
Both of my answers were wrong too. But I agree with the OAs. First one is definitely B (how could I pick 'him helping' ).
The second question is even more interesting. The important thing to note is that 'cowering in the corner of the room' modifies the 'frightened child'. D is definitely wrong because of passive voice. E is wordy. The modifier 'who was' immidiately follows 'frightened child' without comma. It can be removed without changing the meaning of the sentence in any way. But ofcourse all this is after getting to know the official answer. As they say, hindsight is 20-20.
- Vipin
I agree with you for the first question.
However, for the second one I still support ^ E ^. In C ,even if "who was" is omitted, we cannot think that it modifes the child because modifier is in wrong place.
18 Mar 2006, 16:17
<MASK>
Similar topics Replies Last post
Similar
Topics:
2 I want to know when to use 'Having been'? How it is used? It 2 20 Jun 2016, 05:21
26 Researchers have questioned the use of costly and 16 08 Mar 2017, 09:22
In recent years cattle breeders have increasingly used 0 13 Dec 2016, 02:35
In recent years cattle breeders have increasingly used 0 29 Dec 2015, 07:45
230 In recent years cattle breeders have increasingly used 57 22 Jun 2017, 10:38
Display posts from previous: Sort by
<UNMASK>
It is currently 23 Jun 2017, 14:22
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
Events & Promotions
Events & Promotions in June
Open Detailed Calendar
Everyone of us have understood that without him helping us
Author Message
TAGS:
Hide Tags
Director
Joined: 02 Mar 2006
Posts: 575
Location: France
Everyone of us have understood that without him helping us [#permalink]
Show Tags
15 Mar 2006, 16:25
1
This post was
BOOKMARKED
00:00
Difficulty:
(N/A)
Question Stats:
100% (01:05) correct 0% (00:00) wrong based on 12 sessions
HideShow timer Statistics
Everyone of us have understood that without him helping us we would not have succeeded in our program over the past six months
a. Everyone of us have understood that without him helping us
b. Everyone of us has understood that without his helping us
c. Everyone of us have understood that without his help
d. Everyone of us has understood that without him helping us
e. Every single one of us have understood that without him helping us
A thoroughly frightened child was seen by her cowering in the corner of the room.
a. A thoroughly frightened child was seen by her cowering in the corner of the room.
b. Cowering in the corner of the room a thoroughly frightened child was seen by her.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her.
e. She saw a thoroughly frightened child who was cowering in the corner of the room.
Intern
Joined: 20 Jan 2006
Posts: 14
Show Tags
15 Mar 2006, 17:11
1. 'Everyone' is singular , so 'has' is required.
A & C are out.
E is wordy
Between B & D.
usage of 'his helping us' is wrong.
Will opt for D.
2. Modifier quesion.
D is my choice
Manager
Joined: 20 Feb 2006
Posts: 213
Show Tags
15 Mar 2006, 17:18
(1) D for Subject-Verb agreement and
(2) E - in active voice and uses modifier correctly .
Intern
Joined: 20 Jan 2006
Posts: 14
Show Tags
15 Mar 2006, 17:48
Yes, E is the choice for Question #2.
Thanks chuckle.
GMAT Club Legend
Joined: 07 Jul 2004
Posts: 5043
Location: Singapore
Show Tags
<MASK>
Show Tags
15 Mar 2006, 20:00
Q2)
a. awkward
b. awkward
d. Reject -> passive voice
Between c and e, e looks a better choice though c isn't all that bad.
Director
Joined: 17 Sep 2005
Posts: 903
Show Tags
15 Mar 2006, 23:14
SC1: IMO it's D. 'has' is required and 'without him helping us' is correct.
SC2: It has to be E. Simple and concise.
Regards,
Brajesh
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
16 Mar 2006, 02:41
1) ^ D ^
We need "has" after everyone.
a. Everyone of us have understood that without him helping us
b. Everyone of us has understood that without his helping us ---> "his" is wrong here
c. Everyone of us have understood that without his help
e. Every single one of us have understood that without him helping us
2) ^ E ^
a. A thoroughly frightened child was seen by her cowering in the corner of the room. ---> wrong modifier
b. Cowering in the corner of the room a thoroughly frightened child was seen by her. ---> we need a comma after "room"
c. She saw, cowering in the corner of the room, a thoroughly frightened child. ---> wrong modifier
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her. ---> passive
Manager
Joined: 06 Aug 2005
Posts: 90
Location: Minneapolis
Show Tags
16 Mar 2006, 09:29
D for 1st one. (everyone is always singular and its not a SANAM)
E for 2nd one. (active)
Intern
Joined: 05 Feb 2005
Posts: 2
Show Tags
16 Mar 2006, 10:44
in the second question if "cowering in the corner of the room" is the modifier for "she", then is nothing wrong with it and is correct one.
Intern
Joined: 24 Jan 2006
Posts: 10
Show Tags
16 Mar 2006, 12:24
i choose C for question 2.
what's wrong with C?
it's nice and short, and makes sense. i don't see a modifier problem. please elaborate on the incorrectness of C.
_________________
dyin inside but outside you lookin fearless...
www.djcham.com
= )
Senior Manager
Joined: 22 Nov 2005
Posts: 474
Show Tags
16 Mar 2006, 13:53
djcham wrote:
i choose C for question 2.
what's wrong with C?
it's nice and short, and makes sense. i don't see a modifier problem. please elaborate on the incorrectness of C.
C has modifier problem.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
It sounds like She was cowering .... of the room. But it was not intended from original sentence.
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
16 Mar 2006, 14:13
To Ketrin1 and djcham
C)She saw, cowering in the corner of the room, a thoroughly frightened child.
This choice has a modifier problem. It sounds as if the cowering person is "she". However, the person cowering should be "the throughly frightened child"
Manager
Joined: 13 Dec 2005
Posts: 224
Location: Milwaukee,WI
Show Tags
16 Mar 2006, 14:19
1 D everyone requires has ..... and him is correct usage
2. E active,concise and clear
Director
Joined: 02 Mar 2006
Posts: 575
Location: France
Show Tags
18 Mar 2006, 05:56
No one has answered like OA, but Ketrin1 and djcham for Q2.
So the OA is:
Q1. B. Everyone is singular and requires the singular has. The preposition without requires the gerund helping preceded by the possessive his.
Q2. C. This is suspenseful sentence since what "she saw" is held off to the very last word in the sentence. Also, an active verb, saw, is preferable to the passive was seen.
Manager
Joined: 20 Feb 2006
Posts: 213
Show Tags
18 Mar 2006, 10:20
Vow!!! Both of my answers are wrong
I can at least attempt to understand the answer for Question 1 to be B.
But, I still do not understand the answer for Question 2.
Won't the "Cowering" modify "she" and not the "child" in C???
SVP
Joined: 14 Dec 2004
Posts: 1689
Show Tags
18 Mar 2006, 11:01
I disagree on SC2 OA, "E" is the best.
"C" clearly has modifier issue.
What is the source of these questions, karlfurt?
Manager
Joined: 04 Sep 2005
Posts: 141
Location: Fringes of the Boreal, Canada
Show Tags
18 Mar 2006, 12:58
Q2: E doesn't seem right to me. The "who was" is redundant because the past tense is already used in "She saw". The sentence would sound better if "who was" was omitted: eg. "She saw a thoroughly frightened child cowering in the corner of the room."
_________________
"To hell with circumstances; I create opportunities." - Bruce Lee
Senior Manager
Joined: 08 Sep 2004
Posts: 257
Location: New York City, USA
Show Tags
18 Mar 2006, 13:10
Both of my answers were wrong too. But I agree with the OAs. First one is definitely B (how could I pick 'him helping' ).
The second question is even more interesting. The important thing to note is that 'cowering in the corner of the room' modifies the 'frightened child'. D is definitely wrong because of passive voice. E is wordy. The modifier 'who was' immidiately follows 'frightened child' without comma. It can be removed without changing the meaning of the sentence in any way. But ofcourse all this is after getting to know the official answer. As they say, hindsight is 20-20.
- Vipin
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
18 Mar 2006, 16:17
vipin7um wrote:
Both of my answers were wrong too. But I agree with the OAs. First one is definitely B (how could I pick 'him helping' ).
The second question is even more interesting. The important thing to note is that 'cowering in the corner of the room' modifies the 'frightened child'. D is definitely wrong because of passive voice. E is wordy. The modifier 'who was' immidiately follows 'frightened child' without comma. It can be removed without changing the meaning of the sentence in any way. But ofcourse all this is after getting to know the official answer. As they say, hindsight is 20-20.
- Vipin
I agree with you for the first question.
However, for the second one I still support ^ E ^. In C ,even if "who was" is omitted, we cannot think that it modifes the child because modifier is in wrong place.
18 Mar 2006, 16:17
Go to page 1 2 Next [ 35 posts ]
Similar topics Replies Last post
Similar
Topics:
2 I want to know when to use 'Having been'? How it is used? It 2 20 Jun 2016, 05:21
26 Researchers have questioned the use of costly and 16 08 Mar 2017, 09:22
In recent years cattle breeders have increasingly used 0 13 Dec 2016, 02:35
In recent years cattle breeders have increasingly used 0 29 Dec 2015, 07:45
230 In recent years cattle breeders have increasingly used 57 22 Jun 2017, 10:38
Display posts from previous: Sort by
<UNMASK>
It is currently 23 Jun 2017, 14:22
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
Events & Promotions
Events & Promotions in June
Open Detailed Calendar
Everyone of us have understood that without him helping us
Author Message
TAGS:
Hide Tags
Director
Joined: 02 Mar 2006
Posts: 575
Location: France
Everyone of us have understood that without him helping us [#permalink]
Show Tags
15 Mar 2006, 16:25
1
This post was
BOOKMARKED
00:00
Difficulty:
(N/A)
Question Stats:
100% (01:05) correct 0% (00:00) wrong based on 12 sessions
HideShow timer Statistics
Everyone of us have understood that without him helping us we would not have succeeded in our program over the past six months
a. Everyone of us have understood that without him helping us
b. Everyone of us has understood that without his helping us
c. Everyone of us have understood that without his help
d. Everyone of us has understood that without him helping us
e. Every single one of us have understood that without him helping us
A thoroughly frightened child was seen by her cowering in the corner of the room.
a. A thoroughly frightened child was seen by her cowering in the corner of the room.
b. Cowering in the corner of the room a thoroughly frightened child was seen by her.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her.
e. She saw a thoroughly frightened child who was cowering in the corner of the room.
Intern
Joined: 20 Jan 2006
Posts: 14
Show Tags
15 Mar 2006, 17:11
1. 'Everyone' is singular , so 'has' is required.
A & C are out.
E is wordy
Between B & D.
usage of 'his helping us' is wrong.
Will opt for D.
2. Modifier quesion.
D is my choice
Manager
Joined: 20 Feb 2006
Posts: 213
Show Tags
15 Mar 2006, 17:18
(1) D for Subject-Verb agreement and
(2) E - in active voice and uses modifier correctly .
Intern
Joined: 20 Jan 2006
Posts: 14
Show Tags
15 Mar 2006, 17:48
Yes, E is the choice for Question #2.
Thanks chuckle.
GMAT Club Legend
Joined: 07 Jul 2004
Posts: 5043
Location: Singapore
Show Tags
15 Mar 2006, 19:58
Q1)
as vs have --> go with has since everyone is singular
Between b and d, go with d. 'him helping us' is the correct form.
GMAT Club Legend
Joined: 07 Jul 2004
Posts: 5043
Location: Singapore
Show Tags
15 Mar 2006, 20:00
Q2)
a. awkward
b. awkward
d. Reject -> passive voice
Between c and e, e looks a better choice though c isn't all that bad.
Director
Joined: 17 Sep 2005
Posts: 903
Show Tags
15 Mar 2006, 23:14
SC1: IMO it's D. 'has' is required and 'without him helping us' is correct.
SC2: It has to be E. Simple and concise.
Regards,
Brajesh
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
16 Mar 2006, 02:41
1) ^ D ^
We need "has" after everyone.
a. Everyone of us have understood that without him helping us
b. Everyone of us has understood that without his helping us ---> "his" is wrong here
c. Everyone of us have understood that without his help
e. Every single one of us have understood that without him helping us
2) ^ E ^
a. A thoroughly frightened child was seen by her cowering in the corner of the room. ---> wrong modifier
b. Cowering in the corner of the room a thoroughly frightened child was seen by her. ---> we need a comma after "room"
c. She saw, cowering in the corner of the room, a thoroughly frightened child. ---> wrong modifier
d. A thoroughly frightened child, cowering in the corner of the room, was seen by her. ---> passive
Manager
Joined: 06 Aug 2005
Posts: 90
Location: Minneapolis
Show Tags
16 Mar 2006, 09:29
D for 1st one. (everyone is always singular and its not a SANAM)
E for 2nd one. (active)
Intern
Joined: 05 Feb 2005
Posts: 2
Show Tags
16 Mar 2006, 10:44
in the second question if "cowering in the corner of the room" is the modifier for "she", then is nothing wrong with it and is correct one.
Intern
Joined: 24 Jan 2006
Posts: 10
Show Tags
16 Mar 2006, 12:24
i choose C for question 2.
what's wrong with C?
it's nice and short, and makes sense. i don't see a modifier problem. please elaborate on the incorrectness of C.
_________________
dyin inside but outside you lookin fearless...
www.djcham.com
= )
Senior Manager
Joined: 22 Nov 2005
Posts: 474
Show Tags
16 Mar 2006, 13:53
djcham wrote:
i choose C for question 2.
what's wrong with C?
it's nice and short, and makes sense. i don't see a modifier problem. please elaborate on the incorrectness of C.
C has modifier problem.
c. She saw, cowering in the corner of the room, a thoroughly frightened child.
It sounds like She was cowering .... of the room. But it was not intended from original sentence.
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
16 Mar 2006, 14:13
To Ketrin1 and djcham
C)She saw, cowering in the corner of the room, a thoroughly frightened child.
This choice has a modifier problem. It sounds as if the cowering person is "she". However, the person cowering should be "the throughly frightened child"
Manager
Joined: 13 Dec 2005
Posts: 224
Location: Milwaukee,WI
Show Tags
16 Mar 2006, 14:19
1 D everyone requires has ..... and him is correct usage
2. E active,concise and clear
Director
Joined: 02 Mar 2006
Posts: 575
Location: France
Show Tags
18 Mar 2006, 05:56
No one has answered like OA, but Ketrin1 and djcham for Q2.
So the OA is:
Q1. B. Everyone is singular and requires the singular has. The preposition without requires the gerund helping preceded by the possessive his.
Q2. C. This is suspenseful sentence since what "she saw" is held off to the very last word in the sentence. Also, an active verb, saw, is preferable to the passive was seen.
Manager
Joined: 20 Feb 2006
Posts: 213
Show Tags
18 Mar 2006, 10:20
Vow!!! Both of my answers are wrong
I can at least attempt to understand the answer for Question 1 to be B.
But, I still do not understand the answer for Question 2.
Won't the "Cowering" modify "she" and not the "child" in C???
SVP
Joined: 14 Dec 2004
Posts: 1689
Show Tags
18 Mar 2006, 11:01
I disagree on SC2 OA, "E" is the best.
"C" clearly has modifier issue.
What is the source of these questions, karlfurt?
Manager
Joined: 04 Sep 2005
Posts: 141
Location: Fringes of the Boreal, Canada
Show Tags
18 Mar 2006, 12:58
Q2: E doesn't seem right to me. The "who was" is redundant because the past tense is already used in "She saw". The sentence would sound better if "who was" was omitted: eg. "She saw a thoroughly frightened child cowering in the corner of the room."
_________________
"To hell with circumstances; I create opportunities." - Bruce Lee
Senior Manager
Joined: 08 Sep 2004
Posts: 257
Location: New York City, USA
Show Tags
18 Mar 2006, 13:10
Both of my answers were wrong too. But I agree with the OAs. First one is definitely B (how could I pick 'him helping' ).
The second question is even more interesting. The important thing to note is that 'cowering in the corner of the room' modifies the 'frightened child'. D is definitely wrong because of passive voice. E is wordy. The modifier 'who was' immidiately follows 'frightened child' without comma. It can be removed without changing the meaning of the sentence in any way. But ofcourse all this is after getting to know the official answer. As they say, hindsight is 20-20.
- Vipin
Director
Joined: 27 Feb 2006
Posts: 622
Show Tags
18 Mar 2006, 16:17
vipin7um wrote:
Both of my answers were wrong too. But I agree with the OAs. First one is definitely B (how could I pick 'him helping' ).
The second question is even more interesting. The important thing to note is that 'cowering in the corner of the room' modifies the 'frightened child'. D is definitely wrong because of passive voice. E is wordy. The modifier 'who was' immidiately follows 'frightened child' without comma. It can be removed without changing the meaning of the sentence in any way. But ofcourse all this is after getting to know the official answer. As they say, hindsight is 20-20.
- Vipin
I agree with you for the first question.
However, for the second one I still support ^ E ^. In C ,even if "who was" is omitted, we cannot think that it modifes the child because modifier is in wrong place.
18 Mar 2006, 16:17
Go to page 1 2 Next [ 35 posts ]
Similar topics Replies Last post
Similar
Topics:
2 I want to know when to use 'Having been'? How it is used? It 2 20 Jun 2016, 05:21
26 Researchers have questioned the use of costly and 16 08 Mar 2017, 09:22
In recent years cattle breeders have increasingly used 0 13 Dec 2016, 02:35
In recent years cattle breeders have increasingly used 0 29 Dec 2015, 07:45
230 In recent years cattle breeders have increasingly used 57 22 Jun 2017, 10:38
Display posts from previous: Sort by
| 9,991 | 6,699,475 |
# Confused as to what will happen with multiple voltages
Here is (I think) the circuit that forms part of my project:
The value of the resistor (2490 Ohms) is fixed. Vi is a variable voltage (driven by the Arduino PWM output - might be filtered but let's just assume it's a nice, clean DC voltage which will never be greater than 5V). What will Vo be or is it not possible to tell? If it were possible to disconnect the resistor, then I'd do it, but it's inside some other electronics to which I don't have access (a Megasquirt EFI controller).
Also tempted to put a diode in between the Arduino (Vo) output and the node but not sure if this is necessary?
Vo will also be exactly 0 or 5 volts since the arduino output will sink the tiny current flowing through the resistor when it's LOW (0 volts)
The value of Vo depends on the source resistance of the voltage source Vi. If we call this resistance R, then in the absence of a load:
Vo = ((5 * R) + (Vi * 2490))/(2490 + R)
So it all depends on the value of R. If the voltage source is an Arduino pin followed by a simple RC network to smooth out the PWM, then R is the resistor used in the RC network plus about 25 ohms.
Looking at it simply: On the basis that Vo is not required to supply current, the resistor is irrelevant to your calculations. Since Vo is electrically tied to Vi then Vo is always equal to Vi.
This, of course assumes the output impedance of Vi is zero. In reality, if Vi output port has impedance, let us call it Vi(imp), when set low, then the value at Vo will be the algebraic solution of the potential divider set up by your 2490 resistor in series with Vi(imp) and the actual value of Vi(low).
Similarly the value of Vo, with Vi set high, will be dependant upon the actual value of Vi(high) and the impedance of its output when set high.
What will Vo be
Vo = Vi.
Mr Henry,
You are perfectly correct.
My bullsh*t statement is exactly that ; since Vi is the voltage seen at the output terminals and is directly connected to Vo, then Vo = Vi.
The impedance of the output port and the respective Vi(low) and Vi(high) are of no consequence.
Happy new year
If Vo = Vi then that's great. Thanks for all the replies - I think I'll breadboard it and hook up a DVM to check!
jackrae: I hear you. It is always easy to complicate matters.
If Vo = Vi then that's great.
Look at it this way: Vi and Vo are connected via a wire (of zero resistance). So Vi cannot differ from Vo.
Now, if Vi is the specified voltage on that battery with high output resistance, it gets slightly more interesting. But a regular battery / power supply will have output impedance far lower than that resistor's (2490ohm) so even if they differ, it is immaterial.
I hope you are right (and that my simplified circuit is a sufficient analog of the real intended circuit!
spandit:
I hope you are right (and that my simplified circuit is a sufficient analog of the real intended circuit!
You haven't said what your real intended circuit is, so we don't know whether that is true or not.
dc42:
spandit:
I hope you are right (and that my simplified circuit is a sufficient analog of the real intended circuit!
You haven't said what your real intended circuit is, so we don't know whether that is true or not.
Fair point! The node (where we're measuring Vo) is the input to a Megasquirt EFI controller which normally measures the voltage across a potential divider (between the 2490Ohm resistor and a thermistor). I'm giving it a voltage feed from an Arduino instead (the software is taking the average of 4 thermistors and combining them into 1 output whilst also displaying all 4 on an LCD bargraph). It's more or less finished but probably needs a few tweaks
spandit:
Fair point! The node (where we're measuring Vo) is the input to a Megasquirt EFI controller which normally measures the voltage across a potential divider (between the 2490Ohm resistor and a thermistor). I'm giving it a voltage feed from an Arduino instead (the software is taking the average of 4 thermistors and combining them into 1 output whilst also displaying all 4 on an LCD bargraph). It's more or less finished but probably needs a few tweaks
And how are you deriving a voltage feed from the Arduino? If it's from a PWM pin and simple RC smoothing filter, then you need to take the source resistance of the RC filter into account. If you follow the RC filter with an op amp used as a unity-gain buffer, then its output resistance will be low enough to ignore.
| 1,045 | 6,699,476 |
<MASK>
<UNMASK>
<MASK>
Homework Helper
Homework should be posted in the homework forum- and you should show us what you have tried to do. Have you considered L'Hopital's rule?
<MASK>
<UNMASK>
<MASK>
2. lim [sin(2/x)+cos(1/x)]^x
x -> inf
<MASK>
Homework Helper
Homework should be posted in the homework forum- and you should show us what you have tried to do. Have you considered L'Hopital's rule?
<MASK>
<UNMASK>
# L'Hopital's rule for solving limit
mousesgr
1. lim [(1+x)^(1/x) - e ] / x
x ->0
2. lim [sin(2/x)+cos(1/x)]^x
x -> inf
<MASK>
Homework Helper
Homework should be posted in the homework forum- and you should show us what you have tried to do. Have you considered L'Hopital's rule?
<MASK>
Homework Helper
Do you KNOW "L'Hopital's rule"? It is specifically for limit problems where you get things like these.
If you have $$lim_{x->a} \frac{f(x)}{g{x}}$$ where f(a)= 0 and g(a)= 0, then the limit is the same as $$lim_{x->a}\frac{\frac{df}{dx}}{\frac{dg}{dx}}$$.
<MASK>
Homework Helper
mousesgr said:
qs 1 is not 0/0 or inf/inf from
how do consider L'Hopital's rule?
#1 is in form 0 / 0.
Since:
$$\lim_{x \rightarrow 0} (1 + x) ^ {\frac{1}{x}} = e$$
So the numerator will tend to 0 as x approaches 0. The denominator also tends to 0. So it's 0 / 0.
You can use L'Hopital's rule to solve for #1.
-------------------
#2 is $$1 ^ \infty$$
First, you can try to take logs of both sides.
So let $$y = \lim_{x \rightarrow \infty} \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x$$. So:
$$\ln y = \ln \left\{ \lim_{x \rightarrow \infty} \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x \right\} = \lim_{x \rightarrow \infty} \ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x$$
$$= \lim_{x \rightarrow \infty} x \ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] = \lim_{x \rightarrow \infty} \frac{\ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right]}{\frac{1}{x}}$$
This is 0 / 0. So again, you can apply L'Hopital's rule to find the limit.
Viet Dao,
<UNMASK>
# L'Hopital's rule for solving limit
mousesgr
1. lim [(1+x)^(1/x) - e ] / x
x ->0
2. lim [sin(2/x)+cos(1/x)]^x
x -> inf
help...
Last edited:
Homework Helper
Homework should be posted in the homework forum- and you should show us what you have tried to do. Have you considered L'Hopital's rule?
mousesgr
qs 1 is not 0/0 or inf/inf from
how do consider L'Hopital's rule?
Homework Helper
Do you KNOW "L'Hopital's rule"? It is specifically for limit problems where you get things like these.
If you have $$lim_{x->a} \frac{f(x)}{g{x}}$$ where f(a)= 0 and g(a)= 0, then the limit is the same as $$lim_{x->a}\frac{\frac{df}{dx}}{\frac{dg}{dx}}$$.
If you get things like $$0^0$$ or $$\infty^{\infty}$$ (as your second limit), you can take logarithms to reduct to the first case.
Homework Helper
mousesgr said:
qs 1 is not 0/0 or inf/inf from
how do consider L'Hopital's rule?
#1 is in form 0 / 0.
Since:
$$\lim_{x \rightarrow 0} (1 + x) ^ {\frac{1}{x}} = e$$
So the numerator will tend to 0 as x approaches 0. The denominator also tends to 0. So it's 0 / 0.
You can use L'Hopital's rule to solve for #1.
-------------------
#2 is $$1 ^ \infty$$
First, you can try to take logs of both sides.
So let $$y = \lim_{x \rightarrow \infty} \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x$$. So:
$$\ln y = \ln \left\{ \lim_{x \rightarrow \infty} \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x \right\} = \lim_{x \rightarrow \infty} \ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] ^ x$$
$$= \lim_{x \rightarrow \infty} x \ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right] = \lim_{x \rightarrow \infty} \frac{\ln \left[ \sin \left( \frac{2}{x} \right) + \cos \left( \frac{1}{x} \right) \right]}{\frac{1}{x}}$$
This is 0 / 0. So again, you can apply L'Hopital's rule to find the limit.
Viet Dao,
| 1,490 | 6,699,477 |
## Precalculus (6th Edition) Blitzer
$d=-8 \cos \pi t$
Amplitude, $A=-8$ (negative because the object is moving down initially.) Given: $Period =2$ Now, period, $P=\dfrac{2 \pi}{\omega} \implies \omega =\dfrac{2 \pi}{2}=\pi$ So, $d=A \cos \omega t=-8 \cos \pi t$
| 101 | 6,699,478 |
Math111i_GroupQuiz5_Solutions
# Math111i_GroupQuiz5_Solutions - 0) and their business grows...
This preview shows pages 1–2. Sign up to view the full content.
Math 111I Sec 001 Group Quiz 5 Solutions Name Directions: Read each question carefully and answer in the space pro- vided. The use of calculators is allowed, but not necessary. There are 10 points possible. To receive partial credit on any problem work must be shown. 1. Write a function in the form f ( x ) = C · a x for an exponential function with an initial value of 3,700 whose output triples for each unit increase of the input. Solutions. The initial value is C = 3700, and from the description we see that a = 3. So the function follows as f ( x ) = 3700 · 3 x . 2. Determine if the data is linear or exponential. Write the equation of the function x -1 0 1 2 f ( x ) 3.01 5.50 10.07 18.42 Solutions. We check to see if f ( x ) is exponential 5 . 50 3 . 01 1 . 83 10 . 07 5 . 50 1 . 83 18 . 42 10 . 07 1 . 83 So, we notice that f (0) = C and we have that C = 5 . 50 so we have that f ( x ) = 5 . 50(1 . 83) x . 3. A new cell phone company has 400 subscribes their first year (year
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
Unformatted text preview: 0) and their business grows to 600 subscribers the next year (year 1). Write a formula for the number of subscribers, N ( t ), as a function of years in business t , assuming exponential growth. Math 111I Sec 001 Group Quiz 5 Solutions Solutions. Clearly we have that if N ( t ) = Ca t , and we know that N (0) = 400 = C , and we also see that a = 600 400 = 1 . 5 Thus we have that N ( t ) = 400(1 . 5) t . 4. Find the growth/decay factor (assuming exponential growth/decay) for a population that decreases from 11,000,000 to 9,790,000 in one year. Solutions. We have that we decreased over one time period. So the decay factor for each year is a = 11 , 000 , 000 9 , 790 , 000 = 0 . 89...
View Full Document
## Math111i_GroupQuiz5_Solutions - 0) and their business grows...
This preview shows document pages 1 - 2. Sign up to view the full document.
View Full Document
Ask a homework question - tutors are online
| 660 | 6,699,479 |
<MASK>
(d) 30 rpm
<MASK>
Easy explanation: $\frac{Ns-Na}{Nr-Na}=\frac{-Tr}{Ts}$
<MASK>
<UNMASK>
<MASK>
(d) 30 rpm
<MASK>
Easy explanation: $\frac{Ns-Na}{Nr-Na}=\frac{-Tr}{Ts}$
$\frac{60-Na}{0-Na}=\frac{-100}{20}$
<MASK>
<UNMASK>
<MASK>
(b) 10 rpm
<MASK>
(d) 30 rpm
The question was asked in a job interview.
This intriguing question comes from Mechanical Power Transmission in chapter Mechanical Power Transmission of Farm Machinery
by (243k points)
selected by
Correct answer is (b) 10 rpm
Easy explanation: $\frac{Ns-Na}{Nr-Na}=\frac{-Tr}{Ts}$
$\frac{60-Na}{0-Na}=\frac{-100}{20}$
6Na = 60
<MASK>
+1 vote
+1 vote
+1 vote
+1 vote
+1 vote
+1 vote
<UNMASK>
# In an epicyclic gear train, the sun gear on the input shaft is 20 teeth external gear. The planet gear is a 40 teeth external gear and the ring gear is a 100 teeth internal gear. The ring gear is fixed and the sun gear is rotating at 60 rpm counter clockwise. The arm attached to output shaft will rotate at ________
<MASK>
(b) 10 rpm
(c) 20 rpm
(d) 30 rpm
The question was asked in a job interview.
This intriguing question comes from Mechanical Power Transmission in chapter Mechanical Power Transmission of Farm Machinery
by (243k points)
selected by
Correct answer is (b) 10 rpm
Easy explanation: $\frac{Ns-Na}{Nr-Na}=\frac{-Tr}{Ts}$
$\frac{60-Na}{0-Na}=\frac{-100}{20}$
6Na = 60
<MASK>
+1 vote
+1 vote
+1 vote
+1 vote
+1 vote
+1 vote
<UNMASK>
# In an epicyclic gear train, the sun gear on the input shaft is 20 teeth external gear. The planet gear is a 40 teeth external gear and the ring gear is a 100 teeth internal gear. The ring gear is fixed and the sun gear is rotating at 60 rpm counter clockwise. The arm attached to output shaft will rotate at ________
In an epicyclic gear train, the sun gear on the input shaft is 20 teeth external gear. The planet gear is a 40 teeth external gear and the ring gear is a 100 teeth internal gear. The ring gear is fixed and the sun gear is rotating at 60 rpm counter clockwise. The arm attached to output shaft will rotate at ________
(a) 5 rpm
(b) 10 rpm
(c) 20 rpm
(d) 30 rpm
The question was asked in a job interview.
This intriguing question comes from Mechanical Power Transmission in chapter Mechanical Power Transmission of Farm Machinery
by (243k points)
selected by
Correct answer is (b) 10 rpm
Easy explanation: $\frac{Ns-Na}{Nr-Na}=\frac{-Tr}{Ts}$
$\frac{60-Na}{0-Na}=\frac{-100}{20}$
6Na = 60
Na = 10 rpm.
+1 vote
+1 vote
+1 vote
+1 vote
+1 vote
+1 vote
| 737 | 6,699,480 |
Search for a tool
Digital Root
<MASK>
Share
dCode and more
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
<MASK>
Feedback and suggestions are welcome so that dCode offers the best 'Digital Root' tool for free! Thank you!
# Digital Root
## Digital Root Calculator
Treatment of Digits As a single group Separate each number
<MASK>
Digital root calculation uses recursive reduction that consists in repeating the operation of summing/adding digits until the result has only one digit.
<MASK>
## Source code
dCode retains ownership of the "Digital Root" source code. Except explicit open source licence (indicated Creative Commons / free), the "Digital Root" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Digital Root" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Digital Root" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
<MASK>
## Need Help ?
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
<UNMASK>
Search for a tool
Digital Root
<MASK>
Share
dCode and more
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
<MASK>
Feedback and suggestions are welcome so that dCode offers the best 'Digital Root' tool for free! Thank you!
# Digital Root
## Digital Root Calculator
Treatment of Digits As a single group Separate each number
<MASK>
Digital root calculation uses recursive reduction that consists in repeating the operation of summing/adding digits until the result has only one digit.
<MASK>
In mathematics, the calculation of the numerical root is used when casting out 9s.
<MASK>
Example: DCODE = 4,3,15,4,5 and 4+3+15+4+5 = 31 => 3+1 = 4
(variant) DCODE = 4,3,15,4,5 and 4+3+1+5+4+5 = 22 => 2+2 = 4
## Source code
dCode retains ownership of the "Digital Root" source code. Except explicit open source licence (indicated Creative Commons / free), the "Digital Root" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Digital Root" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Digital Root" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
<MASK>
## Need Help ?
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
<UNMASK>
Search for a tool
Digital Root
<MASK>
Share
dCode and more
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
<MASK>
Feedback and suggestions are welcome so that dCode offers the best 'Digital Root' tool for free! Thank you!
# Digital Root
## Digital Root Calculator
Treatment of Digits As a single group Separate each number
<MASK>
Digital root calculation uses recursive reduction that consists in repeating the operation of summing/adding digits until the result has only one digit.
<MASK>
### Why calculating a digital root?
In mathematics, the calculation of the numerical root is used when casting out 9s.
<MASK>
Example: DCODE = 4,3,15,4,5 and 4+3+15+4+5 = 31 => 3+1 = 4
(variant) DCODE = 4,3,15,4,5 and 4+3+1+5+4+5 = 22 => 2+2 = 4
## Source code
dCode retains ownership of the "Digital Root" source code. Except explicit open source licence (indicated Creative Commons / free), the "Digital Root" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Digital Root" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Digital Root" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
<MASK>
The copy-paste of the page "Digital Root" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Digital Root on dCode.fr [online website], retrieved on 2024-09-14, https://www.dcode.fr/recursive-reduction
## Need Help ?
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
<UNMASK>
Search for a tool
Digital Root
Tool to calculate the digital root of a number. The digital root is computed using recursive reduction that consists in performing the sum of all digits in a number and repeat this operation on the result.
Results
Digital Root -
Tag(s) : Number Games, Mathematics
Share
dCode and more
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
<MASK>
Feedback and suggestions are welcome so that dCode offers the best 'Digital Root' tool for free! Thank you!
# Digital Root
## Digital Root Calculator
Treatment of Digits As a single group Separate each number
<MASK>
A numeric root is the sum of the digits of a number repeated until a single digit is obtained.
### How to calculate a digital root?
Digital root calculation uses recursive reduction that consists in repeating the operation of summing/adding digits until the result has only one digit.
<MASK>
This principle is often used in numerology to get a number from the numeric information on an individual (from a date of birth, favorite number, etc.)
Example: 25/12/2000 => 2 + 5 + 1 + 2 + 2 + 0 + 0 + 0 = 12 => 1 + 2 = 3
### Why calculating a digital root?
In mathematics, the calculation of the numerical root is used when casting out 9s.
In numerology, the number root is used for anything and everything.
### How to calculate a digital root for a name?
It is possible to assign a value to a name using the position of letters in the alphabet (see A1Z26 code where A = 1, B = 2, … Z = 26) then make the desired additions and analyze the total.
Example: DCODE = 4,3,15,4,5 and 4+3+15+4+5 = 31 => 3+1 = 4
(variant) DCODE = 4,3,15,4,5 and 4+3+1+5+4+5 = 22 => 2+2 = 4
## Source code
dCode retains ownership of the "Digital Root" source code. Except explicit open source licence (indicated Creative Commons / free), the "Digital Root" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Digital Root" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Digital Root" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
## Cite dCode
The copy-paste of the page "Digital Root" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Digital Root on dCode.fr [online website], retrieved on 2024-09-14, https://www.dcode.fr/recursive-reduction
## Need Help ?
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
<UNMASK>
Search for a tool
Digital Root
Tool to calculate the digital root of a number. The digital root is computed using recursive reduction that consists in performing the sum of all digits in a number and repeat this operation on the result.
Results
Digital Root -
Tag(s) : Number Games, Mathematics
Share
dCode and more
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
Feedback and suggestions are welcome so that dCode offers the best 'Digital Root' tool for free! Thank you!
# Digital Root
## Digital Root Calculator
Treatment of Digits As a single group Separate each number
### What is a digital root? (Definition)
A numeric root is the sum of the digits of a number repeated until a single digit is obtained.
### How to calculate a digital root?
Digital root calculation uses recursive reduction that consists in repeating the operation of summing/adding digits until the result has only one digit.
Example: 789 => 7+8+9 = 24 => 2+4 = 6
This principle is often used in numerology to get a number from the numeric information on an individual (from a date of birth, favorite number, etc.)
Example: 25/12/2000 => 2 + 5 + 1 + 2 + 2 + 0 + 0 + 0 = 12 => 1 + 2 = 3
### Why calculating a digital root?
In mathematics, the calculation of the numerical root is used when casting out 9s.
In numerology, the number root is used for anything and everything.
### How to calculate a digital root for a name?
It is possible to assign a value to a name using the position of letters in the alphabet (see A1Z26 code where A = 1, B = 2, … Z = 26) then make the desired additions and analyze the total.
Example: DCODE = 4,3,15,4,5 and 4+3+15+4+5 = 31 => 3+1 = 4
(variant) DCODE = 4,3,15,4,5 and 4+3+1+5+4+5 = 22 => 2+2 = 4
## Source code
dCode retains ownership of the "Digital Root" source code. Except explicit open source licence (indicated Creative Commons / free), the "Digital Root" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Digital Root" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Digital Root" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
## Cite dCode
The copy-paste of the page "Digital Root" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Digital Root on dCode.fr [online website], retrieved on 2024-09-14, https://www.dcode.fr/recursive-reduction
## Need Help ?
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
| 2,854 | 6,699,481 |
# Convert 3 Inches per hour to Meters per minute (3 in/h to m/min Conversion)
3 inches per hour is equal to 0.001269999 meters per minute.
3 in/h = 0.001269999 m/min
## How to convert 3 inches per hour to meters per minute?
To convert 3 inches per hour to meters per minute, multiply the value in inches per hour by 0.000423333.
You can use the conversion formula :
meters per minute = inches per hour × 0.000423333
To calculate, you can also use our 3 inches per hour to meters per minute converter, which is a much faster and easier option as compared to calculating manually.
## 3 inches per hour is equal to how many meters per minute?
• 3 inches per hour = 0.001269999 meters per minute
• 3.1 inches per hour = 0.0013123323 meters per minute
• 3.2 inches per hour = 0.0013546656 meters per minute
• 3.3 inches per hour = 0.0013969989 meters per minute
• 3.4 inches per hour = 0.0014393322 meters per minute
• 3.5 inches per hour = 0.0014816655 meters per minute
• 3.6 inches per hour = 0.0015239988 meters per minute
• 3.7 inches per hour = 0.0015663321 meters per minute
• 3.8 inches per hour = 0.0016086654 meters per minute
• 3.9 inches per hour = 0.0016509987 meters per minute
## Examples to convert in/h to m/min
Example 1:
Convert 3.2 in/h to m/min.
Solution:
Converting from inches per hour to meters per minute is very easy.
We know that 1 in/h = 0.000423333 m/min.
So, to convert 3.2 in/h to m/min, multiply 3.2 in/h by 0.000423333 m/min.
3.2 in/h = 3.2 × 0.000423333 m/min
3.2 in/h = 0.0013546656 m/min
Therefore, 3.2 inches per hour converted to meters per minute is equal to 0.0013546656 m/min.
Example 2:
Convert 3.8 in/h to m/min.
Solution:
1 in/h = 0.000423333 m/min
So, 3.8 in/h = 3.8 × 0.000423333 m/min
3.8 in/h = 0.0016086654 m/min
Therefore, 3.8 in/h converted to m/min is equal to 0.0016086654 m/min.
For faster calculations, you can simply use our 3 in/h to m/min converter.
## Inches per hour to meters per minute conversion table
Inches per hour Meters per minute
3 in/h 0.001269999 m/min
3.05 in/h 0.00129116565 m/min
3.1 in/h 0.0013123323 m/min
3.15 in/h 0.00133349895 m/min
3.2 in/h 0.0013546656 m/min
3.25 in/h 0.00137583225 m/min
3.3 in/h 0.0013969989 m/min
3.35 in/h 0.00141816555 m/min
3.4 in/h 0.0014393322 m/min
3.45 in/h 0.00146049885 m/min
3.5 in/h 0.0014816655 m/min
3.55 in/h 0.00150283215 m/min
3.6 in/h 0.0015239988 m/min
3.65 in/h 0.00154516545 m/min
3.7 in/h 0.0015663321 m/min
3.75 in/h 0.00158749875 m/min
3.8 in/h 0.0016086654 m/min
3.85 in/h 0.00162983205 m/min
3.9 in/h 0.0016509987 m/min
3.95 in/h 0.00167216535 m/min
| 941 | 6,699,482 |
<MASK>
How do you calculate the thrust required for a fixed wing?
<MASK>
<UNMASK>
<MASK>
How much power do you need to fly?
<MASK>
READ ALSO: Can hackers access my phone through a phone call?
<MASK>
What is Minimum Thrust required? The Minimum Thrust required for a steady, level flight is equal to the sum of drag due to zero lift and drag due to lift and is represented as T=Pdynamic*A* (CD,0+CD,i) or Thrust of an aircraft=Dynamic Pressure*Area* (Zero-lift drag coefficient+Coefficient of drag due to lift).
#### How many pounds of thrust does it take to throw objects?
<MASK>
How do you calculate the thrust required for a fixed wing?
If you are interested in details it depends on the Aspect Ratio (AR) and a few other factors of the wing. The thrust needed for any RC plane i.e. fixed wing can be calculated with thrust formula. T= 0.5 x Rho x (V^2) x S x Coefficient of Lift
<UNMASK>
<MASK>
## How much thrust do you need to lift 1 kg?
<MASK>
How much power do you need to fly?
Power Requirement to Keep a Jet Aircraft in the Air The power required to keep a Boeing 737-300 flying at a constant altitude and speed is 7.2 × 106 watts. The rate of fuel burn necessary for the engines to produce enough power to keep the airplane flying can be calculated from the total power requirement.
How do you calculate thrust for an RC plane?
For Fixed wing you need a thrust of about 70 – 80\% from your all up weight. For example, 1 Kg aircraft you require 750grams thrust. You would require aleast 8 kg thrust at 100\% throttle so that your airplane could fly at a decent speed. ( Recommended thrust is more than its weight.)
### Can airplanes fly with solar power?
Solar unmanned airplane takes to the skies The unmanned solar-powered PHASA-35 airplane aims to fly up to 20 km (66,000 ft) in the sky using only sunlight. The PVs feed the batteries that power the aircraft during the day when sunlight is available, and store energy in a battery pack for night flights.
READ ALSO: Can hackers access my phone through a phone call?
How much thrust does it take to fly a plane?
<MASK>
What is Minimum Thrust required? The Minimum Thrust required for a steady, level flight is equal to the sum of drag due to zero lift and drag due to lift and is represented as T=Pdynamic*A* (CD,0+CD,i) or Thrust of an aircraft=Dynamic Pressure*Area* (Zero-lift drag coefficient+Coefficient of drag due to lift).
#### How many pounds of thrust does it take to throw objects?
By “throwing” them (with of a gun, say) at 3,200 feet per second, you would generate 100 pounds of thrust. 1 … Cite This! Marshall Brain “How Gas Turbine Engines Work” 1 April 2000.
How do you calculate the thrust required for a fixed wing?
If you are interested in details it depends on the Aspect Ratio (AR) and a few other factors of the wing. The thrust needed for any RC plane i.e. fixed wing can be calculated with thrust formula. T= 0.5 x Rho x (V^2) x S x Coefficient of Lift
<UNMASK>
<MASK>
## How much thrust do you need to lift 1 kg?
To accelerate 1 kg at 9.8 m/s² requires 9.8 N of force, not any specific power.
How much power do you need to fly?
Power Requirement to Keep a Jet Aircraft in the Air The power required to keep a Boeing 737-300 flying at a constant altitude and speed is 7.2 × 106 watts. The rate of fuel burn necessary for the engines to produce enough power to keep the airplane flying can be calculated from the total power requirement.
How do you calculate thrust for an RC plane?
For Fixed wing you need a thrust of about 70 – 80\% from your all up weight. For example, 1 Kg aircraft you require 750grams thrust. You would require aleast 8 kg thrust at 100\% throttle so that your airplane could fly at a decent speed. ( Recommended thrust is more than its weight.)
### Can airplanes fly with solar power?
Solar unmanned airplane takes to the skies The unmanned solar-powered PHASA-35 airplane aims to fly up to 20 km (66,000 ft) in the sky using only sunlight. The PVs feed the batteries that power the aircraft during the day when sunlight is available, and store energy in a battery pack for night flights.
READ ALSO: Can hackers access my phone through a phone call?
How much thrust does it take to fly a plane?
<MASK>
What is Minimum Thrust required? The Minimum Thrust required for a steady, level flight is equal to the sum of drag due to zero lift and drag due to lift and is represented as T=Pdynamic*A* (CD,0+CD,i) or Thrust of an aircraft=Dynamic Pressure*Area* (Zero-lift drag coefficient+Coefficient of drag due to lift).
#### How many pounds of thrust does it take to throw objects?
By “throwing” them (with of a gun, say) at 3,200 feet per second, you would generate 100 pounds of thrust. 1 … Cite This! Marshall Brain “How Gas Turbine Engines Work” 1 April 2000.
How do you calculate the thrust required for a fixed wing?
If you are interested in details it depends on the Aspect Ratio (AR) and a few other factors of the wing. The thrust needed for any RC plane i.e. fixed wing can be calculated with thrust formula. T= 0.5 x Rho x (V^2) x S x Coefficient of Lift
<UNMASK>
# How much thrust do you need to lift 1 kg?
## How much thrust do you need to lift 1 kg?
To accelerate 1 kg at 9.8 m/s² requires 9.8 N of force, not any specific power.
How much power do you need to fly?
Power Requirement to Keep a Jet Aircraft in the Air The power required to keep a Boeing 737-300 flying at a constant altitude and speed is 7.2 × 106 watts. The rate of fuel burn necessary for the engines to produce enough power to keep the airplane flying can be calculated from the total power requirement.
How do you calculate thrust for an RC plane?
For Fixed wing you need a thrust of about 70 – 80\% from your all up weight. For example, 1 Kg aircraft you require 750grams thrust. You would require aleast 8 kg thrust at 100\% throttle so that your airplane could fly at a decent speed. ( Recommended thrust is more than its weight.)
### Can airplanes fly with solar power?
Solar unmanned airplane takes to the skies The unmanned solar-powered PHASA-35 airplane aims to fly up to 20 km (66,000 ft) in the sky using only sunlight. The PVs feed the batteries that power the aircraft during the day when sunlight is available, and store energy in a battery pack for night flights.
READ ALSO: Can hackers access my phone through a phone call?
How much thrust does it take to fly a plane?
For a craft weighing x kg you need g*x Newtons of thrust minimum for sustained vertical flight (ref.: high-school physics). In other words for each metric ton of weight you need around 9.81 kN of thrust.
What is minimum thrust required?
What is Minimum Thrust required? The Minimum Thrust required for a steady, level flight is equal to the sum of drag due to zero lift and drag due to lift and is represented as T=Pdynamic*A* (CD,0+CD,i) or Thrust of an aircraft=Dynamic Pressure*Area* (Zero-lift drag coefficient+Coefficient of drag due to lift).
#### How many pounds of thrust does it take to throw objects?
By “throwing” them (with of a gun, say) at 3,200 feet per second, you would generate 100 pounds of thrust. 1 … Cite This! Marshall Brain “How Gas Turbine Engines Work” 1 April 2000.
How do you calculate the thrust required for a fixed wing?
If you are interested in details it depends on the Aspect Ratio (AR) and a few other factors of the wing. The thrust needed for any RC plane i.e. fixed wing can be calculated with thrust formula. T= 0.5 x Rho x (V^2) x S x Coefficient of Lift
| 1,826 | 6,699,483 |
<MASK>
<UNMASK>
<MASK>
Thanks you very much for your help!
-
<MASK>
<UNMASK>
<MASK>
What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$. Any ideas?
Thanks you very much for your help!
-
<MASK>
<UNMASK>
# covariance of integral of Brownian
What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$. Any ideas?
Thanks you very much for your help!
-
<MASK>
<UNMASK>
# covariance of integral of Brownian
What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$. Any ideas?
Thanks you very much for your help!
-
$$\mathbb E(X(t)X(s))=\int_0^t\int_0^s\mathbb E(B(u)B(v))\,\mathrm dv\,\mathrm du=\int_0^t\int_0^s\min\{u,v\}\,\mathrm dv\,\mathrm du$$ Edit: As @TheBridge noted in a comment, the exchange of the order of integration is valid by Fubini theorem, since $\mathbb E(|B(u)B(v)|)\leqslant\mathbb E(B(u)^2)^{1/2}\mathbb E(B(v)^2)^{1/2}=\sqrt{uv}$, which is uniformly bounded on the domain $[0,t]\times[0,s]$ hence integrable on this domain.
Thank you very much :) That's exactly what I did and I got $\frac{1}{2}s^2t$, but shouldn't it give me something "symmetric"? I mean, if I exchange $s$ by $t$ then get the same answer? Since covariance matrices are symmetric. Thanks :) – martin Nov 20 '12 at 22:56
Once you assume that $s\lt t$, the setting is not symmetric with respect to $(s,t)$ anymore. I guess your formula is really $\frac12\min(s,t)^2\max(s,t)$, which is symmetric (but I did not check this was indeed the result). – Did Nov 21 '12 at 6:18
@TheBridge Right, post modified. (Unrelated: beware of the space between @ and user's name.) – Did Nov 21 '12 at 8:57
| 622 | 6,699,484 |
2019-2020-2021 StudyChaCha
#1
September 20th, 2016, 04:01 PM
Unregistered Guest
Model Question Paper For Corporation Bank Clerical Exam
Hii Buddy , Here I m Looking for Previous year Corporation Bank Clerical Recruitment Exam Question Paper , Would you please Provide me Same ?
#2
September 20th, 2016, 05:56 PM
Super Moderator Join Date: May 2011
Re: Model Question Paper For Corporation Bank Clerical Exam
Friend As on your Asking Here I m Giving Sample Questions for Corporation Bank Clerical Recruitment Exam paper :
1. A clock shows the time as 6 a.m. If the minute hand gains 2 minutes every hour, how many minutes
will the clock gain by 9 p.m.?
(a) 30 minutes
(b) 25 minutes
(c) 28 minutes
(d) 34 minutes
2. Find the right number, from the given options, at the place marked by the question mark: 2, 4, 8, 32,
256, ?
(a) 4096
(b) 8192
(c) 512
(d) 1024
3. Find the number missing at question mark:
10, 11, 23, 39, 64, ?, 149
(a) 100
(b) 103
(c) 78
(d) 128
4. A super fast bus of KSRTC starting from 'Trivandrum' and reaches 'Attingal' in 45 minutes with an
average speed of 40 km/hr. If the speed is increased by 10 km/hr how much time it will take to cover
the same distance?
(a) 34 minutes (b) 36 minutes (c) 38 minutes (d) 40 minutes
5. The difference between 6 times and 8 times of a figure is 14. What is the figure?
(a) 12 (b) 9 (c) 7 (d) 6
6. If 92y = 36 what is 9y?
(a) 4 (b) 6 (c) 9 (d) 18
7. One fourth percent of 180 is:
(a) 4.5 (b) 0.45 (c) 0.045 (d) 45
8. A candidate appearing for an examinatio n has to secure 40% marks to pass paper I. But he secured
only 40 marks and failed by 20 marks. What is the maximum mark for paper I?
(a) 100 (b) 200 (c) 180 (d) 150
9. Find the missing number 32, 52, 74, 112, 135
(a) 16 (b) 15 (c) 17 (d) 14
10. If 250 is increased to 300, what is the percentage increase?
(a) 16.67 (b) 20 (c) 23 (d) 17
11. The ratio of 9 seconds to 10 hours is .
(a) 1:40 (b) 1:4000 (c) 9:10 (d) 1:400
12. A person lost 10% when he sold goods at Rs.153. For how much should he sell them to gain 20%?
(a) 204 (b) 250 (c) 240 (d) 210
13. What will be xy if 7862xy is to be divisible by 125?
(a) 25 (b) 00 (c) 75 (d) 50
14. A train of 100 meters long is running at the speed of 36 km per hour. In what time it passes a bridge
of 80 meters long?
(a) 30 seconds (b) 36 seconds (c) 20 seconds (d) 18 seconds
15. If two-third of a bucket is filled in one minute then the time taken to fill the bucket completely will
be .
(a) 90 seconds (b) 70 seconds (c) 60 seconds (d) 100 seconds
16. If a quarter kilogram costs Rs. 60 then how much will cost for 150 grams?
(a) Rs. 30 (b) Rs. 24 (c) Rs. 36 (d) Rs. 40
17. If 3 men or 6 boys can do a piece of work in 20 days then how many days with 6 men and 8 boys
take to do the same work?
(a) 5 (b) 8 (c) 10 (d) 6
18. Find the sum of first 100 natural numbers
(a) 5050 (b) 5005 (c) 9900 (d) 9050
19. Two poles of height 6 meters and 11 meters stand on a plane ground. If the distance between their
feet is 12 meters then find the difference in the distance between their tops:
(a) 12m (b) 5m (c) 13m (d) 11m
(a) 4 (b) 8 (c) 16 (d) 2
21. The solution to x2 +6x+9 = 0 is ..
(a) x1 = + 3, x2 = -3 (b) x1 = 3, x2 = 3
(c) x1 = -3, x2 = -3 (d) No solution
22. What is the chance of getting a 2 or 4 in rolling a die?
(a) 2/3 (b) 1/6 (c) 1/3 (d) 1/2
23. At what rate of simple interest per annum an amount will be doubled in 10 years?
(a) 10% (b) 7.5% (c) 16% (d) 15%
24. Five times an unknown number is 5 less than 50. The unknown number
(a) 10 (b) 11 (c) 9 (d) 5
25. The acute angle between the hour hand and minute hand of a clock at 4 PM
(a) 900 (b) 1200 (c) 1500 (d) 2400
26. Water is filled in a cylindrical vessel in such a way that its volume doubles after every five
minutes. If it takes 30 minutes for the vessel to be full, then the vessel will be one fourth full in
(a) 20 minute (b) 25 minutes
(c) 7 minutes 30 seconds (d) 10 minutes
27. If 10 cats can kill 10 rats in 10 minutes how long will it take 100 cats to kill 100 rats
(a) 1 minutes (b) 10 minute (c) 100 minutes (d) 10000 minutes
28. If 75 % of a number is added to 75, the result is the number itself, then the number is:
(a) 250 (b) 750 (c) 400 (d) 300
29. A school has enough food for 400 children for 12 days. How long will the food last if 80 more
children join them?
(a) 6 days (b) 7 days (c) 10 days (d) 8 days
30. The sum of two consecutive numbers is 55, which is the largest number?
(a) 25 (b) 28 (c) 26 (d) 27
__________________
Message:
Options
Forum Jump StudyChaCha Discussion Forum General Topics Exams MBA / Business Schools Study Abroad and Immigration Consultancy Career and Jobs Questions by Topics Medicine and Health Management
All times are GMT +6.5. The time now is 10:24 PM.
-- Default Style -- Default vBulletin -- Lightweight MBA Discussion - Job Discussion - Contact Us - StudyChaCha - Blog Archives - Forum Archive - Partners : Management Forum | EduVark Top
| 1,731 | 6,699,485 |
<MASK>
<UNMASK>
Home MATHEMATICS TOPIC 2: FRACTIONS ~ MATHEMATICS FORM 1
# TOPIC 2: FRACTIONS ~ MATHEMATICS FORM 1
<MASK>
3. Simply wish to say your article is as astounding. The clearness in your submit is just great and that
i could suppose you are a professional on this subject.
Fine together with your permission allow me to take hold of
your feed to keep up to date with forthcoming post. Thanks one million and please carry on the rewarding work.
0mniartist asmr
<MASK>
7. With havin so much content do you ever run into any issues of plagorism or copyright violation? My website has
a lot of unique content I’ve either authored myself or outsourced but it appears a lot of it is popping
it up all over the internet without my permission. Do you know any
solutions to help reduce content from being
stolen? I’d truly appreciate it. 0mniartist asmr
<MASK>
9. Its like you learn my mind! You seem to understand so much approximately this, such as you wrote the guide
in it or something. I feel that you could do with some p.c.
to pressure the message house a bit, but other than that,
that is great blog. A fantastic read. I’ll certainly be back.
<UNMASK>
Home MATHEMATICS TOPIC 2: FRACTIONS ~ MATHEMATICS FORM 1
# TOPIC 2: FRACTIONS ~ MATHEMATICS FORM 1
<MASK>
1. I don’t know whether it’s just me or if perhaps everybody
else experiencing problems with your site. It appears as though some of
the text on your content are running off the screen. Can someone else please comment and let me know if this is happening to
them as well? This might be a problem with my browser because I’ve had this happen previously.
Cheers 0mniartist asmr
<MASK>
3. Simply wish to say your article is as astounding. The clearness in your submit is just great and that
i could suppose you are a professional on this subject.
Fine together with your permission allow me to take hold of
your feed to keep up to date with forthcoming post. Thanks one million and please carry on the rewarding work.
0mniartist asmr
<MASK>
7. With havin so much content do you ever run into any issues of plagorism or copyright violation? My website has
a lot of unique content I’ve either authored myself or outsourced but it appears a lot of it is popping
it up all over the internet without my permission. Do you know any
solutions to help reduce content from being
stolen? I’d truly appreciate it. 0mniartist asmr
<MASK>
9. Its like you learn my mind! You seem to understand so much approximately this, such as you wrote the guide
in it or something. I feel that you could do with some p.c.
to pressure the message house a bit, but other than that,
that is great blog. A fantastic read. I’ll certainly be back.
<UNMASK>
Home MATHEMATICS TOPIC 2: FRACTIONS ~ MATHEMATICS FORM 1
# TOPIC 2: FRACTIONS ~ MATHEMATICS FORM 1
<MASK>
1. I don’t know whether it’s just me or if perhaps everybody
else experiencing problems with your site. It appears as though some of
the text on your content are running off the screen. Can someone else please comment and let me know if this is happening to
them as well? This might be a problem with my browser because I’ve had this happen previously.
Cheers 0mniartist asmr
2. Hiya! I know this is kinda off topic however I’d figured
I’d ask. Would you be interested in exchanging links or maybe guest authoring a blog post or vice-versa?
My blog discusses a lot of the same subjects as yours and I
feel we could greatly benefit from each other. If you’re interested feel free to send
me an email. I look forward to hearing from you! Fantastic
blog by the way! 0mniartist asmr
3. Simply wish to say your article is as astounding. The clearness in your submit is just great and that
i could suppose you are a professional on this subject.
Fine together with your permission allow me to take hold of
your feed to keep up to date with forthcoming post. Thanks one million and please carry on the rewarding work.
0mniartist asmr
4. Hi there, i read your blog from time to time and i own a similar one and i was
just wondering if you get a lot of spam feedback? If so how do you reduce it, any
plugin or anything you can suggest? I get so much lately it’s driving me insane so any support is very much appreciated.
asmr 0mniartist
5. Hello There. I found your blog using msn. This is a really well written article.
Thanks for the post. I will definitely return. 0mniartist asmr
<MASK>
7. With havin so much content do you ever run into any issues of plagorism or copyright violation? My website has
a lot of unique content I’ve either authored myself or outsourced but it appears a lot of it is popping
it up all over the internet without my permission. Do you know any
solutions to help reduce content from being
stolen? I’d truly appreciate it. 0mniartist asmr
8. Do you have a spam problem on this site; I also am a blogger, and
I was wondering your situation; many of us have created
some nice methods and we are looking to exchange methods with other folks,
why not shoot me an e-mail if interested.
9. Its like you learn my mind! You seem to understand so much approximately this, such as you wrote the guide
in it or something. I feel that you could do with some p.c.
to pressure the message house a bit, but other than that,
that is great blog. A fantastic read. I’ll certainly be back.
<UNMASK>
Home MATHEMATICS TOPIC 2: FRACTIONS ~ MATHEMATICS FORM 1
# TOPIC 2: FRACTIONS ~ MATHEMATICS FORM 1
1580
13
A fraction is a number which is expressed in the form of a/b where a – is the top number called numerator and b– is the bottom number called denominator.
A Fraction
Describe a fraction
A fraction is a number which is expressed in the form of a/b where a – is the top number called numerator and b– is the bottom number called denominator.
Consider the diagram below
The shaded part in the diagram above is 1 out of 8, hence mathematically it is written as 1/8
Example 1
(a) 3 out of 5 ( three-fifths) = 3/5
Example 2
(b) 7 0ut of 8 ( i.e seven-eighths) = 7/8
Example 3
1. 5/12=(5 X 3)/(12 x 3) =15/36
2. 3/8 =(3 x 2)/(8 X 2) = 6/16
Dividing the numerator and denominator by the same number (This method is used to simplify the fraction)
Difference between Proper, Improper Fractions and Mixed Numbers
Distinguish proper, improper fractions and mixed numbers
Proper fraction –is a fraction in which the numerator is less than denominator
Example 4
4/5, 1/2, 11/13
Improper fraction -is a fraction whose numerator is greater than the denominator
Example 5
12/7, 4/3, 65/56
Mixed fraction –is a fraction which consist of a whole number and a proper fraction
Example 6
(a) To convert mixed fractions into improper fractions, use the formula below
(b)To convert improper fractions into mixed fractions, divide the numerator by the denominator
Example 7
Convert the following mixed numbers into improper fractions
Operations on fractions involves addition, subtraction, multiplication and division
• Addition and subtraction of fractions is done by putting both fractions under the same denominator and then add or subtract
• Multiplication
of fractions is done by multiplying the numerator of the first fraction
with the numerator of the second fraction, and the denominator of the
first fraction with the denominator the second fraction.
• For mixed fractions, convert them first into improper fractions and then multiply
• Division of fractions is done by taking the first fraction and then multiply with the reciprocal of the second fraction
• For mixed fractions, convert them first into improper fractions and then divide
Example 11
Find
Solution
Subtraction of Fractions
Subtract fractions
Example 12
Evaluate
Solution
Multiplication of Fractions
Multiply fractions
Example 13
Division of Fractions
Divide fractions
Example 14
Mixed Operations on Fractions
Perform mixed operations on fractions
Example 15
Example 16
Word Problems Involving Fractions
Solve word problems involving fractions
Example 17
1. Musa is years old. His father is 3¾times as old as he is. How old is his father?
2. 1¾of a material are needed to make suit. How many suits can be made from
1. I don’t know whether it’s just me or if perhaps everybody
else experiencing problems with your site. It appears as though some of
the text on your content are running off the screen. Can someone else please comment and let me know if this is happening to
them as well? This might be a problem with my browser because I’ve had this happen previously.
Cheers 0mniartist asmr
2. Hiya! I know this is kinda off topic however I’d figured
I’d ask. Would you be interested in exchanging links or maybe guest authoring a blog post or vice-versa?
My blog discusses a lot of the same subjects as yours and I
feel we could greatly benefit from each other. If you’re interested feel free to send
me an email. I look forward to hearing from you! Fantastic
blog by the way! 0mniartist asmr
3. Simply wish to say your article is as astounding. The clearness in your submit is just great and that
i could suppose you are a professional on this subject.
Fine together with your permission allow me to take hold of
your feed to keep up to date with forthcoming post. Thanks one million and please carry on the rewarding work.
0mniartist asmr
4. Hi there, i read your blog from time to time and i own a similar one and i was
just wondering if you get a lot of spam feedback? If so how do you reduce it, any
plugin or anything you can suggest? I get so much lately it’s driving me insane so any support is very much appreciated.
asmr 0mniartist
5. Hello There. I found your blog using msn. This is a really well written article.
Thanks for the post. I will definitely return. 0mniartist asmr
6. Nice post. I learn something totally new and challenging on blogs I stumbleupon everyday.
It’s always exciting to read content from other authors
and use a little something from their sites. asmr 0mniartist
7. With havin so much content do you ever run into any issues of plagorism or copyright violation? My website has
a lot of unique content I’ve either authored myself or outsourced but it appears a lot of it is popping
it up all over the internet without my permission. Do you know any
solutions to help reduce content from being
stolen? I’d truly appreciate it. 0mniartist asmr
8. Do you have a spam problem on this site; I also am a blogger, and
I was wondering your situation; many of us have created
some nice methods and we are looking to exchange methods with other folks,
why not shoot me an e-mail if interested.
9. Its like you learn my mind! You seem to understand so much approximately this, such as you wrote the guide
in it or something. I feel that you could do with some p.c.
to pressure the message house a bit, but other than that,
that is great blog. A fantastic read. I’ll certainly be back.
| 2,635 | 6,699,486 |
×
×
# In Exercises, find the range, variance, and | Ch 3.3 - 11BSC
ISBN: 9780321836960 18
## Solution for problem 11BSC Chapter 3.3
Elementary Statistics | 12th Edition
• Textbook Solutions
• 2901 Step-by-step solutions solved by professors and subject experts
• Get 24/7 help from StudySoup virtual teaching assistants
Elementary Statistics | 12th Edition
4 5 1 315 Reviews
16
2
Problem 11BSC
In Exercises, find the range, variance, and standard deviation for the given samph data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-2 where we found measures of center. Here we find measures of variation.) Then answer the given questions.
Ghost Prices Listed below are the prices listed for Norton Ghost 14.0 software from these vendors: Newegg, Dell, Buycheapsoftware.com, PC Connection, Walmart, and Overstock.com. When trying to find the best deal, how helpful are the measures of variation?
Step-by-Step Solution:
Step 1 of 3
Solution 11BSC
From the given information, the prices of Norton Ghost 14.0 software from the given vendors are
From the given information, maximum data value is \$71.77 and minimum value is \$48.92. We know that range of the given set of sample values found as follows:
Therefore, the range of the given set of observations is \$22.85.
From the given information,....
Step 2 of 3
Step 3 of 3
##### ISBN: 9780321836960
The full step-by-step solution to problem: 11BSC from chapter: 3.3 was answered by , our top Statistics solution expert on 03/15/17, 10:30PM. The answer to “In Exercises, find the range, variance, and standard deviation for the given samph data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-2 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Ghost Prices Listed below are the prices listed for Norton Ghost 14.0 software from these vendors: Newegg, Dell, Buycheapsoftware.com, PC Connection, Walmart, and Overstock.com. When trying to find the best deal, how helpful are the measures of variation?” is broken down into a number of easy to follow steps, and 85 words. Since the solution to 11BSC from 3.3 chapter was answered, more than 414 students have viewed the full step-by-step answer. This full solution covers the following key subjects: measures, Find, given, buycheapsoftware, listed. This expansive textbook survival guide covers 121 chapters, and 3629 solutions. Elementary Statistics was written by and is associated to the ISBN: 9780321836960. This textbook survival guide was created for the textbook: Elementary Statistics, edition: 12.
#### Related chapters
Unlock Textbook Solution
| 667 | 6,699,487 |
<MASK>
23. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?
a) X(k)=Nck
b) X(k)=ck/N
c) X(k)=N/ck
d) None of the mentioned
<MASK>
25. If W4100=Wx200, then what is the value of x?
a) 2
b) 4
c) 8
d) 16
<UNMASK>
# Discrete Time Signal Analysis
This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Frequency Analysis of Discrete Time Signals-1”.
<MASK>
2. What is the expression for Fourier series coefficient ck in terms of the discrete signal x(n)?
a) 1N∑N−1n=0x(n)ej2πkn/N
b) N∑N−1n=0x(n)e−j2πkn/N
c) 1N∑N+1n=0x(n)e−j2πkn/N
d) 1N∑N−1n=0x(n)e−j2πkn/N
<MASK>
4. The Fourier series for the signal x(n)=cos√2πn exists.
a) True
b) False
5. What are the Fourier series coefficients for the signal x(n)=cosπn/3?
a) c1=c2=c3=c4=0,c1=c5=1/2
b) c0=c1=c2=c3=c4=c5=0
c) c0=c1=c2=c3=c4=c5=1/2
d) none of the mentioned
<MASK>
7. What is the average power of the discrete time periodic signal x(n) with period N?
a) 1N∑Nn=0|x(n)|
b) 1N∑N−1n=0|x(n)|
c) 1N∑Nn=0|x(n)|2
d) 1N∑N−1n=0|x(n)|2
8. What is the equation for average power of discrete time periodic signal x(n) with period N in terms of Fourier series coefficient ck?
a) ∑N−1k=0|ck|
b) ∑N−1k=0|ck|2
c) ∑Nk=0|ck|2
d) ∑Nk=0|ck|
9. What is the Fourier transform X(ω) of a finite energy discrete time signal x(n)?
a) ∑∞n=−∞x(n)e−jωn
b) ∑∞n=0x(n)e−jωn
c) ∑N−1n=0x(n)e−jωn
d) None of the mentioned
<MASK>
11. What is the synthesis equation of the discrete time signal x(n), whose Fourier transform is X(ω)?
a) 2π∫20πX(ω)ejωndω
b) 1π∫2π0X(ω)ejωndω
c) 12π∫2π0X(ω)ejωndω
d) None of the mentioned
<MASK>
13. What is the value of discrete time signal x(n) at n≠0 whose Fourier transform is represented as below?
a) ωcπ.sinωc.nωc.n
b) −ωcπ.sinωc.nωc.n
c) ωc.πsinωc.nωc.n
d) None of the mentioned
14. The oscillatory behavior of the approximation of XN(ω) to the function X(ω) at a point of discontinuity of X(ω) is known as Gibbs phenomenon.
a) True
b) False
15. What is the energy of a discrete time signal in terms of X(ω)?
a) 2π∫π−π|X(ω)|2dω
b) 12π∫π−π|X(ω)|2dω
c) 12π∫π0|X(ω)|2dω
d) None of the mentioned
<MASK>
17. Which of the following has to be performed in sampling rate conversion by rational factor?
a) Interpolation
b) Decimation
c) Either interpolation or decimation
d) None of the mentioned
18. Which of the following operation is performed by the blocks given the figure below?
a) Sampling rate conversion by a factor I
b) Sampling rate conversion by a factor D
c) Sampling rate conversion by a factor D/I
d) Sampling rate conversion by a factor I/D
19. The Nth root of unity WN is given as _____________
a) ej2πN
b) e-j2πN
c) e-j2π/N
d) ej2π/N
<MASK>
21. Which of the following is true?
a) W∗N=1NWN−1
b) WN−1=1NWN∗
c) WN−1=WN∗
d) None of the mentioned
22. What is the DFT of the four point sequence x(n)={0,1,2,3}?
a) {6,-2+2j-2,-2-2j}
b) {6,-2-2j,2,-2+2j}
c) {6,-2+2j,-2,-2-2j}
d) {6,-2-2j,-2,-2+2j}
23. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?
a) X(k)=Nck
b) X(k)=ck/N
c) X(k)=N/ck
d) None of the mentioned
<MASK>
25. If W4100=Wx200, then what is the value of x?
a) 2
b) 4
c) 8
d) 16
<UNMASK>
# Discrete Time Signal Analysis
This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Frequency Analysis of Discrete Time Signals-1”.
1. What is the Fourier series representation of a signal x(n) whose period is N?
a) ∑N+1k=0ckej2πkn/N
b) ∑N−1k=0ckej2πkn/N
c) ∑Nk=0ckej2πkn/N
d) ∑N−1k=0cke−j2πkn/N
2. What is the expression for Fourier series coefficient ck in terms of the discrete signal x(n)?
a) 1N∑N−1n=0x(n)ej2πkn/N
b) N∑N−1n=0x(n)e−j2πkn/N
c) 1N∑N+1n=0x(n)e−j2πkn/N
d) 1N∑N−1n=0x(n)e−j2πkn/N
<MASK>
4. The Fourier series for the signal x(n)=cos√2πn exists.
a) True
b) False
5. What are the Fourier series coefficients for the signal x(n)=cosπn/3?
a) c1=c2=c3=c4=0,c1=c5=1/2
b) c0=c1=c2=c3=c4=c5=0
c) c0=c1=c2=c3=c4=c5=1/2
d) none of the mentioned
<MASK>
7. What is the average power of the discrete time periodic signal x(n) with period N?
a) 1N∑Nn=0|x(n)|
b) 1N∑N−1n=0|x(n)|
c) 1N∑Nn=0|x(n)|2
d) 1N∑N−1n=0|x(n)|2
8. What is the equation for average power of discrete time periodic signal x(n) with period N in terms of Fourier series coefficient ck?
a) ∑N−1k=0|ck|
b) ∑N−1k=0|ck|2
c) ∑Nk=0|ck|2
d) ∑Nk=0|ck|
9. What is the Fourier transform X(ω) of a finite energy discrete time signal x(n)?
a) ∑∞n=−∞x(n)e−jωn
b) ∑∞n=0x(n)e−jωn
c) ∑N−1n=0x(n)e−jωn
d) None of the mentioned
<MASK>
11. What is the synthesis equation of the discrete time signal x(n), whose Fourier transform is X(ω)?
a) 2π∫20πX(ω)ejωndω
b) 1π∫2π0X(ω)ejωndω
c) 12π∫2π0X(ω)ejωndω
d) None of the mentioned
<MASK>
13. What is the value of discrete time signal x(n) at n≠0 whose Fourier transform is represented as below?
a) ωcπ.sinωc.nωc.n
b) −ωcπ.sinωc.nωc.n
c) ωc.πsinωc.nωc.n
d) None of the mentioned
14. The oscillatory behavior of the approximation of XN(ω) to the function X(ω) at a point of discontinuity of X(ω) is known as Gibbs phenomenon.
a) True
b) False
15. What is the energy of a discrete time signal in terms of X(ω)?
a) 2π∫π−π|X(ω)|2dω
b) 12π∫π−π|X(ω)|2dω
c) 12π∫π0|X(ω)|2dω
d) None of the mentioned
<MASK>
17. Which of the following has to be performed in sampling rate conversion by rational factor?
a) Interpolation
b) Decimation
c) Either interpolation or decimation
d) None of the mentioned
18. Which of the following operation is performed by the blocks given the figure below?
a) Sampling rate conversion by a factor I
b) Sampling rate conversion by a factor D
c) Sampling rate conversion by a factor D/I
d) Sampling rate conversion by a factor I/D
19. The Nth root of unity WN is given as _____________
a) ej2πN
b) e-j2πN
c) e-j2π/N
d) ej2π/N
<MASK>
21. Which of the following is true?
a) W∗N=1NWN−1
b) WN−1=1NWN∗
c) WN−1=WN∗
d) None of the mentioned
22. What is the DFT of the four point sequence x(n)={0,1,2,3}?
a) {6,-2+2j-2,-2-2j}
b) {6,-2-2j,2,-2+2j}
c) {6,-2+2j,-2,-2-2j}
d) {6,-2-2j,-2,-2+2j}
23. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?
a) X(k)=Nck
b) X(k)=ck/N
c) X(k)=N/ck
d) None of the mentioned
14. What is the DFT of the four point sequence x(n)={0,1,2,3}?
a) {6,-2+2j-2,-2-2j}
b) {6,-2-2j,2,-2+2j}
c) {6,-2-2j,-2,-2+2j}
d) {6,-2+2j,-2,-2-2j}
25. If W4100=Wx200, then what is the value of x?
a) 2
b) 4
c) 8
d) 16
<UNMASK>
# Discrete Time Signal Analysis
This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Frequency Analysis of Discrete Time Signals-1”.
1. What is the Fourier series representation of a signal x(n) whose period is N?
a) ∑N+1k=0ckej2πkn/N
b) ∑N−1k=0ckej2πkn/N
c) ∑Nk=0ckej2πkn/N
d) ∑N−1k=0cke−j2πkn/N
2. What is the expression for Fourier series coefficient ck in terms of the discrete signal x(n)?
a) 1N∑N−1n=0x(n)ej2πkn/N
b) N∑N−1n=0x(n)e−j2πkn/N
c) 1N∑N+1n=0x(n)e−j2πkn/N
d) 1N∑N−1n=0x(n)e−j2πkn/N
<MASK>
4. The Fourier series for the signal x(n)=cos√2πn exists.
a) True
b) False
5. What are the Fourier series coefficients for the signal x(n)=cosπn/3?
a) c1=c2=c3=c4=0,c1=c5=1/2
b) c0=c1=c2=c3=c4=c5=0
c) c0=c1=c2=c3=c4=c5=1/2
d) none of the mentioned
<MASK>
7. What is the average power of the discrete time periodic signal x(n) with period N?
a) 1N∑Nn=0|x(n)|
b) 1N∑N−1n=0|x(n)|
c) 1N∑Nn=0|x(n)|2
d) 1N∑N−1n=0|x(n)|2
8. What is the equation for average power of discrete time periodic signal x(n) with period N in terms of Fourier series coefficient ck?
a) ∑N−1k=0|ck|
b) ∑N−1k=0|ck|2
c) ∑Nk=0|ck|2
d) ∑Nk=0|ck|
9. What is the Fourier transform X(ω) of a finite energy discrete time signal x(n)?
a) ∑∞n=−∞x(n)e−jωn
b) ∑∞n=0x(n)e−jωn
c) ∑N−1n=0x(n)e−jωn
d) None of the mentioned
10. What is the period of the Fourier transform X(ω) of the signal x(n)?
a) π
b) 1
c) Non-periodic
d) 2π
11. What is the synthesis equation of the discrete time signal x(n), whose Fourier transform is X(ω)?
a) 2π∫20πX(ω)ejωndω
b) 1π∫2π0X(ω)ejωndω
c) 12π∫2π0X(ω)ejωndω
d) None of the mentioned
<MASK>
13. What is the value of discrete time signal x(n) at n≠0 whose Fourier transform is represented as below?
a) ωcπ.sinωc.nωc.n
b) −ωcπ.sinωc.nωc.n
c) ωc.πsinωc.nωc.n
d) None of the mentioned
14. The oscillatory behavior of the approximation of XN(ω) to the function X(ω) at a point of discontinuity of X(ω) is known as Gibbs phenomenon.
a) True
b) False
15. What is the energy of a discrete time signal in terms of X(ω)?
a) 2π∫π−π|X(ω)|2dω
b) 12π∫π−π|X(ω)|2dω
c) 12π∫π0|X(ω)|2dω
d) None of the mentioned
16. Sampling rate conversion by the rational factor I/D is accomplished by what connection of interpolator and decimator?
a) Parallel
c) Convolution
d) None of the mentioned
17. Which of the following has to be performed in sampling rate conversion by rational factor?
a) Interpolation
b) Decimation
c) Either interpolation or decimation
d) None of the mentioned
18. Which of the following operation is performed by the blocks given the figure below?
a) Sampling rate conversion by a factor I
b) Sampling rate conversion by a factor D
c) Sampling rate conversion by a factor D/I
d) Sampling rate conversion by a factor I/D
19. The Nth root of unity WN is given as _____________
a) ej2πN
b) e-j2πN
c) e-j2π/N
d) ej2π/N
<MASK>
21. Which of the following is true?
a) W∗N=1NWN−1
b) WN−1=1NWN∗
c) WN−1=WN∗
d) None of the mentioned
22. What is the DFT of the four point sequence x(n)={0,1,2,3}?
a) {6,-2+2j-2,-2-2j}
b) {6,-2-2j,2,-2+2j}
c) {6,-2+2j,-2,-2-2j}
d) {6,-2-2j,-2,-2+2j}
23. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?
a) X(k)=Nck
b) X(k)=ck/N
c) X(k)=N/ck
d) None of the mentioned
14. What is the DFT of the four point sequence x(n)={0,1,2,3}?
a) {6,-2+2j-2,-2-2j}
b) {6,-2-2j,2,-2+2j}
c) {6,-2-2j,-2,-2+2j}
d) {6,-2+2j,-2,-2-2j}
25. If W4100=Wx200, then what is the value of x?
a) 2
b) 4
c) 8
d) 16
<UNMASK>
# Discrete Time Signal Analysis
This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Frequency Analysis of Discrete Time Signals-1”.
1. What is the Fourier series representation of a signal x(n) whose period is N?
a) ∑N+1k=0ckej2πkn/N
b) ∑N−1k=0ckej2πkn/N
c) ∑Nk=0ckej2πkn/N
d) ∑N−1k=0cke−j2πkn/N
2. What is the expression for Fourier series coefficient ck in terms of the discrete signal x(n)?
a) 1N∑N−1n=0x(n)ej2πkn/N
b) N∑N−1n=0x(n)e−j2πkn/N
c) 1N∑N+1n=0x(n)e−j2πkn/N
d) 1N∑N−1n=0x(n)e−j2πkn/N
3. Which of the following represents the phase associated with the frequency component of discrete-time Fourier series(DTFS)?
a) ej2πkn/N
b) e-j2πkn/N
c) ej2πknN
d) none of the mentioned
4. The Fourier series for the signal x(n)=cos√2πn exists.
a) True
b) False
5. What are the Fourier series coefficients for the signal x(n)=cosπn/3?
a) c1=c2=c3=c4=0,c1=c5=1/2
b) c0=c1=c2=c3=c4=c5=0
c) c0=c1=c2=c3=c4=c5=1/2
d) none of the mentioned
6. What is the Fourier series representation of a signal x(n) whose period is N?
a) ∑∞k=0|ck|2
b) ∑∞k=−∞|ck|
c) ∑0k=−∞|ck|2
d) ∑∞k=−∞|ck|2
7. What is the average power of the discrete time periodic signal x(n) with period N?
a) 1N∑Nn=0|x(n)|
b) 1N∑N−1n=0|x(n)|
c) 1N∑Nn=0|x(n)|2
d) 1N∑N−1n=0|x(n)|2
8. What is the equation for average power of discrete time periodic signal x(n) with period N in terms of Fourier series coefficient ck?
a) ∑N−1k=0|ck|
b) ∑N−1k=0|ck|2
c) ∑Nk=0|ck|2
d) ∑Nk=0|ck|
9. What is the Fourier transform X(ω) of a finite energy discrete time signal x(n)?
a) ∑∞n=−∞x(n)e−jωn
b) ∑∞n=0x(n)e−jωn
c) ∑N−1n=0x(n)e−jωn
d) None of the mentioned
10. What is the period of the Fourier transform X(ω) of the signal x(n)?
a) π
b) 1
c) Non-periodic
d) 2π
11. What is the synthesis equation of the discrete time signal x(n), whose Fourier transform is X(ω)?
a) 2π∫20πX(ω)ejωndω
b) 1π∫2π0X(ω)ejωndω
c) 12π∫2π0X(ω)ejωndω
d) None of the mentioned
12. What is the value of discrete time signal x(n) at n=0 whose Fourier transform is represented as below?
a) ωc
b) -ωc
c) ωc
d) none of the mentioned
13. What is the value of discrete time signal x(n) at n≠0 whose Fourier transform is represented as below?
a) ωcπ.sinωc.nωc.n
b) −ωcπ.sinωc.nωc.n
c) ωc.πsinωc.nωc.n
d) None of the mentioned
14. The oscillatory behavior of the approximation of XN(ω) to the function X(ω) at a point of discontinuity of X(ω) is known as Gibbs phenomenon.
a) True
b) False
15. What is the energy of a discrete time signal in terms of X(ω)?
a) 2π∫π−π|X(ω)|2dω
b) 12π∫π−π|X(ω)|2dω
c) 12π∫π0|X(ω)|2dω
d) None of the mentioned
16. Sampling rate conversion by the rational factor I/D is accomplished by what connection of interpolator and decimator?
a) Parallel
c) Convolution
d) None of the mentioned
17. Which of the following has to be performed in sampling rate conversion by rational factor?
a) Interpolation
b) Decimation
c) Either interpolation or decimation
d) None of the mentioned
18. Which of the following operation is performed by the blocks given the figure below?
a) Sampling rate conversion by a factor I
b) Sampling rate conversion by a factor D
c) Sampling rate conversion by a factor D/I
d) Sampling rate conversion by a factor I/D
19. The Nth root of unity WN is given as _____________
a) ej2πN
b) e-j2πN
c) e-j2π/N
d) ej2π/N
20. Which of the following is true regarding the number of computations requires to compute an N-point DFT?
a) N2 complex multiplications and N(N-1) complex additions
b) N2 complex additions and N(N-1) complex multiplications
c) N2 complex multiplications and N(N+1) complex additions
d) N2 complex additions and N(N+1) complex multiplications
21. Which of the following is true?
a) W∗N=1NWN−1
b) WN−1=1NWN∗
c) WN−1=WN∗
d) None of the mentioned
22. What is the DFT of the four point sequence x(n)={0,1,2,3}?
a) {6,-2+2j-2,-2-2j}
b) {6,-2-2j,2,-2+2j}
c) {6,-2+2j,-2,-2-2j}
d) {6,-2-2j,-2,-2+2j}
23. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?
a) X(k)=Nck
b) X(k)=ck/N
c) X(k)=N/ck
d) None of the mentioned
14. What is the DFT of the four point sequence x(n)={0,1,2,3}?
a) {6,-2+2j-2,-2-2j}
b) {6,-2-2j,2,-2+2j}
c) {6,-2-2j,-2,-2+2j}
d) {6,-2+2j,-2,-2-2j}
25. If W4100=Wx200, then what is the value of x?
a) 2
b) 4
c) 8
d) 16
| 5,930 | 6,699,488 |
Equivalent Fractions
Comparing Fractions with <, >, or =
Multiplying Fractions
100
3/5 = 9/15
What is EQUAL.
100
1/11 ----- 6/11
1/11 > 6/11
100
1 1/4 + 1 3/4
3
100
3 x 4/8 *mixed number*
1 4/8
200
3/10 = 12/40
What is EQUAL.
200
2/3 ----- 2/4
2/3 > 2/4
200
1 8/16 + 1 3/16
2 11/16
200
4 x 5/10 *as a improper fraction*
20/10
300
7/9 = 23/27
What is NOT EQUAL.
300
21/12 ----- 12/6
21/12 < 12/6
300
4 3/5 - 1 1/5
3 2/5
300
5 x 2/7 *as a improper fraction AND mixed number*
10/7 OR 1 3/7
400
4/8 = 12/32
What is NOT EQUAL.
400
1/3 ----- 6/18
1/3 = 8/18
400
2 4/12 + 1 7/12 + 3 1/12
7
400
10 x 1/9 *as a improper fraction AND mixed number*
10/9 OR 1 1/9
500
4/7 = 12/23
What is NOT EQUAL.
500
16/20 ----- 4/3
16/20 < 4/3
500
3 4/7 + 1 3/7 + 1 1/7
6 1/7
500
11 x 3/5 *as a improper fraction AND mixed number*
33/5 or 6 3/5
| 430 | 6,699,489 |
<MASK>
A nucleus of (147N) nitrogen contains 7 protons and 7 neutrons.
<MASK>
= 0.11236 u
<MASK>
∆m = 0.11236 × 931.5 MeV/c2
<MASK>
Eb = 0.11236 × 931.5(MeV/c2)/c2
<MASK>
(5626Fe) nucleus has 26 protons and (56 − 26) = 30 neutrons
<MASK>
Where,
<MASK>
= 0.528461 u
<MASK>
∆m' = (83 × mH + 126 × mn) − m2
<MASK>
= 83.649475 + 127.091790 − 208.980388
<MASK>
<UNMASK>
<MASK>
A nucleus of (147N) nitrogen contains 7 protons and 7 neutrons.
<MASK>
= 0.11236 u
<MASK>
∆m = 0.11236 × 931.5 MeV/c2
<MASK>
Eb = 0.11236 × 931.5(MeV/c2)/c2
<MASK>
(5626Fe) nucleus has 26 protons and (56 − 26) = 30 neutrons
<MASK>
Where,
<MASK>
= 0.528461 u
But 1 u = 931.5 MeV/c2
<MASK>
∆m' = (83 × mH + 126 × mn) − m2
<MASK>
Mass of a neutron, mn = 1.008665 u
<MASK>
= 83.649475 + 127.091790 − 208.980388
<MASK>
<UNMASK>
Class 12 Physics Nuclei Nuclear binding energy per nucleon
<MASK>
• Nuclear binding energy per nucleon is defined as the average energy per nucleon needed to separate a nucleus into its individual constituents.
• It is denoted by Ebn.
• Experimentally there was a graph plotted between binding energy per nucleon and the mass number(A).
• Following are the observations from the graph:-
1. Initially the graph was increasing.This implies that Ebn is very less for lesser mass number.
2. In the middle range the Ebn becomes constant.This means Ebn is independent of mass number.
3. In the end Ebnstarts decreasing.This shows Ebn is less when mass number is more.
<MASK>
A nucleus of (147N) nitrogen contains 7 protons and 7 neutrons.
<MASK>
Mass of a proton, mH = 1.007825 u
<MASK>
Therefore, ∆m = (7 × 1.007825 + 7 × 1.008665 − 14.00307)
= 7.054775 + 7.06055 − 14.00307
= 0.11236 u
<MASK>
∆m = 0.11236 × 931.5 MeV/c2
Hence, the binding energy of the nucleus is given as:
Eb = ∆mc2
Where, c = Speed of light
Eb = 0.11236 × 931.5(MeV/c2)/c2
= 104.66334 MeV
Hence, the binding energy of a nitrogen nucleus is 104.66334 MeV.
<MASK>
(5626Fe) nucleus has 26 protons and (56 − 26) = 30 neutrons
<MASK>
Where,
<MASK>
∆m = ((26 × 1.007825 + 30 × 1.008665) − 55.934939)
= (26.20345 + 30.25995 − 55.934939)
= 0.528461 u
But 1 u = 931.5 MeV/c2
<MASK>
Eb1 = ∆mc2
Where, c =Speed of light
<MASK>
(20983Bi) nucleus has 83 protons and (209 − 83) = 126 neutrons.
Hence, the mass defect of this nucleus is given as:
∆m' = (83 × mH + 126 × mn) − m2
Where,
Mass of a proton, mH = 1.007825 u
Mass of a neutron, mn = 1.008665 u
∆m' = (83 × 1.007825 + 126 × 1.008665 – 208.980388)
= 83.649475 + 127.091790 − 208.980388
= 1.760877 u
But 1 u = 931.5 MeV/c2
<MASK>
Hence, the binding energy of this nucleus is given as:
Eb2 = ∆m'c2
= 1.760877 × 931.5(MeV/c2) x c2
= 1640.26 MeV
<MASK>
<UNMASK>
Class 12 Physics Nuclei Nuclear binding energy per nucleon
<MASK>
• Nuclear binding energy per nucleon is defined as the average energy per nucleon needed to separate a nucleus into its individual constituents.
• It is denoted by Ebn.
• Experimentally there was a graph plotted between binding energy per nucleon and the mass number(A).
• Following are the observations from the graph:-
1. Initially the graph was increasing.This implies that Ebn is very less for lesser mass number.
2. In the middle range the Ebn becomes constant.This means Ebn is independent of mass number.
3. In the end Ebnstarts decreasing.This shows Ebn is less when mass number is more.
Problem: - Obtain the binding energy (in MeV) of a nitrogen nucleus (147N), given m (147N) =14.00307u?
Answer:- Atomic mass of (147N ) nitrogen, m = 14.00307 u
A nucleus of (147N) nitrogen contains 7 protons and 7 neutrons.
<MASK>
Mass of a proton, mH = 1.007825 u
<MASK>
Therefore, ∆m = (7 × 1.007825 + 7 × 1.008665 − 14.00307)
= 7.054775 + 7.06055 − 14.00307
= 0.11236 u
But 1 u = 931.5 MeV/c2
∆m = 0.11236 × 931.5 MeV/c2
Hence, the binding energy of the nucleus is given as:
Eb = ∆mc2
Where, c = Speed of light
Eb = 0.11236 × 931.5(MeV/c2)/c2
= 104.66334 MeV
Hence, the binding energy of a nitrogen nucleus is 104.66334 MeV.
Deriving Nuclear force from Ebn
<MASK>
Problem: - Obtain the binding energy of the nuclei (5626Fe) and (20983Bi) in units of MeV from the following data: m (5626Fe) = 55.934939u and m (20983Bi) = 208.980388 u.
Answer:- Atomic mass of (5626Fe), mFe = 55.934939 u
(5626Fe) nucleus has 26 protons and (56 − 26) = 30 neutrons
<MASK>
Where,
Mass of a proton, mH = 1.007825 u
<MASK>
∆m = ((26 × 1.007825 + 30 × 1.008665) − 55.934939)
= (26.20345 + 30.25995 − 55.934939)
= 0.528461 u
But 1 u = 931.5 MeV/c2
Therefore, ∆m = 0.528461 × 931.5 MeV/c2
<MASK>
Eb1 = ∆mc2
Where, c =Speed of light
Eb1 = 0.528461 × 931.5(MeV/c2)/c2
= 492.26 MeV
<MASK>
Atomic mass of (20983Bi), m2 = 208.980388 u
(20983Bi) nucleus has 83 protons and (209 − 83) = 126 neutrons.
Hence, the mass defect of this nucleus is given as:
∆m' = (83 × mH + 126 × mn) − m2
Where,
Mass of a proton, mH = 1.007825 u
Mass of a neutron, mn = 1.008665 u
∆m' = (83 × 1.007825 + 126 × 1.008665 – 208.980388)
= 83.649475 + 127.091790 − 208.980388
= 1.760877 u
But 1 u = 931.5 MeV/c2
Therefore, ∆m' = 1.760877 × 931.5 (MeV/c2) x c2
Hence, the binding energy of this nucleus is given as:
Eb2 = ∆m'c2
= 1.760877 × 931.5(MeV/c2) x c2
= 1640.26 MeV
Average binding energy per nucleon = (1640.26/209) =7.848MeV
.
<UNMASK>
Class 12 Physics Nuclei Nuclear binding energy per nucleon
Nuclear binding energy per nucleon
• Nuclear binding energy per nucleon is defined as the average energy per nucleon needed to separate a nucleus into its individual constituents.
• It is denoted by Ebn.
• Experimentally there was a graph plotted between binding energy per nucleon and the mass number(A).
• Following are the observations from the graph:-
1. Initially the graph was increasing.This implies that Ebn is very less for lesser mass number.
2. In the middle range the Ebn becomes constant.This means Ebn is independent of mass number.
3. In the end Ebnstarts decreasing.This shows Ebn is less when mass number is more.
Problem: - Obtain the binding energy (in MeV) of a nitrogen nucleus (147N), given m (147N) =14.00307u?
Answer:- Atomic mass of (147N ) nitrogen, m = 14.00307 u
A nucleus of (147N) nitrogen contains 7 protons and 7 neutrons.
Hence, the mass defect of this nucleus, ∆m = 7mH + 7mn − m
Where,
Mass of a proton, mH = 1.007825 u
Mass of a neutron, mn= 1.008665 u
Therefore, ∆m = (7 × 1.007825 + 7 × 1.008665 − 14.00307)
= 7.054775 + 7.06055 − 14.00307
= 0.11236 u
But 1 u = 931.5 MeV/c2
∆m = 0.11236 × 931.5 MeV/c2
Hence, the binding energy of the nucleus is given as:
Eb = ∆mc2
Where, c = Speed of light
Eb = 0.11236 × 931.5(MeV/c2)/c2
= 104.66334 MeV
Hence, the binding energy of a nitrogen nucleus is 104.66334 MeV.
Deriving Nuclear force from Ebn
1. Lighter nuclei: -
1. In the initial part of the graph A(mass number) is less therefore Ebn is also less.As a result lesser energy is required to separate the nucleons.
2. This showsnuclei areunstable.
3. The nuclei are unstable and in order to become stable lighter nuclei combine with each other to form heavier nuclei.
4. Let the energy of heavier nuclei formed is E’bn and of lighter nuclei be Ebn. This implies E’bn> Ebn.
5. Energy is released when 2 lighter nuclei combine together to form a heavier nuclei.
6. This process is known as Nuclear Fusion.
2. For heavier nuclei:-
1. Mass number is very high and Ebn is very less.
2. In order to become stable the heavier nuclei will split into 2 lighter nuclei.
3. Energy associated with heavier nuclei =Ebn and energy associated with 2 lighter nuclei =E’bn.
4. This implies E’bn> Ebn. Energy is released in this process by the heavier nuclei in order to attain stability.
5. This process is known as Nuclear Fission.
3. Constancy of Ebnin the mid-range of A:-
1. In this portion the mass number is increasing due to whichnumber of nucleons also increase.
2. The force which is present between the nucleons is of short range.The strength of the force decreases as the distance increases.
3. The nucleons are getting affected by their nearest neighbouring nucleons and not by the nucleons which are far away.
4. As a result Ebn remains constant.
5. But when there are too many nucleons Ebn suddenly starts decreasing.
Problem: - Obtain the binding energy of the nuclei (5626Fe) and (20983Bi) in units of MeV from the following data: m (5626Fe) = 55.934939u and m (20983Bi) = 208.980388 u.
Answer:- Atomic mass of (5626Fe), mFe = 55.934939 u
(5626Fe) nucleus has 26 protons and (56 − 26) = 30 neutrons
Hence, the mass defect of the nucleus, ∆m = ((26 × mH) + (30 × mn)– mFe)
Where,
Mass of a proton, mH = 1.007825 u
Mass of a neutron, mn = 1.008665 u
∆m = ((26 × 1.007825 + 30 × 1.008665) − 55.934939)
= (26.20345 + 30.25995 − 55.934939)
= 0.528461 u
But 1 u = 931.5 MeV/c2
Therefore, ∆m = 0.528461 × 931.5 MeV/c2
The binding energy of this nucleus is given as:
Eb1 = ∆mc2
Where, c =Speed of light
Eb1 = 0.528461 × 931.5(MeV/c2)/c2
= 492.26 MeV
Average binding energy per nucleon= (492.26/56)=8.76 MeV
Atomic mass of (20983Bi), m2 = 208.980388 u
(20983Bi) nucleus has 83 protons and (209 − 83) = 126 neutrons.
Hence, the mass defect of this nucleus is given as:
∆m' = (83 × mH + 126 × mn) − m2
Where,
Mass of a proton, mH = 1.007825 u
Mass of a neutron, mn = 1.008665 u
∆m' = (83 × 1.007825 + 126 × 1.008665 – 208.980388)
= 83.649475 + 127.091790 − 208.980388
= 1.760877 u
But 1 u = 931.5 MeV/c2
Therefore, ∆m' = 1.760877 × 931.5 (MeV/c2) x c2
Hence, the binding energy of this nucleus is given as:
Eb2 = ∆m'c2
= 1.760877 × 931.5(MeV/c2) x c2
= 1640.26 MeV
Average binding energy per nucleon = (1640.26/209) =7.848MeV
.
| 3,391 | 6,699,490 |
## DSE Math 2016 Paper 1 ANS
Posted: March 27, 2017 in Mathematics
Exam week’s coming, so here it is.
###### Section A (35 marks)
(1) Simplify $\displaystyle\frac{(x^8 y^7)^2}{x^5 y^{-6}}$ and express your answer with positive indices. (3 marks)
Solution
$\displaystyle\frac{(x^8 y^7)^2}{x^5 y^{-6}} = \frac{x^{16}y^{14}}{x^5 y^{-6}} = x^{11}y^{20}$
(2) Make $x$ the subject of the formula $Ax=(4x+B)C$. (3 marks)
Solution
$Ax=4Cx+BC$
$Ax-4Cx=BC$
$\therefore x = \dfrac{BC}{A-4C}$
(3) Simplify $\dfrac{2}{4x-5}+\dfrac{3}{1-6x}$. (3 marks)
Solution
$\displaystyle \frac{2(1-6x)+3(4x-5)}{(4x-5)(1-6x)} = \frac{2-12x+12x-15}{(4x-5)(1-6x)} = \frac{13}{(4x-5)(6x-1)}$
(4) Factorize
(a) $5m-10n$.
(b) $m^2+mn-6n^2$.
(c) $m^2+mn-6n^2-5m+10n$. (4 marks)
Solution
(a) $5(m-2n)$
(b) $(m+3n)(m-2n)$
(c) $(m+3n)(m-2n)-5(m-2n) = (m-2n)(m+3n-5)$
(5) In a recreation club, there are 180 members and the number of male members is 40% more than the number of female members. Find the difference of the number of male members and the number of female members. (4 marks)
Solution
Let $f$ be the number of female members.
$1.4f + f = 180 \quad \Rightarrow \quad f = \dfrac{180}{2.4} = 75$
$\therefore \text{difference} = 1.4f-f=0.4f=0.4(75)=30$
(6) Consider the compound inequality
$x+6 < 6(x+11) \text{ or } x\leq -5 \qquad (*)$
(a) Solve (*).
(b) Write down the greatest negative integer satisfying (*). (4 marks)
Solution
(a) $x+6<6x+66 \text{ or } x\leq -5 \\ \Leftrightarrow 5x>-60 \text{ or } x\leq -5 \\ \Leftrightarrow x>-12 \text{ or } x\leq -5 \\ \Leftrightarrow x\in\mathbb{R}$
(b) $-1$
(7) In a polar coordinate system, $O$ is the pole. The polar coordinate of the points $A$ and $B$ are $(12, 75^\circ)$ and $(12, 135^\circ)$ respectively.
(a) Find $\angle AOB$.
(b) Find the perimeter of $\triangle AOB$.
(c) Write down the number of folds of rotational symmetry of $\triangle AOB$. (4 marks)
Solution
(a) $\angle AOB = 135^\circ - 75^\circ = 60^\circ$
(b) Obviously the triangle is isosceles and hence $\alpha = \beta = 60^\circ$ and hence $\triangle AOB$ is equilateral. Therefore the perimeter is 36.
(c) 3
(8) It is given that $f(x)$ is the sum of two parts, one part varies as $x$ and the other part varies as $x^2$. Suppose that $f(3)=48$ and $f(9)=198$.
(a) Find $f(x)$.
(b) Solve the equation $f(x)=90$. (5 marks)
Solution
(a) Let $f(x) = ax^2+bx$ for some non-zero real numbers $a$ and $b$.
$f(3) = 9a+3b = 48 \Leftrightarrow 3a+b=16 \qquad (1)$
$f(9) = 81a+9b = 198 \Leftrightarrow 9a+b=22 \qquad (2)$
(2) – (1) gives $6a=6 \Rightarrow a=1 \Rightarrow b=13$
$\therefore f(x) = x^2+13x$
(b) $x^2+13x-90=0 \Rightarrow (x+18)(x-5)=0 \Rightarrow x=-18 \text{ or } x=5$
(9) The frequency distribution table and the cumulative frequency distribution table below show the distribution of the heights of the plants in a garden.
(a) Find $x$, $y$, and $z$.
(b) If a plant is randomly selected from the garden, find the probability that the height of the selected plant is less than 1.25 m but not less than 0.65 m.
Solution
(a) $a=2, \therefore x=2+4=6, y=37-15=22, z=37+3=40$
(b) $\dfrac{22-6}{40}=\dfrac{16}{40}=\dfrac{2}{5}$
(10) The coordinates of the points $A$ and $B$ are $(5, 7)$ and $(13,1)$ respectively. Let $P$ be a moving point in the rectangular coordinate plane such that $P$ is equidistant from $A$ and $B$. Denote the locus of $P$ by $\Gamma$.
(a) Find the equation of $\Gamma$. (2 marks)
(b) $\Gamma$ intersects the x-axis and y-axis at $H$ and $K$ respectively. Denote the origin by $O$. Let $C$ be the circle which passes through $O$, $H$, and $K$. Someone claims that the circumference of $C$ exceeds 30. Is the claim correct? Explain your answer. (3 marks)
Solution
(a) $(x-5)^2+(y-7)^2=(x-13)^2+(y-1)^2$
$\Rightarrow -10x+25-14y+49=-26x+169-2y+1$
$\Rightarrow 16x=12y+96$
$\Rightarrow 4x-3y-24=0$
(b) Let $y=0$, then $x=6$, hence $H(6, 0)$
Let $x=0$, then $y=-8$, hence $K(0, -8)$
Since $\triangle HKO$ is a right triangle where $\angle O = 90^\circ$, hence $HK$ is the diameter of the circle $C$.
$\text{Circumference} = \pi d = \pi\sqrt{6^2+8^2} = 10\pi > 30$
Therefore, the claim is correct.
(11) An inverted right circular conical vessel contains some milk. The vessel is held vertically. The depth of milk in the vessel is $12 \text{ cm}$. Peter pours $444\pi\text{ cm}^2$ of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is $16\text{ cm}$.
(a) Express the final volume of milk in the vessel in terms of $\pi$. (3 marks)
(b) Peter claims that the final area of the wet curved surface of the vessel is at least $800\text{ cm}^2$. Do you agree? Explain your answer. (3 marks)
Solution
(a) Let $x$ be the original volume of milk.
$\displaystyle\frac{444\pi+x}{x}=\left(\frac{16}{12}\right)^3=\left(\frac{4}{3}\right)^3$
$27(444\pi+x)=64x$
$27\cdot444\pi=37x$
$x=27\cdot4\cdot3\pi = 81\cdot4\pi=324\pi$
$\therefore \text{Final volume} = 444\pi+324\pi = 768\pi\text{ cm}^3$
(b) $\dfrac{\pi r^2 (16)}{3} = 768 \pi \Rightarrow r^2=144 \Rightarrow r=12\text{ cm}$
$\text{Curved surface area} = \pi r s = \pi(12)\sqrt{12^2+16^2} = 240\pi \approx 754 \text{ cm}^2 < 800\text{ cm}^2$
Therefore, disagree.
(12) The bar chart below shows the distribution of the ages of the children in a group, where $a>11$ and $4. The median of the ages of the children in the group is 7.5.
(a) Find $a$ and $b$. (3 marks)
(b) Four more children now join the group. It is found that the ages of these four children are all different and the range of the ages of the children in the group remains unchanged. Find
(i) the greatest possible median of the ages of the children in the group.
(ii) the least possible mean of the ages of the children in the group. (4 marks)
Solution
(a) Since the median age is 7.5, hence $11+a=11+b+4 \Rightarrow a=b+4$
Now, $a>11 \text{ and } 47 \text{ and } 4.
Since $a,b\in\mathbb{Z}^+ \therefore b=8\text{ or } 9$
$\therefore (a,b) = (12,8)\text{ or }(13,9)$
(b) (i) To maximize the median, we add age 7, 8, 9, 10. Therefore the new median will be 8.
(b) (ii) To minimize the mean, we add age 6, 7, 8, 9.
Case $(a, b)=(12,8)$,
$\text{mean} = \dfrac{12\cdot6+13\cdot7+12\cdot8+9\cdot9+4\cdot10}{12+13+12+9+4} = \dfrac{380}{50}=\dfrac{38}{5}=7.6$
Case $(a, b)=(13,9)$,
$\text{mean} = \dfrac{12\cdot6+14\cdot7+12\cdot8+10\cdot9+4\cdot10}{12+14+12+10+4} = \dfrac{396}{52}=\dfrac{99}{13}=7.\overline{615384}$
(13) In figure 1, $ABC$ is a triangle. $D$, $E$, and $M$ are points lying on $BC$ such that $BD=CE$, $\angle ADC = \angle AEB$ and $DM=EM$.
(a) Prove that $\triangle ACD\cong\triangle ABE$. (2 marks)
(b) Suppose that $AD=15\text{ cm}$, $BD=7\text{ cm}$ and $DE=18\text{ cm}$.
(i) Find $AM$.
(ii) Is $\triangle ABE$ a right-angled triangle? Explain your answer. (5 marks)
Solution
(a) $\angle ADC = \angle AEB \quad \text{(given)}$
$AD=AE \quad \text{(sides opp. equal }\angle\text{s)}$
$BD=CE \quad\text{(given)}$
$BE = BD+DE = CE+DE = CD\quad (BD=CE)$
$\therefore \triangle ACD\cong\triangle ABE \quad\text{(SAS)}$
(b) (i) $AD=AE=15 cm\quad \text{(given)}$
Hence, $\triangle ADE$ is isosceles.
$DM=EM\quad\text{(given)}$
$\therefore AM \perp DE\quad\text{(property of isos }\triangle \text{)}$
$\therefore AM=\sqrt{15^2-9^2} = 12\text{ cm} \quad\text{(Pyth. thm)}$
(b) (ii) $DM=EM=9\text{ cm}$
By Pyth. thm, $AB=\sqrt{12^2+(7+9)^2} = 20\text{ cm}$
$AB^2+AE^2 = 20^2+15^2 = 25^2 = AE^2$
Therefore, $\triangle ABE$ is right-angled.
(14) Let $p(x)=6x^4+7x^3+ax^2+bx+c$, where $a$, $b$, and $c$ are constants. When $p(x)$ is divided by $x+2$ and when $p(x)$ is divided by $x-2$, the two remainders are equal. It is given that $p(x)=(lx^2+5x+8)(2x^2+mx+n)$, where $l$, $m$, and $n$ are constants.
(a) Find $l$, $m$, and $n$. (5 marks)
(b) How many real roots does the equation $p(x)=0$ have? Explain your answer. (5 marks)
Solution
(a) $p(-2)=p(2)$
$6(-2)^4+7(-2)^3+a(-2)^2+b(-2)+c = 6(2)^4+7(2)^3+a(2)^2+b(2)+c$
$-56-2b = 56+2b$
$4b = -112$
$b=-28$
$\therefore p(x)=6x^4+7x^3+ax^2-28x+c$
$(lx^2+5x+8)(2x^2+mx+n) = (2l)x^4+(lm+10)x^3+(ln+5m+16)x^2+(5n+8m)x+8n$
By matching the coefficient,
$2l=6 \Rightarrow l=3$
$lm+10=7 \Rightarrow m=-1$
$5n+8m=-28 \Rightarrow n=-4$
(b) $p(x)=(lx^2+5x+8)(2x^2+mx+n)=(3x^2+5x+8)(2x^2-x-4)=0$
$\therefore (3x^2+5x+8)=0\text{ or }(2x^2-x-4)=0$
For $3x^2+5x+8=0, \quad \Delta=25-4\cdot3\cdot8 = -71\quad \therefore$ no real root.
For $2x^2-x-4=0, \quad \Delta=1+4\cdot2\cdot4 = 33\quad \therefore$ two real roots.
Therefore, $p(x)=0$ has two real roots.
###### Section B (35 marks)
(15) If 4 boys and 5 girls randomly form a queue, find the probability that no boys are next to each other in the queue. (3 marks)
Solution
Consider this : _ G _ G _ G _ G _ G _
There are 6 places to put 4 B’s so there are 6P4 ways to place the boys. Moreover, there are 5! ways to permute those 5 G’s. Therefore,
$\displaystyle \text{P(no boys are next to each other)}=\frac{_6P_4 \cdot 5!}{9!} = \frac{6\cdot5\cdot4\cdot3}{9\cdot8\cdot7\cdot6} = \frac{5}{42}$
(16) In a test, the mean of the distribution of the scores of a class of students is 61 marks. The standard scores of Albert and Mary are -2.6 and 1.4 respectively. Albert gets 22 marks. A student claims that the range of the distribution is at most 59 marks. Is the claim correct? Explain your answer. (3 marks)
Solution
$Z=\dfrac{X-\mu}{\sigma} \quad \Rightarrow\quad -2.6 = \dfrac{22-61}{\sigma} \quad\Rightarrow\quad \sigma = 15$
So, the standard deviation is 15. Now, let’s calculate what Mary gets on the test.
$X=Z\sigma+\mu = 1.4\cdot 15 + 61 = 82$
$82-22 = 60$, which is greater than 59. Therefore the claim is false.
(17) The 1st term and the 38th term of an arithmetic sequence are 666 and 555 respectively. Find
(a) the common difference of the sequence. (2 marks)
(b) the greatest value of $n$ such that the sum of the first $n$ term of the sequence is positive. (3 marks)
Solution
(a) $t_{38} =t_1 + 37d$
$\therefore d=\dfrac{t_{38}-t_1}{37} = \dfrac{555-666}{37} = \dfrac{-111}{37}=-3$
(b) $S_n = \dfrac{(2a+(n-1)d)n}{2} > 0$
$\displaystyle \frac{(2(666)+(n-1)(-3))n}{2}>0$
Since $n$ is positive integer, we can divide both side by $n$ without switching the sign.
$2(666)-3(n-1)>0$
$n-1 < 2(222)$
$n < 445$
Therefore, the greatest such $n$ is 444.
(18) Let $f(x) = \dfrac{-1}{3}x^2 + 12x - 121$.
(a) Using the method of completing the square, find the coordinates of the vertex of the graph of $y = f(x)$. (2 marks)
(b) The graph of $y=g(x)$ is obtained by translating the graph of $y=f(x)$ vertically. If the graph of $y=g(x)$ touches the x-axis, find $g(x)$. (2 marks)
(c) Under a transformation, $f(x)$ is changed to $latex \dfrac{-1}{3}x^2 – 12x – 121$. Describe the geometric meaning of the transformation. (2 marks)
Solution
(a) $\displaystyle f(x) = \frac{-1}{3}x^2 + 12x - 121$
$= -\frac13 (x^2-36x+18^2-18^2) - 121$
$= -\frac13(x-18)^2+\frac{18^2}{3}-121$
$= -\frac13(x-18)^2 -13$
Therefore, the vertex is $(18, -13)$.
(b) Basically translate the graph of $f(x)$ 13 units upward.
$\therefore g(x) = -\frac13(x-18)^2$
(c) Since $\displaystyle\frac{-1}{3}x^2-12x-121=\frac{-1}{3}(-x)^2+12(-x)-121 = f(-x)$
Therefore, the transformation is reflecting the graph of f(x) horizontally along the y-axis.
(19) Figure 2 shows a geometric model $ABCD$ in the form of tetrahedron. It is given that $\angle BAD=86^\circ, \angle CBD=43^\circ, AB=10\text{ cm}, AC=6\text{ cm}, BC=8\text{ cm} \text{ and } BD=15\text{ cm}$.
(a) Find $\angle ABD$ and $CD$. (4 marks)
(b) A craftsman claims that the angle between $AB$ and the face $BCD$ is $\angle ABC$. Do you agree? Explain your answer. (2 marks)
Solution
(a) By Sine Law, $\displaystyle \frac{\sin\angle ADB}{AB}=\frac{\sin\angle BAD}{BD}$
$\displaystyle \frac{\sin\angle ADB}{10}=\frac{\sin 86^\circ}{15}$
$\therefore \angle ADB = \sin^{-1} \dfrac{10\sin 86^\circ}{15}$
$\therefore \angle ABD = 180^\circ - 86^\circ - \sin^{-1} \dfrac{2\sin 86^\circ}{3} = 52.31439868 \approx 52.3^\circ \text{ (3 s.f.)}$
By Cosine Law, $CD^2=AC^2+AD^2-2AC\cdot AD \cos\angle CBD$.
$\therefore CD = \sqrt{8^2+15^2-2\cdot8\cdot15\cdot\cos 43^\circ} = 10.65246974 \approx 10.7 \text{ cm (3 s.f.)}$
(b) By Sine law, $\displaystyle \frac{AD}{\sin\angle ABD}=\frac{BD}{\sin\angle BAD}$
$\therefore AD=\dfrac{15\sin\angle ABD}{\sin 86^\circ} \approx 11.8996447\ldots$
$AC^2+CD^2=6^2+10.65246974^2=149.4751116\neq AD^2=141.6015440$
Therefore, do not agree.
(20) $\triangle OPQ$ is an obtuse-angled triangle. Denote the in-centre and the circumcentre of $\triangle OPQ$ by $I$ and $J$ respectively. It is given that $P$, $I$, and $J$ are collinear.
(a) Prove that $OP=PQ$. (3 marks)
(b) A rectangular coordinate system is introduced so that the coordinates of $O$ and $Q$ are $(0,0)$ and $(40,30)$ respectively while the y-coordinate of $P$ is 19. Let $C$ be the circle which passes through $O$, $P$, and $Q$.
(i) Find the equation of $C$.
(ii) Let $L_1$ and $L_2$ be two tangents to $C$ such that the slope of each tangent is $\dfrac34$ and the y-intercept of $L_1$ is greater than that of $L_2$. $L_1$ cuts the x-axis and y-axis at $S$ and $T$ respectively while $L_2$ cuts the x-axis and the y-axis at $U$ and $V$ respectively. Someone claims that the area of the trapezium $STUV$ exceeds 17000. Is the claim correct? Explain your answer. (9 marks)
Solution
(a) Since $P$, $I$, and $J$ are collinear, we can draw a line passing through all three points and meet the line segment $OQ$ at $K$. Since $I$ is incenter, $PI$ is angle bisector of $\angle OPQ$, hence $\angle OPK = \angle QPK$. Since $J$ is circumcenter, then $PK$ is perpendicular bisector of $OQ$ and so $\angle PKO = \angle PKQ = 90^\circ$. $PK = PK$ obviously by common sides. Hence, $\triangle PKO \cong \triangle PKQ$ by ASA. Therefore the corresponding sides of the two congruent triangles are congruent and so $OP=PQ$.
(b) (i) Let $P=(x, 19)$. Since $OP = PQ$,
$\sqrt{x^2+19^2} = \sqrt{(x-40)^2+(19-30)^2}$
$x^2+361=x^2-80x+1600+121$
$80x=1360$
$x=17$
Let circle $C: x^2+y^2+Dx+Ey+F=0$
Substitute $O(0,0)$ into circle C yields $F=0$. Hence,
$P(17, 19): 17^2+19^2+17D+19E=0 \Rightarrow 650+17D+19E \qquad (1)$
$Q(40, 30): 40^2+30^2+40D+30E=0 \Rightarrow 250+4D+3E=0 \qquad (2)$
3(1)-19(2):
$1950+51D-4750-76D=0$
$\therefore D=-112$
$\therefore E=66$
Therefore, the equation of circle C is $x^2+y^2-112x+66y=0$
(b) (ii) Since $L_1$ and $L_2$ have slope of $\dfrac34$, They both have the equation of the form: $y=\dfrac34x+b$
They are both tangents to the circle hence:
$x^2+y^2-112x+66y =0 \quad\text{and}\quad y=\dfrac34x+b$
$x^2+\left(\frac34x+b\right)^2-112x+66\left(\frac34x+b\right)=0$
$x^2+\frac{9}{16}x^2+\frac{3b}{2}x+b^2-112x+\frac{99}{2}x+66b=0$
$16x^2+9x^2+24bx+16b^2-1792x+792x+1056b=0$
$25x^2+(24b-1000)x+16b^2+1056b=0$
Since they are tangents to the circle,
$\Delta = (24b-1000)^2-4(25)(16b^2+1056b)=0$
$576b^2-48000b+1000000-1600b^2-105600b=0$
$1024b^2+153600b-1000000=0$
$16b^2+2400b-15625=0$
$(4b+625)(4b-25)=0$
$b=-\dfrac{625}{4} \text{ or } b=\dfrac{25}{4}$
$\displaystyle\therefore \quad L_1: y=\frac34x+\frac{25}{4} \quad\text{and}\quad L_2: y =\frac34x-\frac{625}{4}$
$L_1:$ Let $y=0 \Rightarrow x=-\dfrac{25}{3} \Rightarrow S\left(-\dfrac{25}{3},0\right) \Rightarrow T\left(0,\dfrac{25}{4}\right)$
$L_2:$ Let $y=0 \Rightarrow x=\dfrac{625}{3} \Rightarrow U\left(\dfrac{625}{3},0\right) \Rightarrow V\left(0,-\dfrac{625}{4}\right)$
$\displaystyle \therefore ST=\sqrt{\left(\frac{25}{3}\right)^2+\left(\frac{25}{4}\right)^2}=25\sqrt{\frac{1}{9}+\frac{1}{16}}=\frac{25}{12}$
$\displaystyle \therefore UV=\sqrt{\left(\frac{625}{3}\right)^2+\left(\frac{625}{4}\right)^2}=625\sqrt{\frac{1}{9}+\frac{1}{16}}=\frac{3125}{12}$
$C: x^2+y^2-112x+66y=0$
$\Rightarrow x^2-112x+56^2+y^2+66y+33^2=56^2+33^2=4225$
$\Rightarrow (x-56)^2+(y+33)^2=65^2 \Rightarrow r=65 \Rightarrow d=130$
$\displaystyle\therefore \text{Area} = \frac{(ST+UV)d}{2} = \left(\frac{125}{12}+\frac{3125}{12}\right)65 =17604.1\overline{6}>17000$
Therefore, the claim is correct.
## 777, 365, and 21
Posted: March 26, 2017 in Mathematics
The number “689” has been the laughingstock of the HK chief executive for the past 4 years. “689” basically means 冇柒用, so in response, this year’s chief executive will receive “777” votes just to emphasis that she is very 好有柒用. It’s not one 7 or two 7s, but three 7s in a row, how nice!
Actually 777 is quite an interesting number. In gambling, it is a lucky number to most people because “777” is a jackpot number. In computer science, “777” means you have been granted read, write, and executable permission on a file (in UNIX or UNIX-like operating system such as Linux, BSD, Mac OSX, etc). To most HKers, 777, 365, and 21 mean the number of vote of today’s HK chief executive election. To me, I have another interpretation.
$777 = 7 \cdot 111 = 7 \cdot 3 \cdot 37$
$365 = 5 \cdot 73$
$21 =3 \cdot 7$
$\therefore 777 \cdot 365 \cdot 21 = 7 \cdot 3 \cdot 37 \cdot 5 \cdot 73 \cdot 3 \cdot 7$
Math is beautiful, isn’t it?
###### Remarks
The joke has made into wikipedia LOL
## Fibonacci Number & Golden Ratio
Posted: March 22, 2017 in Mathematics
Today I saw this funny picture of Donald Trump and then I know why everybody says Golden Ratio can be found all over nature. Yes it is living among us. You can’t escape from it, not even from the president of America.
So, what is Golden Ratio?
In Euclid’s Elements, Book VI Definition 2, Euclid wrote (in Greek of course):
Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν
Or in English,
A straight-line is said to have been cut in extreme and mean ratio when as the whole is to the greater segment so the greater (segment is) to the lesser.
Or in plain English, it basically says that when you divide a line segment into two uneven pieces, the ratio of the whole line segment to the longer piece equals the ratio of the longer piece to the shorter piece. And this ratio is called the Golden Ratio (often denote as $\phi$).
$\displaystyle\frac{a+b}{a} = \frac{a}{b} = \phi \quad\Leftrightarrow\quad 1+\frac{1}{\phi}=\phi \quad\Leftrightarrow\quad \phi^2=\phi+1$
After solving the quadratic equation, you will get:
$\displaystyle\phi = \frac{1+\sqrt5}{2} \approx 1.61803398874989\ldots$
Now, let’s consider the Fibonacci sequence.
$F_n = 1,1,2,3,5,8,13,21,34,55,89,\ldots$
and examine some nth power of the golden ratio.
$\phi^1 = \phi$
$\phi^2 = \phi+1$
$\phi^3 = \phi^2+\phi = 2\phi + 1$
$\phi^4 = 2\phi^2+\phi = 2(\phi+1) + \phi = 3\phi+2$
$\phi^5 = 3\phi^2+2\phi = 3(\phi+1) + 2\phi = 5\phi + 3$
$\phi^6 = 5\phi^2+3\phi = 5(\phi + 1) + 3\phi = 8\phi + 5$
This suggests that
$x^2=x+1 \quad \Rightarrow \quad x^n = F_n x + F_{n-1} \quad\forall n\in\mathbb{Z}^+$
Let’s define $F_0 = 0$, then
$x^1 = F_1 x + F_0 = x$
$x^2 = F_2 x + F_1 = x+1$
Assume $x^k = F_k x + F_{k-1}$, then
$x^{k+1} = x^k\cdot x = F_k x^2 + F_{k-1} x \\= F_k(x+1)+F_{k-1}x = (F_k+F_{k-1})x+F_k \\ = F_{k+1} x + F_k$
Hence, by math induction, $x^n = F_n x + F_{n-1} \quad \forall n\in\mathbb{Z}^+ \text{ where } x^2=x+1$.
Now, go back to the equation $x^2 = x+1$, the two roots are
$\displaystyle\phi = \frac{1+\sqrt5}{2} \quad\text{and}\quad \Phi=\frac{1-\sqrt5}{2}$
Hence,
$\phi^n = F_n\phi + F_{n-1} \qquad \text{(1)}$
$\Phi^n = F_n\Phi + F_{n-1} \qquad \text{(2)}$
(1) – (2) yields
$\displaystyle F_n = \frac{\phi^n - \Phi^n}{\phi - \Phi} = \frac{\phi^n - \Phi^n} {\sqrt5}$
Or
$\displaystyle F_n = \frac{1}{\sqrt5}\left(\left(\frac{1+\sqrt5}{2}\right)^n-\left(\frac{1-\sqrt5}{2}\right)^n\right)$
###### Remarks
Since
$\displaystyle\Phi = 1-\phi \quad\text{and}\quad \phi = 1+\frac{1}{\phi}$
then
$\displaystyle F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt5} = \frac{\phi^n-(-\frac{1}{\phi})^n}{\sqrt5}$
$\displaystyle\lim_{n\to\infty}\frac{F_{n+1}}{F_n} = \lim_{n\to\infty} \frac{\phi^{n+1}-(-\frac{1}{\phi})^{n+1}}{\sqrt5} \cdot \frac{\sqrt5}{\phi^n-(-\frac{1}{\phi})^n}$
As $\displaystyle n\to\infty, \quad -\frac{1}{\phi^{n+1}} \text{ and } -\frac{1}{\phi^n} \to 0$
$\displaystyle \therefore \lim_{n\to\infty} \frac{F_{n+1}}{F_n} = \phi$
As $n\to\infty, \quad \displaystyle -\frac{1}{\phi^n} \to 0$
$\displaystyle \therefore F_n \approx \frac{\phi^n}{\sqrt5}$ as n is large.
The last approximation comes in handy when you want to find the nth Fibonacci number, let’s say the 30th Fibonacci number.
$\displaystyle F_{30} \approx \frac{\phi^{30}}{\sqrt5} \approx 832040.00000024\ldots$
So, you can compute the 30th Fibonacci number to be 832040 in seconds with any normal scientific calculator.
## Einstein’s Five House Riddle
Posted: March 15, 2017 in Mathematics
This riddle is claimed to be written by Einstein when he was a boy. He claimed that only 2% of the people can solve it. Are you that 2% who can solve it?
There are five houses in five different colors in a row. In each house lives a person with a different nationality. The five owners drink a certain type of beverage, smoke a certain brand of cigar, and keep a certain pet. No owners have the same pet, smoke the same brand of cigar, or drink the same beverage.
The question is: Who owns the fish?
Hints:
1. The Brit lives in the red house.
2. The Swede keeps dogs as pets.
3. The Dane drinks tea.
4. The green house is on the left of the white house.
5. The green house’s owner drinks coffee.
6. The owner who smokes Pall Mall rears birds.
7. The owner of the yellow house smokes Dunhill.
8. The owner living in the center house drinks milk.
9. The Norwegian lives in the first house.
10. The owner who smokes blends lives next to the one who keeps cats.
11. The owner who keeps horses lives next to the man who smokes Dunhill.
12. The owner who smokes BlueMaster drinks beer.
13. The German smokes Prince.
14. The Norwegian lives next to the blue house.
15. The man who smokes blend has a neighbor who drinks water.
| 8,463 | 6,699,491 |
# $2 . \mathrm{II} . 6 \mathrm{D} \quad$
Define the Wronskian $W(x)$ associated with solutions of the equation
$\frac{d^{2} y}{d x^{2}}+p(x) \frac{d y}{d x}+q(x) y=0$
and show that
$W(x) \propto \exp \left(-\int^{x} p(\xi) d \xi\right) .$
Evaluate the expression on the right when $p(x)=-2 / x$.
Given that $p(x)=-2 / x$ and that $q(x)=-1$, show that solutions in the form of power series,
$y=x^{\lambda} \sum_{n=0}^{\infty} a_{n} x^{n} \quad\left(a_{0} \neq 0\right)$
can be found if and only if $\lambda=0$ or 3 . By constructing and solving the appropriate recurrence relations, find the coefficients $a_{n}$ for each power series.
You may assume that the equation is satisfied by $y=\cosh x-x \sinh x$ and by $y=\sinh x-x \cosh x$. Verify that these two solutions agree with the two power series found previously, and that they give the $W(x)$ found previously, up to multiplicative constants.
[Hint: $\left.\cosh x=1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\ldots, \quad \sinh x=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\ldots .\right]$
| 371 | 6,699,492 |