# Related rates weight of duckling

1. May 5, 2013

### syeh

1. The problem statement, all variables and given/known data
Suppose that during the first year after its hatching, the weight of a duck increases at a rate proportional to its weight. The duck weighed 2 pounds when hatched, and 3.5 lbs at age 4 months. How many lbs will it weight at 6 months?

A) 4.2 lbs
B) 4.6 lbs
C) 4.8 lbs
D) 5.6 lbs
E) 6.5 lbs

2. Relevant equations

3. The attempt at a solution

I assumed this was a linear equation and used the points (0, 2) and (4, 3.5) to find the slope, .375, and the equation of the line to be y=.375x+2. Then i plugged in 6 and got y(6)=.375(6)+2 = 4.25. But the answer is 4.6.

Is the graph not a linear equation? Maybe since it says "weight increases at a rate proportional to its weight", it is an exponential function?? Then, how would I find f(6)??

2. May 5, 2013

### rock.freak667

If dw/dt is the rate of change of weight, then

dw/dt ∝ w such that dw/dt = kw where k is a constant.

You will need to solve this DE to get w(t).

Then they gave you two conditions, so that you can solve for your constants.

3. May 5, 2013

### syeh

What do you mean, dw/dt=kw? by using (0,2) and (4,3.5), how would i solve for w(t)??

4. May 5, 2013

### SteamKing

Staff Emeritus
Read the problem statement carefully. "the weight of a duck increases at a rate proportional to its weight."

Thus, let w(t) = the weight of the duck at time t.

the rate of change in weight with respect to time is dw/dt.

This rate is proportional to the weight of the duck, thus dw/dt = kw, where k = constant of proportionality.

Do you know about separation of variables?

5. May 5, 2013

### syeh

ok, I see. So i took dw/dt = kw and got
∫1/w dw = ∫k dt
lnw = kt + C

Using (0,2) to find C:
ln2 = C

Using (4, 3.5) to find k:
ln3.5 = 4.5k + ln2
4.5k = ln(1.75)
k=1.124

So, lnw= 0.124t + ln2
w=2e^(0.124t)

to plug in 6 months:
w(6)= 4.218

How come I didnt get the correct answer, 4.6 lbs?

6. May 5, 2013

### haruspex

4 became 4.5?

7. May 5, 2013

Thank you
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}