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# Further X2 Volumes Problems

(a)

## Figure 1 below shows a scale model of the volcano Mt Rekrap.

The base of the model is elliptical in shape with the axes 60 cm by 40 cm reducing
uniformly to a circle of radius 12 cm at the top.
The hollow core of the model has circular cross sections with a circle of radius 6 cm at the
base rising uniformly to a circle also of radius 12 cm at the top.
The model is 24 cm high.
Figure 2 shows the top view of the cross sectional area of the volcano.

Figure 1

24 cm
h cm

40 cm

60 cm
Figure 2

3
h
4

(i)

## Show that at height h, the length of the semi-major axis is given by a = 30

(ii)

Show that the area of the cross sectional slice at height h is given by

A=
9024 448h + 3h 2
16
You may assume that the area of an ellipse with semi-major axis a and semi-minor
axis b is given by ab .

(iii)

## Find the volume of the scale model of Mt Rekrap.

2
4

2
[6336]

7
7
QUESTION 5. (Continued)
QUESTION 5. (Continued)
HSC 1998
(b)
(b)

Marks
Marks
9

a
a

Z
Z

A
A

a
a

Y
Y

W
W

C
C

R
R

S
S

x
x

P
P

Q
Q
x
x

X
X

b
b

B
B

The diagram shows a sandstone solid with rectangular base ABQP of length
The diagram shows a sandstone solid with rectangular base ABQP of length
b metres and width a metres. The end PQRS is a square, and the other end ABC
b metres and width a metres. The end PQRS is a square, and the other end ABC
is an equilateral triangle. Both ends are perpendicular to the base.
is an equilateral triangle. Both ends are perpendicular to the base.
Consider the slice of the solid with face WXYZ and thickness x metres, as
Consider the slice of the solid with face WXYZ and thickness x metres, as
shown in the diagram. The slice is parallel to the ends and AW = BX = x metres.
shown in the diagram. The slice is parallel to the ends and AW = BX = x metres.
(i) Find the height of the equilateral triangle ABC.
(i) Find the height of the equilateral triangle ABC.
(ii) Given that the triangles CRS and CYZ are similar, find YZ in terms of a,
(ii) Given that the triangles CRS and CYZ are similar, find YZ in terms of a,
b and x .
b and x .
(iii) Let the perpendicular height of the trapezium WXYZ be h metres. Show
(iii) Let the perpendicular height of the trapezium WXYZ be h metres. Show
that
that

((

))

a
x
h = a 3 + 2 3 x .
h = 2 3 + 2 3 b .
2
b
(iv) Hence show that the cross-sectional area of WXYZ is given by
(iv) Hence show that the cross-sectional area of WXYZ is given by

[[((

]]

a22
a
2 3 x + b 3 (b + x ) .
4 b22 2 3 x + b 3 (b + x ) .
4b
(v) Find the volume of the solid.
(v) Find the volume of the solid.

))

9
9

SGS 6A q23

## (b) The region bounded by arc AB and the chord AB is rotated

about the chord AB. Show that the solid formed has volume
103
r 3 cubic units.
6 2

EXTENS ION

## 23. [A First Principles Approach] The region bounded by y = x2

and y = x is rotated about the line y = x to generate a solid.
In this question you will find the volume of this solid by first
CHAPTER 6: Volumes
approximating a small portion with a disk.

y ' x2
y'x

"h
r

&
Let P (x, x2) be a typical point on the parabola. From x to x + x
P#x\$ x %
(b) 4
we will approximate the area between the curve and the line16(a)
with
y
x3
"x
1
a rectangle. One corner of this rectangle is at P and a side is on
the line y Exercise
= x. Let6A
the(Page
dimensions
27) of this rectangle be r h as
"
shown in the
diagram.
!)#
)#
1(a)
(ii) 3
(b)(i) 9
y
14
5
x
(a) Show that h (ii)
= 2 x and find r as a function of x.
!" "
#
(b) When the rectangle is rotated about the line y = x it generates a cylindrical prism
with volume V ."Find V
& .

Chapter Six

y\$x %1

## (d) Hence show that

!" the volume
" (i) x of the solid is
2(a)

30 2

(b) 8
15

6B The Method
ofy \$Cylindrical
Shells
x
#
#

## Inevitably when" tackling yvolume

\$'#'! x problems there will be
SGS
2012
THSC q16
b result
certain
questions
which
in awkward integrals. As
an example, consider the
solid
generated
when the region
x
"
#
2
between y = 4x x and y = x is rotated about the
y-axis, as shown on the right. The volume
can be found
32
(b)(i) 5
3(a)
(ii) 8
by the methods ofySection 6A since each
cross-section is
(ii)
an annulus. However the situation is complicated by the
(
fact that the formula for the outer radius changes. Below
! is router = y whilst above y = 3 it is
router = 2 + 4 y. Hence
(i) the integral must be split
into two parts. (Try this as an exercise.)

## Exercise 6B (Page 32)

1(a) y

(b)

y
router
(

)
(

router
( )
2(a)

#x x
(b) 128
5

4(a)

2882
2
2 2
2
19(a) 43 ab
(b) 2 ab
(c) 2 abc
3
9
16
92
20(a) AD = 5 , CD = 5
(c) 15 cm
1
2

2 2
23(a) (x x ) (b) V = (x x ) x
2
2

2 2
(c) V = (x x )
2
17(b)

(b) 128
5

y
Chapter 6: Volumes

volumes

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c 2014

3(a)

128
3888
(b)
3
5
3
9
(b)
5
14
8
(b) 8
3
256
(b) 192
15
5
8
(b) 8
3

(c) 10

The diagram above shows (the parabola y = 3 + 2x x2 and its4(a)reflection in the y-axis.
5(a) same height y when rotated about
The vertical strips shown will generate cylindrical shells of the
6(a)
the line x = 1.
6
5 7
7(a)shells is 4 y
(i)
Show that the sum
of the
areas of these two cylindrical
1
48
7
2
4

6(a)

(ii)

(b)

(c) 10

9(a)

(b) 16
3

7
Find
the volume of the solid formed when the region bounded* by the parabolas
8 32
3 x-axis is rotated about the line x = 1.
and the
9 256
15
44

10(a) 8
(b) 224
(c) 8
c.u.
3
15

"
3

11(a) 9
(b) 15
14
7
8
(b) 8
3
296
13(b) 15
32a3
112a3
14(a)
(b)
15
15
15(b) 8
3

12(a)

!&
(c)

128a3
15

"

&

41

Marks
Question 8 (continued)
HSC 2002 q8
38

CHAPTER 6: Volumes

(b)

## 12. In the diagram on the rightKis a cylindrical wedge ABCD. The N

height of the cylinder is equal to the diameter of its base. Let the
L
M
radius of the base be r units.
(a) Show that the typical triangular cross-section shaded has area
(r 2 x2 ) square units.

(b) Hence show that the wedge has volume 43 r 3 cubic units.

A
(x, y)

B
!r
!r

## 13. The diagram on the right shows a cylindrical drinking glass of

B glass
interior radius 4 cm and perpendicular height 12 cm. The
a
is filled with water which is then drunk slowly until half of the
bottom of the glass is exposed. Use the methods
of the previous
F
question to find the volume of water aremaining.

y
r
C
r
x

m
12 c

4 cm

A
In the diagram,
ABshows
and CD
are line
length Its
2a in horizontal
4 m planes
14. The diagram
to the right
a large
tanksegments
of depth 6ofmetres.
apart. The
of CD is
vertically
at aits
distance
base and
top are2asquares
withmidpoint
sides of 2E metres
and
4 metresabove the midpoint F
of
AB,
and
AB
lies
in
the
SouthNorth
direction,
while
CD lies in the WestEast
respectively. A typical square cross-section of side length x metres
direction.
is shown
h metres above the base.
6m
(a) Show that x =

+ 2.

## The rectangle3KLMN is the horizontal cross-section of the tetrahedron ABCDhat

(b) Hence
find xthe
capacity
of the tank.
distance
from
the midpoint
P of EF (so PE = PF = a).
2m

(i)

## 15. A rubbish skip

onofa the
building
site KLMN.
has a rectangular base 6 metres by 3 metres, and its
area
rectangle
perpendicular height is 2 metres. Its sides are trapezia that slope outwards from bottom
to top. The open top is a rectangle 7 metres by 4 metres.
(ii) Find the volume of the tetrahedron ABCD.
!
"!
"
(a) Show that a rectangular cross-section h metres above the base has area h2 + 6 h2 + 3
square metres.

SGS
6C q16
(b) Hence
find the capacity of the skip.

4
2

End of paper

16.

C
Q

P
U

T
A

CD

O
S

AB
R

The first diagram shows two identical intersecting cylinders of radius r. The second
diagram is the view from above. Their axes AB and CD intersect at 90 at the point O.
The third diagram shows the solid which is common to both cylinders, bounded at its
widest point by the horizontal square P QRS.
(a) The typical square cross-section T U V W shown is parallel to the square P QRS and
y units above it. Find an expression for the area of this typical cross-section of the
42
CHAPTER 6: Volumes
solid in terms of y.

SGS NOTES

14

(b) Hence find the volume that is common to the two intersecting cylinders.
10(a)

(b) 27
2

y
#

## SGS Mathematics Year 12

Chapter 6: Volumes

volumes
"

5/4/14

c 2014

45 23 m3
2
2
3
16(a) 4(r y ) (b) 16
3 r
2
2
4 3
18(a) a x
(b) 3 a

1 2
19(a) 23 a (b) ax
(e) 12
a b(5 + 2 3) m
b2

R
20(a)(ii) (1 e
) (c)(i) R 2 (ii) (1
2
R2
(d) (1 e
) 4I 2 (1 e2R ) (e
15(b)
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