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Volume of Pyramids and Cones Worksheet - Page 3

Volume of Pyramids and Cones Worksheet
  • Page 3
 21.  
Which of the following statements is/are false?
I. Volume of a cone is one-third of the product of the base area and height.
II. Volume of a cylinder is the product of base area and height.
III. Volume of a cylinder is the sum of its base area and its height.
IV. Volume of a cone is one-fourth the product of its base area and height.
a.
II and III only
b.
I and II only
c.
III and IV only
d.
II and IV only


Solution:

Volume of a cone is one-third of the product of the base area and height.

Volume of a cylinder is the product of base area and height.

Therefore, the statements I and II are true.

Hence, the statements III and IV are false.


Correct answer : (3)
 22.  
A cone and a cylinder have the same radius and same height. Which of the following is false for the volume of cone and cylinder?
a.
Volume of the cone is lesser than the volume of the cylinder.
b.
Volume of the cone is 1 3 times the volume of the cylinder.
c.
Volume of the cone is same as the volume of the cylinder.
d.
Volume of the cylinder is 3 times the volume of the cone.


Solution:

Since the cone and the cylinder have the same radius, they have same base area.

Volume of the cone = 13(base area × height)

= 3 × volume of a cone
Volume of a cylinder = base area × height = 3(13 × base area × height)

Therefore, the statement 'volume of the cone is same as the volume of the cylinder' is false.


Correct answer : (3)
 23.  
Find the volume of the figure shown, if AB = 8 ft and CO = C′ O = 18 ft.


a.
94π ft3
b.
190π ft3
c.
192π ft3
d.
96π ft3


Solution:

From the figure, the base diameter of each cone, d = AB = 8 ft
and the height of each cone, h = CO = 18 ft.

= 82
The base radius of the cone, r = diameter / 2
[Substitute diameter = 8 ft.]

= 4 ft
[Divide numerator and denominator by 3.]

Volume of each cone, V = 1 / 3πr2h
[Formula.]

= 13 × π × 42 × 18
[Substitute r = 4 and h = 18.]

= 96π ft3
[Simplify.]

The volume of the figure = 2 × V
[Since the figure contains two identical cones.]

= 2 × 96π ft3
[Substitute, V = 96π ft3.]

= 192π ft3

The volume of the figure is 192π ft3.


Correct answer : (3)
 24.  
Find the volume of the rectangular pyramid shown.

a.
140 cm3
b.
120 cm3
c.
40 cm3
d.
100 cm3


Solution:

From the figure, the length, width, and height of the rectangular pyramid are 5 cm, 4 cm, and 6 cm.

Volume of a rectangular pyramid = 1 / 3 × length × width × height

= 1 / 3 × 5 × 4 × 6
[Substitute the values.]

= 120 / 3 cm3

= 40 cm3
[Simplify.]

The volume of the rectangular pyramid is 40 cm3.


Correct answer : (3)
 25.  
What is the base area of a rectangular pyramid, whose height is 15 cm and volume is 125 cm3?
a.
10 cm2
b.
25 cm2
c.
8 cm2
d.
none of these


Solution:

The volume of the rectangular pyramid = 1 / 3 × base area × height
[Volume formula.]

The base area of the rectangular pyramid = 3 × VolumeHeight

= 3 × 12515
[Substitute the values.]

= 375 / 15
[Multiply 3 by 125.]

= 25
[Divide.]

The base area of the rectangular pyramid is 25 cm2.


Correct answer : (2)
 26.  
What is the base width of a rectangular pyramid, if its length, height, and volume are 3 cm., 2 cm., and 12 cm3 respectively?
a.
1 cm
b.
2 cm
c.
11 cm
d.
6 cm


Solution:

Volume of the rectangular pyramid = 1 / 3 × length × width × height
[Volume formula.]

The base width of the rectangular pyramid = 3  × Volumelength × height

= 3 × 123 × 2
[Substitute the values.]

= 36 / 6
[Simplify.]

= 6
[Divide the numerator and the denominator by 6.]

So, the base width of the rectangular pyramid is 6 cm.


Correct answer : (4)
 27.  
Find the volume of the rectangular pyramid, whose base length, diagonal of the base and height of the pyramid are 4 cm, 5 cm, and 6 cm respectively.

a.
70 cm3
b.
24 cm3
c.
60 cm3
d.
none of these


Solution:

The rectangular base consists of two right triangles.

According to the right triangle property, 42 + w2 = 52, where w is the width of the rectangular base.

16 + w2 = 25

w2 = 25 - 16

w2 = 9
[Subtract 16 from 25.]

w = 9

w = 3 cm.

So, the width of the rectangular base = 3 cm.

Volume of the rectangular pyramid = (1 / 3) × length × width × height

= (13) × 4 × 3 × 6
[Substitute the values.]

= 723
[Multiply 4, 3 and 6.]

= 24 cm.3
[Divide.]

Volume of the rectangular pyramid is 24 cm.3.


Correct answer : (2)
 28.  
What is the base area and volume of the rectangular pyramid whose length, width, and height are 6 cm, 4 cm, and 5 cm respectively?

a.
24 cm3 and 120 cm2
b.
24 cm2 and 40 cm3
c.
24 cm2 and 100 cm3
d.
none of these


Solution:

The base area of the rectangular pyramid = length × width

= 6 × 4
[Substitute the values.]

= 24 cm2
[Multiply 6 by 4.]

The volume of a rectangular pyramid = 1 / 3 × base area × height

= 1 / 3 × 24 × 5
[Substitute the values.]

= 120 / 3
[Multiply 24 by 5.]

= 40 cm3
[Divide.]

The base area and volume of the rectangular pyramid are 24 cm2 and 40 cm3.


Correct answer : (2)
 29.  
Find the height of the cone whose volume is 480π cm.3 and base radius is 12 cm.


a.
20 cm
b.
13 cm
c.
10 cm
d.
None of the above


Solution:

Let h be the height of the cone and r be the base radius of the cone.

Volume of the cone, v = (1 / 3)πr2h
[Volume formula.]

The height of the cone, h = 3vπ r2

3×480ππ (12)2
[Substitute v = 480π and r = 12]

= 10 cm

The height of the cone is 10 cm.


Correct answer : (3)
 30.  
What is the base radius of a cone, whose volume is 12π ft3 and height is 4 ft?


a.
2 ft
b.
6 ft
c.
5 ft
d.
3 ft


Solution:

Let r be the radius of the cone.

Volume of the cone, V = (1 / 3r2h
[Formula.]

r = 3Vπh
[Rewrite the formula.]

= 3 × 12ππ × 4
[Substitute V = 12π and h = 4.]

= 3 ft
[Simplify.]

The base radius of the cone is 3 ft.


Correct answer : (4)

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