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Volume of Pyramids and Cones Worksheet - Page 4

Volume of Pyramids and Cones Worksheet
  • Page 4
 31.  
What is the volume of the figure shown?

a.
25π in3
b.
40π in3
c.
30π in3
d.
26π in3


Solution:

From the figure, the base radius of each cone, r = 3 in.
and the height of each cone, h = 5 in.

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

= 1 / 3 × π × 32 × 5
[Substitute r = 3 and h = 5.]

= 15π in.3
[Simplify.]

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

= 2 × 15π
[Substitute V = 15π.]

= 30π in.3

The volume of the figure is 30π in.3.


Correct answer : (3)
 32.  
Find the volume of the figure.

a.
60 in.3
b.
40 in.3
c.
120 in.3
d.
20 in.3


Solution:

From the figure, the length, width and height of the rectangular pyramid are 5 in., 4 in., and 3 in.

The base area of the rectangular pyramid = length × width

= 5 × 4
[Substitute the values.]

= 20 in.2
[Multiply.]

The volume of the rectangular pyramid = (1 / 3) × base area × height

= (13) × 20 × 3
[Substitute the values.]

= 20 in.3
[Simplify.]

The volume of the figure = 2 × volume of the rectangular pyramid

= 2 × 20
[Substitute volume = 20.]

= 40 in.3
[Multiply.]

The volume of the figure is 40 in.3


Correct answer : (2)
 33.  
Find the volume of the cone in the figure.


a.
11π cm3
b.
14π cm3
c.
13π cm3
d.
15π cm3


Solution:

From the figure, the base radius of the cone, r = 3 cm. and height, h = 5 cm.

Volume of the cone = 1 / 3πr2h
[Formula.]

= 13x π x 32 x 5
[Substitute base radius, r = 3 cm. and height, h = 5 cm.]

= 45π3
[Divide numerator and denominator by 3.]

= 15π
[Simplify.]

The volume of the cone is 15π cm.3.


Correct answer : (4)
 34.  
What is the volume of the cone whose diameter is 6 ft and slant height is 5 ft?

a.
13 ft3
b.
11 ft3
c.
14 ft3
d.
12π ft3


Solution:

The base radius of the cone, r = diameter / 2 = 6 / 2 = 3 ft
[Since the diameter of the cone is 6 ft.]

From the figure, AB2 + BC2 = AC2
[Pythagorean theorem.]

h2 + 32 = 52
[Substitute AB = h, BC = 3 ft, and AC = 5 ft.]

9 + h2 = 25
[Evaluate powers.]

h2 = 25 - 9
[Subtract 9 from both sides.]

h2 = 16
[Subtract 9.]

h = 16 = 4
[Take square root on both sides.]

The height of the cone is 4 ft.

The volume of the cone = 1 / 3πr2h
[Volume formula.]

= 1 / 3 × π × 32 × 4
[Substitute r = 3 and h = 4.]

= 12π ft3
[Simplify the expression.]

The volume of the cone is 12π ft3.


Correct answer : (4)
 35.  
What is the volume of the cone whose diameter is 6 ft and slant height is 5 ft?

a.
11π ft3
b.
14π ft3
c.
13π ft3
d.
12π ft3


Solution:

= 62
= 3 ft
The base radius of the cone, r = diameter / 2
[Since the diameter of the cone is 6 ft.]

From the figure, AB2 + BC2 = AC2
[Pythagorean theorem.]

h2 + 32 = 52
[Substitute AB = h, BC = 3 ft, and AC = 5 ft.]

h2 + 9 = 25
[Evaluate powers.]

h2 = 25 - 9
[Subtract 9 from both sides.]

h2 = 16
[Subtract.]

h = √16
h = 4
[Take square root on both sides.]

The height of the cone is 4 ft.

The volume of the cone = 1 / 3πr2h
[Volume formula.]

= 13x π x 32 x 4
[Substitute r = 3 and h = 4.]

= 12π ft3
[Simplify the expression.]

The volume of the cone is 12π ft3.


Correct answer : (4)
 36.  
If the radius of the base of a right circular cone is 6r mm. and its height is equal to the radius of the base, find its volume in mm.3.
a.
67πr3
b.
74πr3
c.
72πr3
d.
77πr3


Solution:

Height of the cone = base radius of the cone = 6r mm.

Volume of a cone = 1 / 3πr2h
[Formula.]

= 13 x π x (6r)2 x 6r
[Substitute the values.]

= 72πr3
[Simplify.]

The volume of the right circular cone = 72πr3 mm.3.


Correct answer : (3)
 37.  
What is the volume of a cone, if its radius is 3 cm and its curved surface area is 47.1 cm2? (Round the answer to the nearest whole number.)
a.
113 cm3
b.
141 cm3
c.
38 cm3
d.
47 cm3


Solution:

Curved surface area of cone = πrl
[Formula.]

πrl = 47.1
[Since curved surface area of cone is 47.1 cm2.]

3.14 x 3 x l = 47.1
[Substitute the values of π and r.]

l = 5
[Simplify.]

Slant height of the cone(l) = 5.

(Height of cone)2 = (slant height of cone)2 - (radius)2.

(Height of cone)2 = (5)2 - (3)2.
[Substitute the values.]

Height of cone = 4 cm
[Take the square root of each side.]

Volume of cone = 1 / 3πr2h
[Formula.]

= 1 / 3 x 3.14 x 32 x 4
[Substitute the values.]

= 37.68
[Simplify.]

38
[Round the answer to the nearest whole number.]

Volume of the cone = 38 cm3.


Correct answer : (3)
 38.  
The base of a pyramid is a right triangle and two sides containing the right angle are 4 ft and 4 ft. If height of the pyramid is 15 ft, then find the volume of the pyramid.
a.
35 ft3
b.
40 ft3
c.
49 ft3
d.
46 ft3


Solution:

Base of the pyramid = right triangle

Base area of the pyramid = area of right triangle = 1 / 2 × base × height = 1 / 2× 4 × 4 = 8 ft2

Volume of the pyramid = 1 / 3 × base area × height
[Formula.]

= 13 × 8 × 15
[Substitute the values.]

= 40
[Simplify.]

Volume of the pyramid = 40 ft3.


Correct answer : (2)
 39.  
The height of a right circular cone is 12 cm. If its volume is 1024π cm.3, what is the slant height of the cone?
a.
20 cm
b.
29 cm
c.
25 cm
d.
16 cm


Solution:

Volume of a cone = 13πr2h
[Formula.]

1024π = 13 x π x r2 x 12
[Substitute the values.]

r2 = 256
[Simplify.]

Slant height of cone (l) = √(r2 + h2).

= √ (256 + 122)
[Substitute the values.]

= √ (400) = 20
[Simplify.]

Slant height of the cone = 20 cm.


Correct answer : (1)
 40.  
State whether the statement is true or false. "The volume of a cone is equal to one third the volume of a cylinder with the same base radius and the same height."
a.
True
b.
False


Solution:

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

Volume of cylinder = πr2h
[Formula.]

Volume of cone = 1 / 3πr2h
[Formula.]

= 13 x πr2h

= 13 x volume of cylinder
[Since volume of cylinder is πr2h.]

Volume of cone = 1 / 3 x volume of cylinder.

So, the statement is true.


Correct answer : (1)

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