Volume of Pyramids and Cones Worksheet - Page 5

Volume of Pyramids and Cones Worksheet
• Page 5
41.
The circumference of the base of a cone of height 12 in. is 44 in. Find the volume of the cone. [Use π = $\frac{22}{7}$]
 a. 611. in.3 b. 630. in.3 c. 616 in.3 d. 620. in.3

Solution:

Circumference of the base of a cone = 2πr
[Formula.]

r = 44 in.
[Since circumference of a cone is 44 in.]

2 x 227 x r = 44

r = 7 in.
[Simplify.]

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

= 13 x 227 x 72 x 12
[Substitute the values.]

= 616
[Simplify.]

Volume of the cone = 616 in.3.

42.
A tent is in the shape of a cylinder with a conical top. The radius of the base of the tent is 9 m. The height of the cylindrical part is 16 m. and that of the conical part is 25 m. Find the volume of air that occupies the tent. (Round the answer to one decimal place.)

 a. 6203.9 m3 b. 6173.9 m3 c. 6193.9 m3 d. 6188.9 m3

Solution:

Volume of cylindrical part = πr2h
[Formula.]

= 3.14 x 92 x 16
[Substitute the values.]

= 4069.44
[Simplify.]

Volume of the cylindrical part = 4069.44 m.3.

Volume of Conical part = 1 / 3πr2h
[Formula.]

= 13 x 3.14 x 92 x 25
[Substitute the values.]

= 2119.50
[Simplify.]

Volume of the cone = 2119.50 m.3.

= (2119.50 + 4069.44) m.3 = 6188.94 m.3
Volume of air that occupies the tent = volume of cylindrical part + volume of conical part

6188.9 m.3
[Round the answer to one decimal place.]

Volume of air that occupies the tent = 6188.9 m.3.

43.
The area of the base of a cone is 15 in.2 Its volume is 195 in.3 What is the height of the cone?
 a. 49 in. b. 34 in. c. 44 in. d. 39 in.

Solution:

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

= 13 x πr2 x h

Volume of cone = 1 / 3 x base area of cone x h
[The base of a cone is a circle.]

195 = 13 x 15 x h
[Substitute the values.]

h = 39 in.
[Multiply each side with 3 / 15 .]

Height of the cone = 39 in.

44.
A pyramid is placed on a cube whose each edge measures 12 inches such that the total height of the cube and the pyramid is 18 inches. What is the total volume of the solid?

 a. 2024 in.3 b. 2016 in.3 c. 1996 in.3 d. 2028 in.3

Solution:

Height of a cube = length of each edge of a cube = 12 in.

18 = height of the pyramid + 12
Total height of the figure = height of pyramid + height of cube
[Substitute the values.]

Height of the pyramid = 6 in.
[Subtract 12 from each side.]

Volume of the cube = s3 = 123 = 1728 in.3.

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

= 13 x 122 x 6
[Substitute the values.]

= 288
[Simplify.]

Volume of the pyramid = 288 in.3
[Simplify.]

= 1728 + 288 = 2016
Volume of the figure = volume of the cube + volume of the pyramid

Volume of the figure = 2016 in.3.

45.
The base of a pyramid is a right triangle and two sides containing the right angle are 8 ft and 6 ft. If height of the pyramid is 6 ft, then find the volume of the pyramid.
 a. 48 ft3 b. 43 ft3 c. 57 ft3 d. 54 ft3

Solution:

Base of the pyramid = right triangle

Base area of the pyramid = area of right triangle = 1 / 2× base × height = 1 / 2× 8 × 6 = 24 ft2.

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

= 13 × 24 × 6
[Substitute the values.]

= 48
[Simplify.]

Volume of the pyramid = 48 ft3.

46.
The height of a right circular cone is 12 cm. If its volume is 1024π cm3, what is the slant height of the cone?
 a. 29 cm b. 16 cm c. 25 cm d. 20 cm

Solution:

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

1024π = 13 × π × r2 × 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.

47.
State whether the statement is true or false. "Volume of a cone is equal to one third of the volume of a cylinder with the same radius of base 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 × πr2h

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

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

So, the statement is true.

48.
The circumference of the base of a cone of height 18 in. is 88 in. Find the volume of the cone. (Use π = $\frac{22}{7}$)
 a. 3691.2 in.3 b. 3710.8 in.3 c. 3696 in.3 d. 3700.8 in.3

Solution:

Circumference of the base of a cone = 2πr
[Formula.]

r = 88 in.
[Since circumference of a cone is 88 in.]

2 × 227 × r = 88

r = 14 in.
[Simplify.]

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

= 13 × 227 × 142 × 18
[Substitute the values.]

= 3696
[Simplify.]

Volume of the cone = 3696 in.3

49.
The area of the base of a cone is 14 in.2. Its volume is 154 in.3. What is the height of the cone?
 a. 28 in. b. 33 in. c. 38 in. d. 43 in.

Solution:

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

= 13 × πr2 × h

Volume of cone = 1 / 3 × base area of cone × h
[Base of the cone is circle.]

154 = 13 × 14 × h
[Substitute the values.]

h = 33 in.
[Multiply each side with 3 / 14 .]

Height of the cone = 33 in.

50.
A pyramid is placed on a cube of 12 cm edge as in the figure such that the total height of the cube and the pyramid is 15 cm. What is the total volume of the solid?

 a. 1852 cm3 b. 1884 cm3 c. 1872 cm3 d. 1880 cm3

Solution:

Height of a cube = length of each edge of a cube = 12 cm

15 = height of the pyramid + 12
Total height of the figure = height of pyramid + height of cube
[Substitute the values.]

Height of the pyramid = 3 cm
[Subtract 12 from each side.]

Volume of the cube = s3 = 123 = 1728 cm3.

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

= 13 × 122 × 3
[Substitute the values.]

= 144
[Simplify.]

Volume of the pyramid = 144 cm3
[Simplify.]

= 1728 + 144 = 1872
Volume of the figure = volume of the cube + volume of the pyramid