Answered: The right circular cylinder and cone…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter9: Surfaces And Solids

Section9.CT: Test

Problem 12CT

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Question

Start by finding the volume of the cone. What is the volume of a pyramid or cone with a base of area B and height h?

The base of the cone is a circle. What is the area of a circle of radius r?

Then find the volume of the circular cone by substituting the height of the cone and the area of its base into the equation for the volume.

Next, find the volume of the cylinder. What is the volume of a prism or cylinder with a base of area B and height h?

The base of the cylinder is a circle. What is the area of a circle of radius r?

Substitute the height, 2r, and the base area,

πr2,

into the equation for the volume of a cylinder.

Now, find the volume of the sphere. What is the volume of a sphere with radius r?

Therefore, the volumes of the cone, the sphere, and the cylinder are

23πr3,

43πr3,

and

2πr3

respectively.

Now that the expressions for the volumes have been found, they can be compared algebraically. Since the goal is to prove that the volumes of the cone, the sphere, and the cylinder are in the ratio 1 to 2 to 3, divide all three of the volumes by the volume of the cone, which is the quantity that corresponds to the 1 in the ratio.

Divide the volume of the cone by the volume of the cone. The result is ?

Divide the volume of the sphere by the volume of the cone. The result is ?

Divide the volume of the cylinder by the volume of the cone. The result is ?

Thus, it has been verified that ratio of the volume of the cone to the volume of the sphere to the volume of the cylinder is 1 to 2 to 3.