ISEE Upper Level Quantitative : How to find the volume of a cone

Example Questions

Example Question #1 : Cones

The height of Cone B is three times that of Cone A. The radius of the base of Cone B is one-half the radius of the base of Cone A.

Which is the greater quantity?

(a) The volume of Cone A

(b) The volume of Cone B

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

Let  be the radius and height of Cone A, respectively. Then the radius and height of Cone B are  and , respectively.

(a) The volume of Cone A is .

(b) The volume of Cone B is

.

Since , the cone in (a) has the greater volume.

Example Question #1 : Cones

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

Possible Answers:

Correct answer:

Explanation:

The volume of a cone with base radius  and height  is

The height  is three times this, or . Therefore, the formula becomes

Set this volume equal to one and solve for :

This is the radius in yards; since the radius in inches is requested, multiply by 36.

Example Question #1 : How To Find The Volume Of A Cone

The height of a given cylinder is one half the height of a given cone. The radii of their bases are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Possible Answers:

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

Call  the radius of the base of the cone and  the height of the cone. The cylinder will have bases of radius  and height .

In the formula for the volume of a cylinder, set  and :

In the formula for the volume of a cone, set  and :

, so

,

meaning that the cylinder has the greater volume.

Example Question #2 : Cones

The radius of the base of a given cone is three times that of each base of a given cylinder. The heights of the cone and the cylinder are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Possible Answers:

(a) is the greater quantity

(b) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

If we let  be the radius of each base of the cylinder, then  is the radius of the base of the cone. We can let  be their common height.

In the formula for the volume of a cylinder, set  and :

In the formula for the volume of a cone, set  and :

, so . The cone has the greater volume.