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

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Example Questions

Example Question #44 : Solid Geometry

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) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

Let r,h be the radius and height of Cone A, respectively. Then the radius and height of Cone B are $\frac{1}{2}r$ and , respectively.

(a) The volume of Cone A is $\frac{1}{3} \pi r^{2} h$ .

(b) The volume of Cone B is

$\frac{1}{3} \pi \left ( \frac{1}{2} r\right ) ^{2} \cdot 3 h = \frac{1}{3} \cdot \frac{1}{4} \cdot 3\cdot \pi r ^{2} h= \frac{1}{4} r ^{2} h$ .

Since $\frac{1}{4} r ^{2} h < \frac{1}{3} r ^{2} h$ , the cone in (a) has the greater volume.

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Example Question #3 : Volume Of A Cone

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:

$\frac{ \sqrt[3]{ \pi^{2} }}{ 3 \pi}$

$\frac{ 12 \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ 3 \sqrt[3]{ \pi^{2} }}{ \pi}$

Correct answer:

$\frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

Explanation:

The volume of a cone with base radius and height is

$V =\frac{1}{3} \pi r^{2}h$

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

$V =\frac{1}{3} \pi r^{2} \cdot 3r$

$V = \pi r^{3}$

Set this volume equal to one and solve for :

$\pi r^{3} = 1$

$\pi r^{3} \div \pi = 1 \div \pi$

$r^{3}=\frac{ 1 }{ \pi}$

$r=\sqrt[3]{\frac{ 1 }{ \pi}} ={\frac{ \sqrt[3] {1} }{ \sqrt[3]{ \pi}}} ={\frac{ 1}{ \sqrt[3]{ \pi}}}={\frac{1 \cdot \sqrt[3]{ \pi^{2} }}{ \sqrt[3]{ \pi} \cdot \sqrt[3]{ \pi^{2}}} ={\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}}$

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

$\frac{ \sqrt[3]{ \pi^{2} }}{ \pi} \times 36 = \frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

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Example Question #1 : Cones

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:

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

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

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 $\frac{1}{2} H$ .

In the formula for the volume of a cylinder, set r = R and $h = \frac{1}{2} H$ :

$V = \pi r^{2}h$

$V = \pi R ^{2} \cdot \frac{1}{2}H = \frac{1}{2} \pi R ^{2}H$

In the formula for the volume of a cone, set r = R and h = H :

$V = \frac{1}{3} \pi r^{2}h$

$V = \frac{1}{3} \pi R^{2}H$

$\frac{1}{2} > \frac{1}{3}$ , so

$\frac{1}{2} \pi R ^{2}H > \frac{1}{3} \pi R ^{2}H$ ,

meaning that the cylinder has the greater volume.

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Example Question #1 : 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

(a) and (b) are equal

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

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 r = R and h = H :

$V = \pi r^{2}h$

$V = \pi R ^{2}H$

In the formula for the volume of a cone, set r = 3 R and h = H :

$V = \frac{1}{3} \pi r^{2}h$

$V = \frac{1}{3} \pi (3R)^{2}H = \frac{1}{3} \pi (9R^{2})H = 3\pi R ^{2}H$

3 > 1 , so $3\pi R ^{2}H > \pi R ^{2}H$ . The cone has the greater volume.

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

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #44 : Solid Geometry

Example Question #3 : Volume Of A Cone

Example Question #1 : Cones

Example Question #1 : Cones

All ISEE Upper Level Quantitative Resources

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