How to find the volume of a cone - ISEE Upper Level (grades 9-12) Quantitative Reasoning

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Question

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

Answer

Let 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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Question

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

Answer

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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Question

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

Answer

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 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 and :

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

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

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

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

Answer

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 :

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

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

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

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

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

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

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