Right circular Cylinder A has a base radius of r and a height of h. Right circular Cylinde… — SAT

Right circular Cylinder $A$ has a base radius of $r$ and a height of $h$ . Right circular Cylinder $B$ has a base radius that is $3$ times the base radius of Cylinder $A$ , and a height that is $\frac{1}{2}$ the height of Cylinder $A$ . What is the ratio of the volume of Cylinder $B$ to the volume of Cylinder $A$ ?

A
$\frac{3}{2}$
B
$\frac{9}{4}$
$\frac{9}{2}$ Answer
D
$18$

Answer

The ratio of the volume of Cylinder

B

to the volume of Cylinder

A

\frac{9}{2}

The volume of a cylinder is given by the formula

V = \pi r^2 h

. For Cylinder

A

, the volume is

V_A = \pi r^2 h

. For Cylinder

B

, the radius is

3r

and the height is

\frac{1}{2}h

. Substituting these values gives

V_B = \pi (3r)^2 \left(\frac{1}{2}h\right) = \frac{9}{2}\pi r^2 h

. The ratio of the volume of Cylinder

B

to the volume of Cylinder

A

is therefore the ratio of nine to two.

Step-by-Step Solution

Write the volume formula for Cylinder

A

in terms of its radius

r

and height

h

V_A = \pi r^2 h

To establish a baseline volume for Cylinder

A

using the standard formula for the volume of a right circular cylinder.

Express the radius and height of Cylinder

B

in terms of

r

and

h

, and substitute them into the volume formula.

The radius of Cylinder

B

3r

and the height is

\frac{1}{2}h

. Thus,

V_B = \pi (3r)^2 \left(\frac{1}{2}h\right) = \pi (9r^2) \left(\frac{1}{2}h\right) = \frac{9}{2}\pi r^2 h

To find the volume of Cylinder

B

expressed in terms of the variables

r

and

h

Divide the volume of Cylinder

B

by the volume of Cylinder

A

to find the ratio.

\frac{V_B}{V_A} = \frac{\frac{9}{2}\pi r^2 h}{\pi r^2 h} = \frac{9}{2}

To calculate the ratio of the volume of Cylinder

B

to Cylinder

A

Key Concept

Dimensional scaling of the volume of a cylinder

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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