Question

Two similar cylinders have volumes of 512π in.3 and 64π in.3 . By comparing volume, what is the scale factor of the small cylinder to the larger cylinder?(1 point)

Answer 1

To find the scale factor of the small cylinder to the larger cylinder based on their volumes, we can use the formula for the volume of similar cylinders. The volumes of similar shapes are related by the cube of the scale factor.

Let \( V_1 = 512\pi \) in³ be the volume of the larger cylinder, and \( V_2 = 64\pi \) in³ be the volume of the smaller cylinder. We can find the scale factor \( k \) using the following relationship:

\[
\frac{V_2}{V_1} = k^3
\]

Substituting the volumes into the equation gives:

\[
\frac{64\pi}{512\pi} = k^3
\]

The \(\pi\) cancels out:

\[
\frac{64}{512} = k^3
\]

Simplifying the fraction:

\[
\frac{64}{512} = \frac{1}{8}
\]

Now we have:

\[
k^3 = \frac{1}{8}
\]

To find \( k \), we take the cube root of both sides:

\[
k = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}
\]

Thus, the scale factor of the small cylinder to the larger cylinder is \( \frac{1}{2} \).