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?

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Answer 1

To find the scale factor of the small cylinder to the larger cylinder based on their volumes, we start by identifying the volumes of both cylinders given in the problem:

- Volume of the larger cylinder: \( V_1 = 512\pi \, \text{in}^3 \)
- Volume of the smaller cylinder: \( V_2 = 64\pi \, \text{in}^3 \)

The volumes of similar three-dimensional shapes (like cylinders) are related to the cube of the scale factor (k) of their corresponding linear dimensions. Therefore, we can express the relationship between the volumes in terms of the scale factor as follows:

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

Substituting in the volumes:

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

The \(\pi\) terms cancel out:

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

Next, we simplify \(\frac{64}{512}\):

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

So, 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 smaller cylinder to the larger cylinder is:

\[
\boxed{\frac{1}{2}}
\]