Question

Two similar cylinders have volumes of
512
π

in.
3
and
64
π

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 using their volumes, we can use the relationship between the volumes of similar three-dimensional objects. The ratio of their volumes is equal to the cube of the ratio of their linear dimensions (scale factor).

Let the volume of the larger cylinder be \( V_L = 512\pi \) in³ and the volume of the smaller cylinder be \( V_S = 64\pi \) in³.

First, we find the ratio of the volumes:

\[
\text{Ratio of volumes} = \frac{V_S}{V_L} = \frac{64\pi}{512\pi} = \frac{64}{512} = \frac{1}{8}
\]

Since the ratio of the volumes is \( \frac{1}{8} \), we set this equal to the cube of the scale factor \( k \):

\[
\frac{V_S}{V_L} = k^3
\]

Thus, we have:

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

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

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

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