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)
what is the scale factor of the small cylinder to the larger cylinder?

Answer 1

To find the scale factor of the smaller cylinder to the larger cylinder based on their volumes, we start by defining the volumes of the two cylinders:

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

Next, we find the ratio of the volumes:

\[
\text{Volume ratio} = \frac{V_2}{V_1} = \frac{64\pi}{512\pi} = \frac{64}{512} = \frac{1}{8}
\]

The scale factor for the dimensions of similar shapes relates to the cube root of the volume ratio. Thus, we calculate the scale factor \( k \) as follows:

\[
k = \sqrt[3]{\frac{V_2}{V_1}} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}
\]

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