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 \( \frac{1}{2} \).
1 answer