The volumes of the two cylinders are given as follows:
- Volume of the larger cylinder: \( V_1 = 512\pi , \text{in}^3 \)
- Volume of the smaller cylinder: \( V_2 = 64\pi , \text{in}^3 \)
To find the scale factor of the small cylinder to the larger cylinder, we can compare their volumes directly. The scale factor \( k \) is given by the ratio of the volumes:
\[ k = \left( \frac{V_2}{V_1} \right)^{1/3} \]
Calculating the ratio of the volumes:
\[ \frac{V_2}{V_1} = \frac{64\pi}{512\pi} = \frac{64}{512} = \frac{1}{8} \]
Now, we find the cube root:
\[ k = \left( \frac{1}{8} \right)^{1/3} = \frac{1}{2} \]
Therefore, the scale factor of the small cylinder to the larger cylinder is \( \frac{1}{2} \).