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