To find the scale factor of the small cylinder to the larger cylinder, we can use the relationship between the volumes of similar cylinders. The volume of similar solids is proportional to the cube of their linear dimensions.
Let the volumes of the larger and smaller cylinders be:
- Volume of larger cylinder, \( V_1 = 512\pi , \text{in}^3 \)
- Volume of smaller cylinder, \( V_2 = 64\pi , \text{in}^3 \)
The scale factor \( k \) can be calculated from the ratio of the volumes:
\[ \frac{V_1}{V_2} = \frac{512\pi}{64\pi} = \frac{512}{64} = 8 \]
Since the volumes of similar solids relate to the cube of the scale factor:
\[ \frac{V_1}{V_2} = k^3 \]
We set this equal to 8:
\[ k^3 = 8 \]
To find \( k \), we take the cube root of both sides:
\[ k = \sqrt[3]{8} = 2 \]
Thus, the scale factor of the small cylinder to the larger cylinder is the reciprocal of the scale factor from the larger to the smaller:
\[ \text{Scale factor of small to large} = \frac{1}{k} = \frac{1}{2} \]
Therefore, the scale factor of the small cylinder to the larger cylinder is \( \frac{1}{2} \).