To find the scale factor between the two similar cylinders based on their volumes, we use the relationship between the volumes of similar three-dimensional figures.
The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
For two similar cylinders, the volumes compare as follows: \[ \frac{V_1}{V_2} = k^3 \] where \( k \) is the scale factor between the two cylinders.
Given:
- Volume of the larger cylinder \( V_1 = 512\pi , \text{in}^3 \)
- Volume of the smaller cylinder \( V_2 = 64\pi , \text{in}^3 \)
First, we can simplify the volume ratio: \[ \frac{V_1}{V_2} = \frac{512\pi}{64\pi} = \frac{512}{64} = 8 \]
Now, we set up the equation: \[ k^3 = 8 \]
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: \[ \frac{1}{k} = \frac{1}{2} \]
Therefore, the scale factor of the small cylinder to the larger cylinder is \( \frac{1}{2} \).