To find the scale factor of the small cylinder to the large cylinder by comparing their volumes, we can use the ratio of their volumes.
Let the volume of the large cylinder be \( V_L = 512\pi \) in³ and the volume of the small cylinder be \( V_S = 64\pi \) in³.
The ratio of the volumes is:
\[ \text{Scale Factor} = \frac{V_S}{V_L} = \frac{64\pi}{512\pi} \]
The \(\pi\) cancels out, so we have:
\[ \text{Scale Factor} = \frac{64}{512} \]
Now, simplifying this ratio:
\[ \frac{64}{512} = \frac{1}{8} \]
So, the scale factor of the small cylinder to the large cylinder is \(\frac{1}{8}\).
If you're interested in the linear scale factor (the ratio of their corresponding dimensions such as height or diameter), you would take the cube root of the volume ratio:
\[ \text{Linear Scale Factor} = \sqrt[3]{\frac{V_S}{V_L}} = \sqrt[3]{\frac{64}{512}} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \]
Thus, the scale factor of the small cylinder to the large cylinder is:
- Volume scale factor: \(\frac{1}{8}\)
- Linear scale factor: \(\frac{1}{2}\)
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