\frac{V_{\text{cyl}}}{V_{\text{cube}}} = \frac{36\pi r^3}{216r^3} = \frac{36\pi}{216} = \frac{\pi}{6}
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- The volume of a cube is $ V_{\text{cube}} = s^3 $. For the cube, $ s = 6r $, so:
- V_{\text{cube}} = (6r)^3 = 216r^3
- The ratio of the cylinder's volume to the cube's volume is:
- Thus, the ratio is $ \boxed{\frac{\pi}{6}} $.
- Question: A pharmacologist is modeling a spherical drug capsule of radius $ r $ and a hemispherical cap of radius $ r $ attached to a cylindrical urodose container of height $ 2r $ and radius $ r $. What is the total volume of the composite object?
- Solution: The volume of a hemisphere is $ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 $.
