Problem
(5) A sphere has been placed inside a cylinder with the same dimensions; both have a radius of \( 6 \mathrm{~cm} \), and a height of \( 12 \mathrm{~cm} \). How many cubic centimeters of space does the sphere NOT fill up? (A) \( 432 \pi \mathrm{cm}^{3} \) (B) \( 72 \pi \mathrm{cm}^{3} \) (C) \( 288 \pi \mathrm{cm}^{3} \) (D) \( 144 \pi \mathrm{cm}^{3} \)
Solution
Step 1 ：Find the volume of the cylinder: \( V_{cylinder} = \pi r^2 h = \pi (6)^2 (12) \)
Step 2 ：Find the volume of the sphere: \( V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (6)^3 \)
Step 3 ：Subtract the sphere volume from the cylinder volume: \( V_{cylinder} - V_{sphere} = 288 \pi \mathrm{cm}^{3} \)
