2003 AIME I Problems/Problem 5
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures by by units. Given that the volume of this set is where and are positive integers, and and are relatively prime, find
The set can be broken into several parts: the big parallelepiped, external parallelepipeds that each share a face with the large parallelepiped and have a height of , the spheres (one centered at each vertex of the large parallelepiped), and the cylinders connecting each adjacent pair of spheres.
- The volume of the parallelepiped is cubic units.
- The volume of the external parallelepipeds is .
- There are of the spheres, each of radius . Together, their volume is .
- There are of the cylinders, so complete cylinders can be formed. Their volumes are , , and , adding up to .
The combined volume of these parts is . Thus, the answer is .
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