Question

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\frac{m + n\pi}{p},$ where $m,$ $n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$

答案

Solution:

The region consists of the box itself, six slabs of thickness $1$ projecting outward from the faces, quarter-cylinders of radius $1$ along the twelve edges, and eighth-spheres of radius $1$ at the eight corners. The box has volume $3 \cdot 4 \cdot 5 = 60,$ and the slabs total $2(3 \cdot 4 + 3 \cdot 5 + 4 \cdot 5) = 94.$

The four quarter-cylinders along edges parallel to each dimension combine into a full cylinder, so the cylinders total $\pi \cdot 1^2 (3 + 4 + 5) = 12\pi.$ The eight octants combine into one unit sphere of volume $\frac{4\pi}{3}.$

The total volume is $60 + 94 + 12\pi + \frac{4\pi}{3} = 154 + \frac{40\pi}{3} = \frac{462 + 40\pi}{3},$ so $m + n + p = 462 + 40 + 3 = 505.$

2003 AIME I Problem 5

Solution:

Problem 5 in Other Years