NMSR PUZZLES
November 2005 Solution
By John Geohegan
The November Bonus Puzzle asked for the volume remaining after drilling a hole six inches long straight through a solid sphere. The answer is 113.1 (36p) cubic inches, which is the volume of a sphere having a radius of 3 inches. Surprisingly, the answer doesn't depend on the size of the sphere other than requiring its minimum size. A sphere larger than 3" radius R requires a hole with radius equal to the square root of (R2 - 9) to obtain the required length of 6 inches. The volume of the hole is then 6p times (R2 - 9) and the volume of the truncated spherical segment (with its hole filled) is 6p times (R2 - 3), which can be found with the calculus or from a handbook which lists such formulas. The difference (e.g. the sphere with the hole excavated) is 36p, and is independent of the sphere's radius.