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Hole in the Sphere
(solution)
If you want to avoid the calculations in Solution 1, simply take a look
directly at Solution 2 below it.
Solution 1
Let R be the radius of the sphere. As the illustration indicates,
the radius of the cylindrical hole will then be the square root of R2
- 9, and the altitude of the spherical caps at each end of the cylinder
will be R - 3. To determine the residue after the cylinder and caps have
been removed, we add the volume of the cylinder,
6π(R2
- 9), to twice the volume of the spherical cap, and subtract the
total from the volume of the sphere, 4πR3/3.
The volume of the cap is obtained by the following formula, in which A
stands for its altitude and r for its radius:
πA(3r2 + A2)/6.
When this computation is made, all terms obligingly cancel out except
36π
- the volume of the residue in cubic inches. In other words, the residue
is constant regardless of the hole's diameter or the size of the sphere!
Solution 2
John W. Campbell, Jr., editor of Astounding Science Fiction, was
one of several readers who solved the sphere problem quickly by reasoning
adroitly as follows: The problem would not be given unless it has a unique
solution. If it has a unique solution, the volume must be a constant which
would hold even when the hole is reduced to zero radius. Therefore the
residue must equal the volume of a sphere with a diameter of six inches,
namely 36π.
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