Begin with a unit cube and inscribe a sphere in it. Then inscribe 8
small spheres in the empty corners so that each of them is tangent
to the sides of the cube and also tangent to the large central sphere. Find the volume of the region enclosed by the cube but outside the 9 spheres.
Without checking Mr. Lord:
Sphere diameter = 1
The distance across the cube from opposite corners is sqrt(3).
Distance from sphere shell to any corner is (sqrt(3) - 1))/2
Let D = diameter of smaller cube.
So: (sqrt(3) - 1))/2 = D + [D(sqrt(3) - 1)]/2
= D (sqrt(3) + 1)/2
(sqrt(3) - 1))/2 = D (sqrt(3) + 1)/2
D = (sqrt(3) - 1) / (sqrt(3) + 1) ~ 0.26795
Total Volume = (4*pi/3) * (0.5^3 + 8 * 0.1339746^3)
= 4.18879 * (0.144238) = 0.6042 cubit units
Empty space = 0.3958 cubic units
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Posted by hoodat
on 2025-01-24 18:08:34 |
solution
