A cube of side length 10 and a sphere having the same volume as the cube have the same center. Find the volume of the sphere that lies outside the cube.
The cube has a volume of 10^3=1000. Let the radius of the sphere be R. Then the sphere has a volume of (4/3)*pi*R^3. These are equal: 1000 = (4/3)*pi*R^3.
This simplifies to R = 5*cbrt[6/pi]. Let h be the thickness of one of the spherical caps extending from the cube. Then h = 5*cbrt[6/pi] - 5.
A formula for the volume of a spherical cap in terms of R and h is given at https://mathworld.wolfram.com/SphericalCap.html by V = (pi/3)*h^2*(3R-h).
Then the volume needed for this problem is six times that: 6*(pi/3)*(5*cbrt[6/pi] - 5)^2*(3*5*cbrt[6/pi] - (5*cbrt[6/pi] - 5)).
Then after some tedious arithmetic the volume equals 3000 - 750*cbrt[36*pi] + 250*pi = 158.416.
Edited on September 9, 2020, 1:03 pm
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