Question

Find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius 2.

Answer 1

A little investigation will lead you to the formula that if a hole of radius r is drilled through a sphere of radius R, the remaining volume is just 4pi/3 (R^2 - r^2)^(3/2)

In this case, that would be 4pi/3 * 3√3 = 4√3 pi
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You might also look up the napkin ring problem, where it is shown that if a hole of length h is drilled through a sphere, the remaining volume is independent of the radius of the sphere!

In this case, h/2 = R^2 - r^2 = √3, so h = 2√3

The remaining volume is pi/6 h^3 = pi/6 * 24√3 = 4√3 pi

Answer 2

thanks!