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Here we see a cross section of the famous
napkin ring. It is a sphere with a cylindrical hole drilled through the middle.
The length of the hole is 3 (some units, inches, centimeters, whatever). I
forget the diameter of the sphere. What is the volume still remaining (the
sphere minus what has been removed)?
Hint: It may help you to know that the volume of a sphere is (4 pi r^3) / 3, where r^3 means r cubed. The volume of a cylinder is pi R^2 L. And the volume of the spherical cap (two of which were removed with the cylinder) is pi h (3 R^2 + h^2). By the way, a spherical cap is a simple kind of spherical segment. A spherical segment is the portion of a sphere between two parallel planes. A spherical segment can be thought of as a larger spherical cap minus a smaller spherical cap.
Solution: Choosing some radius r, it takes some algebra to simplify the following:
V(ring)= V(sphere) - V(cylinder) - 2V(cap)
In the end, we find that the volume is pi L^3 /6. In other words, we don't need to know the radius of the sphere; all we need to know is the length of the cylinder. Well the length was 3, so the answer to our problem is 9 pi / 2.
There is tricky way to solve this. We could have guessed that the answer depends solely on the length of the cylinder, since that is the only information given. So, what if the radius of the sphere is 1.5 (diameter 3), and the volume of the cylinder is 0? Then we just apply the formula for the volume of a sphere: (4 pi 27/8) / 3 = 9 pi / 2, which is the right answer.