A cylindrical hole is bored centrally through a ball, you are told the length of the hole, now find the volume of the remaining material. The surprising thing is that the data is sufficient - you do not need to know the radius of the ball.
Can this problem be solved, if so, what is the solution?
You know the volume of the hole, and the intact sphere.
You have enough information to compute the diameter of the hole, and so the volumes of the two spherical caps that are needed in addition to a cylinder the same size as the hole to complete the sphere.
So yes, you have enough information to solve this problem.
I tried to solve it but I don't succeed.
Mathstud, what is the "h" in the exponent "2h" at the end of your formula?
CaptainBlack, how do you know the volume of the hole and the intact sphere at the outset?
Can you give examples of solutions?
The problem comes from the website of W. W. Sawyer.
where r is the radius of the sphere
where h is the height of the hole and the radius of the cylinder, and we also know:
Then the total volume of the caps is:
(you will need to check the volume of the caps as I changed variables on the fly so there could easily be an error)
If this has all gone right when you simplify that it should be independent of .
It can be shown, by carefully integrating the volume of the "caps" on either end of such a hole, that this is, in fact, independent of the radius of the sphere, as claimed in the problem.