An inverted cone of radius a and height h is filled with water. A sphere made of a material denser than water is placed in the cone. Find the radius of the sphere, r, that will displace the largest volume of water.
Since the problem is cylindrically symmetric, I believe it can be reduced to a circle and triangle and a difference of areas. But that's as far as I am willing to drop my olive.
[Later: no this is wrong. Area further from the central axis is worth more volume than area close to the central axis, when revolved. It is the integral of the area element x 2 pi r that must be maximized, where r is a cylindrical coordinate. It _is_ a 3D problem. My comment above is wrong.]
Edited on September 15, 2019, 4:13 pm