Imagine a flat-bottomed cylinderal container with a circular cross section of radius 4 in. a marble with radius 0<r<4 inches is placed in the bottom of the can. what is the radius of the bottom that requires the most water to cover it.
well, clearly, the marble has a height of 2r, so the water must be that deep to cover it.
Since the cylinder is said to have a radius of 4, I think you want the radius of the marble that requires the most water to cover it.
The volume of water is
v = π*16(2r) - 4/3 π r^3
= 32πr - 4/3 πr^3
dv/dr = 32π - 4πr^2
dv/dr=0 when r = √8
cute problem. A larger marble requires deeper water, but it takes up more of the cylinder, decreasing the volume of water.