Find the dimensions of the largest cylinder that can be cut from a solid sphere of radius a.
if you cut off the cylinder's ends so that it is a right circular cylinder, then looking at a cross-section of the sphere through the cylinder,
radius of sphere = a
radius of cylinder = r
height of cylinder = 2√(a^2-r^2)
the volume v of the cylinder is
v = πr^2 * h
v = πr^2 * 2√(a^2-r^2)
dv/dr = 2πr(a^2 - 3r^2)/√(a^2-r^2)
dv/dr = 0 where a^2 = 3r^2, or r = a/√3
so the cylinder has height 2√(a^2-r^2) = a√(8/3)