To estimate an answer for the first problem: consider two cylinders with zero volume, the one equatorial (r=R) and the other polar (r=0). Both correspond to zer

To estimate an answer for the first problem: consider two cylinders with zero volume, the one equatorial (r=R) and the other polar (r=0). Both correspond to zero volume. Hence the maximum volume occurs (hand wave) for a height somewhere between 0 and 2R. Not knowing better, the initial estimate for h is to bisect this interval, to give h_est = R (compare 1 with 2/sqrt(3) ~= 1.15). It would be nice to construct a first order correction term. Any ideas?