jmc

Question

Given a cylinder of fixed volume V, the total surface area (including the two circular ends) is minimized for a radius of R and height H. Find HR.

Accepted Answer

For radius r and height h, the volume is given by r^2 h = V, and the total surface area is given by A = 2 r^2 + 2 rh.By AM-GM, align* A &= 2 r^2 + 2 rh &= 2 r^2 + rh + rh & 3 [3](2 r^2)( rh)( rh) &= 3 [3]2 ^3 r^4 h^2. align*Since r^2 h = V, r^2 h = V. Then 3 [3]2 ^3 r^4 h^2 = 3 [3]2 ^3 V^2^2 = 3 [3]2 V^2.Equality occurs when 2 r^2 = rh, so hr = 2.