A container open at the top is a right circular cylinder having a surface (side and lower base) area of 108 square centimeters. What should the radius and altitude be in order to provide the largest possible volume.

Answer 1

so you have
πr^2 + 2πrh = 108
so h = (108-πr^2)/(2πr)
Now, the volume is
v = πr^2 h = πr^2 (108-πr^2)/(2π) = 54r^2 - π/2 r^4
so max vomume occurs when
dv/dr = 0
dv/dr = 108r - 2πr^3 = 2r(54-πr^2)
dv/dr=0 at r=√(54/π) = 3√(6/π)
That makes the height h = (108-πr^2)/(2πr) = 27/(πr)