An open right circular cylinder tank of given volume stands with vertical axis. The material of its bottom costs twice as much per unit area as that of the sides. Find the most economical proportion.

with radius r and height h, we have

πr^2 h = v, so
h = v/(πr^2)
Without loss of generality, we can let the unit cost of the sides be 1. So the total cost is
c = 1*2πrh + 2*πr^2 = 2πr(v/(πr^2)) + 2πr^2 = 2v/r + 2πr^2
dc/dr = 4πr - 2v/r^2
dc/dr=0 when r^3 = v/(2π)