an open right circular cylindrical tank of given volume stands with vertical axis. the material of its bottom costs twice as much per unit area as that for the sides. find the most economical proportion?
let the volume be V, where V is a constant
V = πr^2h
h = V/r^2
Cost = 2(πr^2) + 1(2πrh)
= 2πr^2 + 2Vπ/r
d(Cost)/dr = 4πr - 2Vπ/r^2 = 0 for min of Cost
4πr = 2Vπ/r^2
r^3 = 2Vπ/(4π) = V/4
form the ratio h : r
= V/r^2 : r
= V/r^3
= V/(V/4) = 4
so the ration of h:r should be 4:1