We are going to manufacture a cylindrical bottle which can c

We are going to manufacture a cylindrical bottle which can contain exactly 3pi (liters) of water. It has a metal top and bottom, but cardboard sides. Metal costs 1.5 times as expensive as cardboard. For what radius r and height h is the cost minimized?

Please explain

Solution

Volume V = r²h, so rh = V/r

Cost C =1.5*(2r²) + 2rh = 3r² + 2V/r

dC/dr = 6r - 2V/r² = 0

r = [V/(3)]^1/3

Given V = 3 liters = 3*10^-3 m³

so r = 0.1 m, h = V/(r²) = 0.3 m