You are designing a new cylindrical can to hold 1 liter (1000 cubic centimeters) of Jake's Special Paint. Find the dimensions that will minimize the cost of metal to manufacture the can.
$\displaystyle V={\pi}r^{2}h=1000$..........[1]
$\displaystyle S=2{\pi}rh+2{\pi}r^{2}$..........[2]
Solve [1] for, say, h and sub into [2]:
$\displaystyle h=\frac{1000}{{\pi}r^{2}}$
$\displaystyle S=2{\pi}r(\frac{1000}{{\pi}rh})+2{\pi}r^{2}=2{\pi} r^{2}+\frac{2000}{r}$
This is what is to be minimized. Differentiate, set to 0 and solve for r. Then h will follow.
I believe you will find that the minimum is achieved when the diameter equals the height.