# Thread: Designing a cyclinder with minimal surface area

1. ## Designing a cyclinder with minimal surface area

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.

2. $V={\pi}r^{2}h=1000$..........[1]

$S=2{\pi}rh+2{\pi}r^{2}$..........[2]

Solve [1] for, say, h and sub into [2]:

$h=\frac{1000}{{\pi}r^{2}}$

$S=2{\pi}r(\frac{1000}{{\pi}rh})+2{\pi}r^{2}=2{\pi} r^{2}+\frac{2000}{r}$

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.

3. is the derivative:

4 * pi * r - 2000 * r ^ -2

4. or simplified...

4*pi*r-(2000/x^2)