Volume of cylinder:
\(\displaystyle V\, =\, 1000cm^3\, =\, \pi r^2h\)
Surface area of cylinder: Top+Bottom+Shaft
\(\displaystyle A\, =\, 2(\pi r^2)\, +\, 2\pi r h\)
You want to minimize the surace area, so solve for either r or h in the volume formula (I will demonstrate solving for h).
\(\displaystyle h\, =\, \frac{1000cm^3}{\pi r^2}\)
Then...
\(\displaystyle A\, =\, 2(\pi r^2)\, +\, 2\pi r \frac{1000cm^3}{\pi r^2}\, =\, 2 \pi r^2\, +\, \frac{2000cm^3}{r}\)
Notice A is a function of r now. You can take the derivative of A with respect to r and set equal to zero to get an r that will give you a minimum area. Use that r in the volume equation to get the height.
