Answer
$\frac{h}{r}=\frac{8}{\pi}\approx2.55$
Work Step by Step
Step 1. The cylinder has a base radius of $r$ and a height of $h$, its volume can then be expressed as $V=\pi r^2h=1000cm^3$ which gives $h=\frac{1000}{\pi r^2}$
Step 2. The total area formula given in this case is $A=8 r^2+2\pi rh=8 r^2+\frac{2000}{r}$
Step 3. To minimize the area, take the derivative of the above equation to get $A'=16 r-\frac{2000}{ r^2}$
Step 4. Let $S'=0$ to get $r^3=\frac{2000}{16}=5^3, r=5cm$, $h=\frac{1000}{\pi 5^2}=\frac{40}{\pi}=\frac{8}{\pi}r$ or $\frac{h}{r}=\frac{8}{\pi}\approx2.55$
Step 5. As $S''=16+\frac{4000}{r^3}\gt0$, the function is concave up and we have a minimum area for the $h,r$ values above.
