A closed Cylinder container needs to have a volume of 128pie dm3. Find the dimensions if the total area must be a minimum.
The surface area is,
$\displaystyle 2\pi r^2+2\pi rh$
The volume is,
$\displaystyle \pi r^2 h=128\pi$
Thus,
$\displaystyle h=\frac{128}{r^2}$
Substitute that into first equation,
$\displaystyle 2\pi r^2+2\pi r\left( \frac{128}{r^2} \right)$
Thus,
$\displaystyle 2\pi r^2+\frac{256 \pi}{r}$
In these problems it is safe to assume that the solution appears at the critical point.
Thus, derivative,
$\displaystyle 4\pi r-256 \pi r^{-2}=0$
Divide by $\displaystyle 4\pi$,
$\displaystyle 1-64 r^{-2}=0$
Solve for positive root,
$\displaystyle r=8$