# optimization problem(minimum total area, given the volume)

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• Nov 28th 2006, 06:38 AM
dcfi6052
optimization problem(minimum total area, given the volume)
A closed Cylinder container needs to have a volume of 128pie dm3. Find the dimensions if the total area must be a minimum.:D
• Nov 28th 2006, 07:09 AM
ThePerfectHacker
Quote:

Originally Posted by dcfi6052
A closed Cylinder container needs to have a volume of 128pie dm3. Find the dimensions if the total area must be a minimum.:D

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,
$\displaystyle r=8$
• Nov 28th 2006, 07:50 AM
dcfi6052
:D :cool:
• Nov 28th 2006, 08:15 AM
CaptainBlack
Quote:

Originally Posted by dcfi6052
:D :cool:

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