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

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- Nov 28th 2006, 06:38 AMdcfi6052optimization 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 AMThePerfectHacker
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 AMdcfi6052
:D :cool:

- Nov 28th 2006, 08:15 AMCaptainBlack