# Math Help - maximize volume, minimize surface area

1. ## maximize volume, minimize surface area

Given a volume of 1700 cm^3, minimize the surface area of a cylinder.

S= $2 \pi rh+2 \pi r^2$

V= $\pi r^2h=1700$

so, for substitutions:
$h=1700/ \pi r^2$, which I can put into the S equation to get:
$S=2 \pi r (1700/r^2) +2 \pi r^2$

And I think I can simplify the above equation to:
$3400 \pi /r +2 \pi r^2$

and so is the derivative of this equal to:
$-3400 \pi /r^2 +4 \pi r$

I wasn't completely sure if the above equation was right. But continuing on, I would set the derivative equal to 0 and solve for r.

Any help would be greatly appreciated!-I know this is a long post, sorry.

2. You missed the pi there:

$S=2 \pi r (1700/r^2) +2 \pi r^2$

It should be:

$S=2 \pi r \left(\dfrac{1700}{\pi r^2}\right) +2 \pi r^2$

3. I can't believe I missed that! Thank you. So to simplify that equation:

$3400/r + 2 \pi r^2$

So the new derivative is:
$-3400/r^2 +4 \pi r$

and using my graphing calc, there is a zero at 6.4677966...I guess thats right

4. so I assume the above value I got for r is the minimum? then I just substitute it into the other equation and solve for h...right?

5. Yes, but why 'assume' while you can be sure?

Remember that if $\dfrac{d^2S}{dr^2} > 0$, the stationary point is a minimum.