A closed cylindrical tank is to be built using 924m^2 of metal. Calculate the radius and height if the volume of the cylinder is to be a maximum.

I just cant get my head around these problems. (Headbang)

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- Nov 28th 2010, 08:00 AMmanexaA closed cylindrical tank
A closed cylindrical tank is to be built using 924m^2 of metal. Calculate the radius and height if the volume of the cylinder is to be a maximum.

I just cant get my head around these problems. (Headbang) - Nov 28th 2010, 08:10 AMskeeter
$\displaystyle 2\pi r^2 + 2\pi rh = 924$

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

solve for h in terms of r in the first equation ... sub the result into the volume equation ... find dV/dr, find the critical values of r, then determine the value of r that maximizes the volume.

use that value for r to calculate h. - Nov 29th 2010, 05:15 AMmanexa
Hey skeeter

Thanks for the help.

I still don't get it though. I think I will just have to accept that some things in maths are beyond me. (Worried) - Nov 29th 2010, 04:24 PMskeeter
don't give up so easy. this problem is not as bad as you think.

this first equation is the total surface area of a cylinder ...

$\displaystyle 2\pi r^2 + 2\pi rh = 924$

$\displaystyle h = \frac{924 - 2\pi r^2}{2\pi r} = \frac{462 - \pi r^2}{\pi r}$

sub this into the volume formula for h ...

$\displaystyle V = \pi r^2 \cdot \frac{462 - \pi r^2}{\pi r}$

$\displaystyle V = 462r - \pi r^3$

take the derivative w/r to r and set the result equal to zero ...

$\displaystyle \frac{dV}{dr} = 462 - 3\pi r^2 = 0$

$\displaystyle r = \sqrt{\frac{154}{\pi}}$

sub this value into the equation where h is in terms of r ... you should see that h = 2r ; the max volume for a fixed surface area occurs when the cylinder's side profile is a square.