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.
$\displaystyle 2\pi r^2 + 2\pi rh = 924$
$\displaystyle V = \pi r^2 h$
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.
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.