1. ## Optimization (word problem)

Ok, so the problem is about Optimization and I'm not really sure how to approach it.

Minimum Cost An industrial tank (two hemispheres and a circular cylinder) must have a volume of 3000 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize cost.

Sadly I have only gotten the volume formula

$4/3$π $r^3+$π $r^2h=3000$

Any help would be appreciated. I'm not looking for a solution, but rather for a process. Thanks in advance.

2. We must minimize the cost of the material, so we need the surface area.

The surface area of the sphere is $4{\pi}r^{2}$

The surface area of the cylindrical portion is $2{\pi}rh$

$S=4{\pi}r^{2}+2{\pi}rh$

Use the volume formula to solve for a variable and sub into the surface area formula so that it is in terms of one variable.

Then, you can optimize. Don't forget to take into account the cost conditions given.

3. ah ok!

I feel so stupid I didn't realize it was a surface area question