Hello, Parrishguy!

I'll get you started . . .

Suppose you are a pharmacist who manufactures pill capsules.

The pill is shaped like a cylinder with a hemisphere on both ends.

The pill must have a volume of 1000 . maybe 1000 mm³ ?

The surface area of the hemispheres cost twice as much as the surface area of the cylinder.

Find the dimensions which will minimize the cost of the pill. Code:

* * *
* *
* *
* *
* - - - - + - - - - *
| |
| |
h | | h
| |
| |
* - - - - + - - - - *
r r
* *
* *
* *
* * *

The hemispheres and the cylinder have radius

The cylinder has height

The volume of the two hemispheres is: .

The volume of the cylinder is: .

The total volume is 1000: . .[1]

The surface area of the cylinder is: . mm².

If it costs dollars per mm², its cost is: . dollars.

The surface area of the two hemispheres is: . mm².

Since it costs dollars per mm². its cost is: . dollars.

The total cost is: . .[2]

Substitute [1] into [2]: .

This simplifies to: .

. . And *that* is the function we must minimize.