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 $\displaystyle {\color{red}\rlap{///}}\text{mg}$ . 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:

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r r
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The hemispheres and the cylinder have radius $\displaystyle r.$

The cylinder has height $\displaystyle h.$

The volume of the two hemispheres is: .$\displaystyle \tfrac{4}{3}\pi r^3$

The volume of the cylinder is: .$\displaystyle \pi r^2h$

The total volume is 1000: .$\displaystyle \tfrac{4}{3}\pi r^3 + \pi r^2h \:=\:1000 \quad\Rightarrow\quad h \:=\:\frac{3000-4\pi r^3}{3\pi r^2} $ .[1]

The surface area of the cylinder is: .$\displaystyle 2\pi rh$ mm².

If it costs $\displaystyle k$ dollars per mm², its cost is: .$\displaystyle 2\pi krh$ dollars.

The surface area of the two hemispheres is: .$\displaystyle 4\pi r^2$ mm².

Since it costs $\displaystyle 2k$ dollars per mm². its cost is: .$\displaystyle 8\pi kr^2$ dollars.

The total cost is: .$\displaystyle C \;=\;8\pi kr^2 + 2\pi krh$ .[2]

Substitute [1] into [2]: .$\displaystyle C \;=\;8\pi kr^2 + 2\pi kr\left(\frac{3000-4\pi r^3}{3\pi r^2}\right) $

This simplifies to: .$\displaystyle C \;=\;\tfrac{16}{3}\pi kr^2 + 2000kr^{-1}$

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