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## Homework Statement

## Homework Equations

Maxima/minima are where the first derivative is 0

Volue of a hemisphere: ##~\displaystyle V=\frac{2}{3}\pi r^3##

Area of a hemisphere: ##~\displaystyle A=2\pi r^2##

## The Attempt at a Solution

$$A=2\pi rh+2\pi r^2=2\pi[h+r],~~V=\pi r^2h+\frac{2}{3}\pi r^3~\rightarrow~h=\frac{2r}{3}-\frac{V}{\pi r^2}$$

$$A=2\pi r\left[ \frac{2r}{3}-\frac{V}{\pi r^2}+r \right]=2\pi r\left[ \frac{5}{3}r-\frac{V}{\pi r^2} \right]$$

$$A'=2\pi \left[ \frac{5}{3}r-\frac{V}{\pi r^2}+r\left( \frac{5}{3}+\frac{2V}{\pi}r^{-3}\right)\right]$$

$$A'=0:~2\pi\left[ \frac{10}{3}r-\frac{V}{\pi r^2}+\frac{2V}{\pi r^3}\right]=0$$

$$2\pi\left[ \frac{10\pi r^3-3Vr+6V}{3\pi r^3} \right]=0$$

$$\Rightarrow~10\pi r^3=3V(r-2)$$

I express V in terms of ##~\xi=\frac{h}{r}~##:

$$V=\pi r^3\xi+\frac{2}{3}\pi r^3$$

$$10\pi r^3=3\left[ \pi r^3\xi+\frac{2}{3}\pi r^3 \right](r-2)$$

Blocked end