Hello, Super Mallow!

2) A window is in the form of a rectangle surmounted by a semicircle.

The rectangle is of clear glass, whereas the semicircle is of tinted glass

that transmits only half as much light per unit area as the clear glass.

The total perimeter is fixed.

Find the proportions of the window that will admit the most light. Code:

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r r
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2r

The radius of the semicircle is $\displaystyle r.$

The width of the rectangle is $\displaystyle 2r$, its height is $\displaystyle h.$

The perimeter of the semicircle is: .$\displaystyle \pi r$

The perimeter of the rectangle is: .$\displaystyle 2r + 2h$

The total perimeter is a constant, $\displaystyle P.$

Hence, we have: .$\displaystyle \pi r + 2r + 2h \:=\:P\quad\Rightarrow\quad h \:=\:\frac{P - \pi r - 2r}{2}$ . **[1]**

The area of the rectangle is: .$\displaystyle 2rh$

It admits a certain amount of light per squre unit.

Its light admission is: .$\displaystyle L_1\:=\:2rh$

The area of the semicircle is: .$\displaystyle \frac{1}{2}\pi r^2$

It admits half as much light per square unit.

Its light admission is: .$\displaystyle L_2 \:=\:\frac{1}{2}\left(\frac{1}{2}\pi r^2\right) \:=\:\frac{1}{4}\pi r^2$

The total light admitted is: .$\displaystyle L \:=\:\frac{1}{4}\pi r^2 + 2rh$ . **[2]**

Substitute [1] into [2]: .$\displaystyle L \;=\;\frac{1}{4}\pi r^2 + 2r\left(\frac{P - \pi r - 2r}{2}\right)$

And we have: .$\displaystyle L \;=\;-\left(\frac{3\pi}{4} + 2\right)r^2 + Pr$

. . and *that* is the function we must maximize . . .