Question

A window consists of a rectangular piece of clear glass with a semicircular piece of
colored glass on top; the colored glass transmits only 1/2 as much light per unit area as the the clear
glass. If the distance from top to bottom (across both the rectangle and the semicircle) is 2 meters and
the window may be no more than 1.5 meters wide, find the dimensions of the rectangular portion of the
window that lets through the most light.

Answer 1

Let 𝑟 be a radius of a semicircle, $H$ be a distance from top to bottom. Then the horizontal side of the rectangle is equal 2𝑟, vertical side of the rectangle is equal 𝐻 − 𝑟. Effective area of the window is

$$S=S_{rect}+kS_{semi}=2r(H-r)+k\cdot{1\over 2}\pi r^2$$

Given

$$H=2\ m, k={1\over 2}, 0<2r\leq 1.5\ m$$

Substitute

$$S=S(r)=2r(2-r)+{1\over 2}\cdot{1\over 2}\pi r^2$$

$$S=S(r)=4r-2r^2+{1\over 4}\pi r^2, 0<r\leq0.75$$

Find the critical number(s)

$$S'=(4r-2r^2+{1\over 4}\pi r^2)'=4-4r+ {1\over2}\pi r$$

$$S'=0=>4-4r+ {1\over2}\pi r=0$$ $$r={8\over 8-\pi}$$

First derivative test

If $0<r<\dfrac{8}{8-\pi}, S'>0, S$ increases.

If $r>\dfrac{8}{8-\pi}, S'<0, S$ decreases.

The function $S(r)$ has a local maximum at $r=\dfrac{8}{8-\pi}.$

Since the function $S$ has the only extremum, then the function $S$ has the absolute maximum $r=\dfrac{8}{8-\pi}.$

$$\dfrac{8}{8-\pi}>1.5$$

Hence we have to take $r=0.75 m$

The width of the rectangular portion of the window  is $2r=1.5m.$

The height of the rectangular portion of the window  is $2-0.75m=1.25m$

The dimensions of the rectangular portion of the window that lets through the most light are

$$1.5m\times1.25m$$