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Find the dimensions of a window that admits maximum amount of light

by RoRi

Consider a window in the shape of semicircle resting on top of a rectangle (with the length of the diameter of the semicircle equal to the length of the base of the rectangle). The rectangular portion is clear glass, while the semicircular portion is colored glass which admits half as much light per square foot as the clear glass. The perimeter of the window is P. Find the dimensions of the window in terms of the perimeter P that admit the most light.

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Let r denote the length of the radius of the semicircle. Then, the perimeter is given by the equation,

    \[ P = 2r + 2x + \pi r \quad \implies \quad x = \frac{ P - 2r - \pi r}{2}. \]

The total amount of light coming through the window is the area of the rectangular portion times 1, plus the area of the semicircular region times 1/2:

    \[ L = 2rx + \frac{1}{2} \left( \frac{\pi r^2}{2} \right) = 2rx + \frac{\pi r^2}{4}. \]

Plugging in our expression for x we get an expression for the total amount of light in terms of the radius (and the constant P),

    \begin{align*} L &= 2r \left (\frac{P - 2r - \pi r}{2} \right) + \frac{\pi r^2}{4} \\ &= Pr - 2r^2 - \pi r^2 + \frac{\pi r^2}{4} \\ &= Pr - 2r^2 - \frac{3 \pi r^2}{4}. \end{align*}

Differentiating with respect to r we have

    \[ L' = P - 4r - \frac{3 \pi r}{2}. \]

Setting this equal to 0 and solving for r,

    \[ P - 4r - \frac{3 \pi r}{2} = 0 \quad \implies \quad r = \frac{2P}{8 + 3 \pi}. \]

Since the length of the base of the rectangular region is 2r, the base of the rectangle has length

    \[ \text{Base } = 2r = \frac{4P}{8 + 3 \pi}. \]

Then, using the expression above for x, the height of the rectangular region, we have,

    \begin{align*} x &= \frac{P - 2r - \pi r}{2} \\ &= \frac{P - \frac{4P}{8 + 3 \pi} - \frac{2P \pi}{8 + 3 \pi}}{2} \\ &= \frac{P}{2} - \frac{2P}{8 + 3 \pi} - \frac{P \pi}{8 + 3 \pi} \\ &= \frac{(8 + 3 \pi)P - 4P - 2\pi P}{16 + 6 \pi} \\ &= \frac{P(4 + \pi)}{16 + 6 \pi}. \end{align*}

These are the requested dimensions in terms of the perimeter.

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