A window is in the form of a rectangle surmounted by semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window for maximum air flow through the window.
Let width of window $= \ell$
Then diameter of semicircle $= \ell$
Let height of window $= b$
Perimeter of the window $= P = \ell + 2b + \dfrac{\pi \ell}{2}$
Given: Perimeter of window $= 10 \; m$
$\implies$ $\ell + 2b + \dfrac{\pi \ell}{2} = 10$
i.e. $\ell \left(2+\pi\right) + 4b = 20$
i.e. $b = 5 - \dfrac{\ell \left(2+\pi\right)}{4}$ $\;\; \cdots$ (1)
Area of the window $=A= b \; \ell + \dfrac{\pi}{2} \times \dfrac{\ell^2}{4}$ $\;\; \cdots$ (2)
Substituting the value of b from equation (1) in equation (2) gives
$A = \ell \left[5- \dfrac{\ell \left(2+\pi\right)}{4}\right] + \dfrac{\pi \ell^2}{8}$
i.e. $A= 5 \ell - \dfrac{\pi}{4} \ell^2 -\dfrac{1}{2}\ell^2 + \dfrac{\pi}{8} \ell^2$
i.e. $A = 5 \ell - \dfrac{\pi}{8}\ell^2 - \dfrac{1}{2}\ell^2$
i.e. $A = 5 \ell - \dfrac{\ell^2}{2}\left(\dfrac{\pi}{4}+1\right)$
For maximum air flow, $\dfrac{dA}{d\ell} = 0$
i.e. $5-\ell\left(\dfrac{\pi+4}{4}\right) = 0$
$\implies$ $\ell = \dfrac{20}{\pi+4}$
Substituting the value of $\ell$ in equation (1) gives
$b = 5 - \left(\dfrac{\pi + 2}{4}\right) \left(\dfrac{20}{\pi + 4}\right)$
i.e. $b = 5 - 5 \left(\dfrac{\pi + 2}{\pi + 4}\right)$
i.e. $b = \dfrac{5\pi + 20 - 5\pi -10}{\pi + 4}$
i.e. $b = \dfrac{10}{\pi + 4}$
$\therefore$ Length of the window $\ell = \dfrac{20}{\pi + 4} \; m$; breadth of the window $=b=\dfrac{10}{\pi + 4} \; m$