Question

A window is in the shape of a rectangle surmounted by a semi circle. If the perimeter of the window is of fixed length L then the maximum area of the window is possible for what value of a

• A. $\frac{2L}{\pi +8}$
• B. $\frac{L^2}{\pi +8}$
• C. $\frac{2L}{\pi +4}$
  
  
• D. $\frac{L^2}{8\pi +4}$

Answer 1

The correct option is C $\frac{2L}{\pi +4}$

Perimeter of window $=a+2b+\frac{\pi a}{2}$
$L=\frac{a}{2}(2+\pi )+2b$

$Area=\frac{1}{2}\frac{\pi a^2}{4}+ab$
$=a(\frac{\pi }{8}a+b)$
$=a(\frac{\pi }{8}a+\frac{L}{2}-\frac{a}{4}(2+\pi \left)\right)$

$\frac{dA}{da}=(\frac{-\pi }{8}a+\frac{L}{2}-\frac{a}{2})+a(\frac{-\pi }{8}-\frac{1}{2})$

$=\frac{-\pi }{4}a-a+\frac{L}{2}$
$a=\frac{2L}{\pi +4}$