Maximum light through the window

[marutkpadhy]

A window is in the form of rectangle surmounted by a semi circle. The total perimeter of the window is 10m. Find the dimensions of the window to admit maximum light through the whole opening.

 

[MarkFL]

What property of the window is it that determines the amount of light that comes through?

 

[marutkpadhy]

Area

 

[MarkFL]

Area

Correct! :D

So, we need to find the area of the window, which is the sum of a rectangle and a semi-circle, and somehow get the given perimeter involved. Suppose we let $x$ be the width of the rectangle and $y$ be the height. If the semi-circle sits on top of the rectangle, what then is its radius?