Question

A Norman window is a window wherein one side of the rectangle is replaced by a semicircle. A Norman window is to be made out of a plastic beam, whose available length is  . Find out the height of the lower rectangle of the Norman window such that when the window is installed into the wall, maximum air circulation is possible. 

 

The Norman window would look like this.

Let the radius of the semicircular shape in red be  and the height of the rectangular base be .

We can equate the perimeter of the window as . 

Substitutine the value of the perimeter, we get .

Isolating the height, we get .    

  

Then, we can express the area of the window as.

Substituting the expression for the height, we get.

Simplifying the equation, we have .

To get the maximum, we need to get the derivative of this function. That is, .

Equating the derivative to zero, we get .

Solvng for , we get .