Express the area of the window as a function?

Question

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 24 feet, express the area A of the window as a function of the width x (across the base) of the window.

Answer 1

Let y = height of the rectangular portion of the window and let 

P = perimeter of the window.  

 

Then  P = x + 2y  + length of semicircular portion.  The semicircular portion has radius x/2.

 

So, P = x + 2y + (1/2)(circumference)

 

      P = x + 2y  + (1/2)(2)π(x/2)

 

      P = x + 2y +πx/2 = 24

 

2y = 24 - x - πx/2

 

y = 12 - x/2 - πx/4 = 12 - (2 + π)x/4

 

A = xy + (1/2)π(x/2)²

 

    = x(12 - (2 + π)x/4) + πx²/8

 

    = 12x -(2 + π)x²/4 + πx²/8

 

     = 12x - (4 + 2π)x²/8 + πx²/8

 

    = 12x - 4x²/8 - πx²/8

 

    = 12x -(4 + π)x²/8