A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle What is the area of the window with a perimeter of 45 feet?

To find the area of a Norman window with a perimeter of 45 feet, let the width of the rectangle be ( w ) and the height be ( h ). The semicircle has a radius of ( r = \frac{w}{2} ), so the perimeter can be expressed as ( P = w + 2h + \pi r = w + 2h + \frac{\pi w}{2} ). Setting this equal to 45 feet and solving for ( h ) gives ( h = \frac{45 - w - \frac{\pi w}{2}}{2} ). The area ( A ) is then given by ( A = wh + \frac{\pi r^2}{2} = wh + \frac{\pi (w/2)^2}{2} ), which can be maximized by substituting ( h ) back into the area formula.