Calculating the Area of a Norman Window

A Norman window is an architectural feature that has a distinctive shape, consisting of a rectangle surmounted by a semicircle. If the perimeter of the window is known, it is possible to express the area of the window as a function of the width of the rectangular portion.

Let's consider a Norman window with the following parameters:

• The perimeter of the window is 37 feet.
• The width of the rectangular portion is represented by the variable x.

To find the area of the window, we need to express the height and width of the window in terms of the width x.

The height of the rectangular portion is given by: h = (37 - π * x) / (2 + π)

The radius of the semicircular portion is half the width, or x/2.

Therefore, the area of the window is the sum of the area of the rectangle and the area of the semicircle: A = x * h + (π * (x/2)^2) / 2

Substituting the expression for h, we get: A = x * (37 - π * x) / (2 + π) + (π * (x/2)^2) / 2

Simplifying the expression, we arrive at the final formula for the area of the Norman window: A = (37x - (π/2) * x^2) / (2 + π)

This equation allows us to calculate the area of the Norman window as a function of the width x, given that the perimeter of the window is 37 feet.