A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 32 ft, express the area A of the window as a function of the width x of the window.

To solve this problem, let's break it down step by step.

First, let's understand the shape of the window. A Norman window consists of a rectangle surmounted by a semicircle. So, we can find the area of the window by finding the area of the rectangle and adding it to the area of the semicircle.

Let's start with the rectangle. The width of the rectangle is given as x. Since it is a rectangle, the height of the rectangle will be half of x to accommodate the semicircle on top. So, the height of the rectangle is (1/2)x.

The formula to find the area of a rectangle is given by A_rect = length × width. Therefore, the area of the rectangle is A_rect = x × (1/2)x = (1/2)x^2.

Now, let's move on to the semicircle. The semicircle sits on top of the rectangle, and its diameter is equal to the width of the rectangle (x). The formula to find the area of a semicircle is A_semi = (π × r^2) / 2, where r is the radius of the semicircle.

Since the diameter is x, the radius of the semicircle is half of the diameter, which is x/2. Substituting this into the area formula, we get A_semi = (π × (x/2)^2) / 2 = (πx^2) / 8.

Now, to find the total area of the Norman window (A), we add the area of the rectangle and the area of the semicircle:
A = A_rect + A_semi = (1/2)x^2 + (πx^2) / 8.

So, the area A of the window can be expressed as a function of the width x as follows:
A(x) = (1/2)x^2 + (πx^2) / 8.

Please note that I have assumed that the width given (x) is for the rectangle, and the height of the rectangle is half of the width.

nevermind i figured this one out too.