A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 10 ft, express the area A of the window as a function of the width x of the window. A(x)=

Step 1 ：A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 10 ft, we are asked to express the area A of the window as a function of the width x of the window.

Step 2 ：The perimeter of the window is the sum of the lengths of all its sides. For a Norman window, this includes the width of the rectangle (x), the height of the rectangle (h), and the circumference of the semicircle (which is half the circumference of a full circle with diameter x).

Step 3 ：The formula for the perimeter P of the window is therefore: \(P = x + 2h + 0.5 * \pi * x\)

Step 4 ：We know that P = 10 ft, so we can solve this equation for h: \(10 = x + 2h + 0.5 * \pi * x\) => \(h = (10 - x - 0.5 * \pi * x) / 2\)

Step 5 ：The area A of the window is the sum of the area of the rectangle and the area of the semicircle. The area of the rectangle is width times height (x * h), and the area of the semicircle is half the area of a full circle with radius x/2 (0.5 * \pi * (x/2)^2).

Step 6 ：So the formula for the area A of the window is: \(A = x * h + 0.5 * \pi * (x/2)^2\)

Step 7 ：We can substitute the expression for h from above into this formula to express A as a function of x: \(x = x\), \(h = -0.25*\pi*x - x/2 + 5\), \(A = 0.125*\pi*x**2 + x*(-0.25*\pi*x - x/2 + 5)\)

Step 8 ：Final Answer: The area A of the window as a function of the width x of the window is \(\boxed{0.125\pi x^2 + x(-0.25\pi x - \frac{x}{2} + 5)}\)