A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 40 ft, express the area A of the window as a function of the width x of the window. A(x)

A(x) = 20x - ⅛x²(4 + π)

Step-by-step explanation:

Let us denote the height of the rectangular part of the Norman window as "y"

Also, let P be overall perimeter of the Norman window.

Since the rectangular part is surmounted by a semi-circle, we can say the total perimeter is;

P = x + 2y + circumference of semi-circle.

Since the part x of the rectangle also serves as the diameter of the semi-circle, it means the radius of the semi-circle is; r = x/2

Now, circumference of a semi-circle is given by the formula;

C = 2πr

Thus;

C = 2π(x/2)

Let's plug in 2π(x/2) for circumference of semi-circle in the total perimeter equation to get;

P = x + 2y + (½•2π(x/2))

This gives;

P = x + 2y + π(x/2)

We are told that perimeter of the window is 40 ft.

Thus;

40 = x + 2y + π(x/2)

Rearranging gives;

2y = 40 - x - (πx/2)

Divide through by 2 to get;

y = 20 - ½x - ¼πx

y = 20 - (2 + π)x/4

Now, we want to express the area A of the window as a function of the width x of the window.

Area of the rectangular portion is;

A_rect = xy

Area of semi-circle part is;

A_semi = ½π(x/2)²

Thus, total area is;

A_t = xy + ½π(x/2)²

From earlier, we saw that y = 20 - (2 + π)x/4. Thus;

A(x) = x(20 - (2 + π)x/4) + ⅛πx²

A(x) = 20x - ¼(2 + π)x² + ⅛πx²

A(x) = 20x - ⅛(4 + 2π)x² + ⅛πx²

A(x) = 20x - ⅛(4x²) - ⅛(2π)x² + ⅛πx²

A(x) = 20x - ⅛(4x²) - ⅛x²π

A(x) = 20x - ⅛x²(4 + π)

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