Chapter 1.1, Problem 62E

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8th Edition

James Stewart

ISBN: 9781305270343

Textbook Problem

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area *A* of the window as a function of the width *x* of the window.

To determine

**To find:** The function of the area of the window in terms of the width of the window.

Explanation

**Given:**

The window is in the shape of a rectangle surmounted by a semicircle. The Perimeter of the window is 30 ft and the width of the window is denoted as *x* ft.

**Formula used:**

Area of the semi-circle or radius *r*,

Area of the rectangle of length *l* and breadth *b* is

**Calculation:**

Let the height of the window be *h* ft.

The diameter of the semi-circle part is *x* ft as the diameter is same as the width of the window.

Thus, is the radius of the semi-circle part is

Area of the rectangular part of width *x* and height *h* is

Area of the semi-circle part of radius

The area of the window is the sum of the area of the rectangular part and the area of the semi-circle part.

Thus, the perimeter of the window is

Perimeter of the rectangular part of width *x* and height *h* is

Perimeter of the semi-circle part of radius

The perimeter of the window is the sum of the perimeter of the rectangle part and the perimeter of the semi-circle part.

Thus, the perimeter of the window is

That is,

Since the perimeter of the window is 30 ft, the height of the window is computed as follows:

Substitute the value of *h* in

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