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Q. 71

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Precalculus Mathematics for Calculus

Found in: Page 479

Book edition 7th Edition

Author(s) James Stewart, Lothar Redlin, Saleem Watson

Pages 948 pages

ISBN 9781337067508

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Short Answer

Area of a Sector of a Circle Three circles with radii 1, 2, and 3 ft are externally tangent to one another, as shown in the figure. Find the area of the sector of the circle of radius 1 that is cut off by the line segments joining the center of that circle to the centers of the other two circles.

The area of sector is $0 .785 f t^{2}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The radius of circle $r = 1 ft$

Step 2. Write the concept.

Before applying the formula for area of sector we need to find the central angle theta which the sector subtends.

As, shown in the figure

$A B = 5 f t$ , $B C = 3 f t$ and $A B = 4 f t$

Applying Pythagoras theorem $A C^{2} = A B^{2} + B C^{2}$

Also, the area of sector is given by $A = \frac{1}{2} r^{2} θ$

Step 3. Determining the values.

Putting the values of Ab, BC and AB in Pythagoras Theorem we find the central angle of the sector as $90^{\circ}$ .

Therefore the area of sector is:

$A = \frac{1}{2} r^{2} 90 A = 0.5 \times 1 \times \frac{90}{180} π A = 0.785 f t^{2}$

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