Explanation Exercise 59: The area inside the corral at which the goat can graze is ( a 2 − 2 ) cos − 1 ( a 2 ) + π − a 2 4 − a 2 . Given: Goat is grazing in a grassy region which lies along the boundary of the concrete lab of unit radius. The goat is tied to the circumference with rope of length 0 ≤ a ≤ 2 . Calculation: Assume that the goat is tied at the origin. Note that every time the goat grazes it describes a part of circle with radius a . Therefore, the area in which the cow can graze is bounded with circle r = a . Now assuming that the goat is tied at the origin the corral describes a circle of radius 1 on the left side of origin along the negative side of the x-axis. Therefore, the center of the concrete slab has polar coordinates ( 1 , π ) with respect to origin. Now polar equation of the circle centered at ( 1 , π ) and with radius 1 is r = − 2 cos θ . Therefore, the corral is represented in the polar coordinates as r = − 2 cos θ . In the polar coordinates system the above situation can be denoted as shown below in Figure 1. In Figure 1, the shaded region is the required region. Observe that shaded region is nothing but the difference between the complete area of the circle r = a and the area of the region from Exercise 59. The area of the circle r = a is π a 2 . Therefore, required area is calculated as follows. A = a 2 π − ( ( a 2 − 2 ) cos − 1 ( a 2 ) + π − a 2 4 − a 2 ) = a 2 π − ( a 2 − 2 ) cos − 1 ( a 2 ) − π + a 2 4 − a 2 Hence, area of the grassy region in which goat can graze is a 2 π − ( a 2 − 2 ) cos − 1 ( a 2 ) − π + a 2 4 − a 2 . Let us verify the result for a = 0 and a = 2 . Note that, at a = 0 the goat is at the origin so the area is 0

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 12.3, Problem 86E

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To find: The area of the region in which the goat can graze and verify the answer for cases with a=0 and a=2.

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Chapter 12.3, Problem 85E

Chapter 12.3, Problem 87E

Chapter 12 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Prob. 3QC Ch. 12.1 - Prob. 4QC Ch. 12.1 - Prob. 5QC Ch. 12.1 - Prob. 6QC Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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The area of the region in which the goat can graze and verify the answer for cases with a = 0 and a = 2 .

Solution Summary: The author explains how the goat can graze in a grassy region along the boundary of the concrete lab of unit radius.

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To find: The area of the region in which the goat can graze and verify the answer for cases with a=0 and a=2.

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Chapter 12 Solutions