(A) Explanation Given: The area of the yard is 20,000 square feet. Calculation: Let x be the width and y be the length of the required fence. One length is the side of the building. So, the perimeter is L = 2 x + y and the area is x y . It is given that the area of the yard is 20,000 square feet. Therefore, x y = 20 , 000 y = 20 , 000 x Substitute y = 20 , 000 x in L = 2 x + y . L = 2 x + 20 , 000 x Therefore, the length of the fence is given by the function L ( x ) = 2 x + 20 , 000 x . STEP 1: Analyze L ( x ) . The function L ( x ) = 2 x + 20 , 000 x is not defined when x = 0 . So, the domain of the function L ( x ) = 2 x + 20 , 000 x is ( 0 , ∞ ) . The x -intercept of L ( x ) = 2 x + 20 , 000 x can be found by substituting L ( x ) = 0 . 2 x + 20 , 000 x = 0 x = 0 Therefore, the x -intercept is x = 0 . The y -intercept of L ( x ) = 2 x + 20 , 000 x can be found by substituting x = 0 . But, at x = 0 the function is not defined. Therefore, there are no y -intercepts. For very large values of x , L ( x ) ≈ 2 x . Therefore, y = 2 x is an oblique asymptote. STEP 2: Analyze L ′ ( x ) . Evaluate L ′ ( x ) as follows. L ′ ( x ) = d d x ( 2 x + 20 , 000 x ) = 2 ( 1 ) + 20 , 000 ( − 1 x 2 ) = 2 − 20 , 000 x 2 Therefore, L ′ ( x ) = 2 − 20 , 000 x 2 . There exists partition numbers where the function L ′ ( x ) = 0 or where L ′ ( x ) is not defined. So, equate the function L ′ ( x ) = 2 − 20 , 000 x 2 to 0. 2 − 20 , 000 x 2 = 0 20 , 000 x 2 = 2 x 2 = 10 , 000 x = 100 Therefore, the partition number is x = 100 . So, the domain of the function can be divided into 4 intervals as ( 1 , 100 ) and ( 100 , ∞ ) . Let x = 1 be a test number in the interval ( 1 , 100 ) . Substitute x = 1 in L ′ ( x ) = 2 − 20 , 000 x 2 L ′ ( 1 ) = 2 − 20 , 000 1 2 = 2 − 20 , 000 = − 19 , 998 < 0 Since L ′ ( x ) < 0 , the function is decreasing in the interval ( 100 , ∞ ) (B) To determine To find: The dimensions of the rectangle that requires the minimum average cost.

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134674148

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 11.4, Problem 84E

(A)

To determine

To sketch: The graph of the function L=2x+20000x.

(B)

To determine

To find: The dimensions of the rectangle that requires the minimum average cost.

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Chapter 11.4, Problem 83E

Chapter 11.4, Problem 85E

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Chapter 11 Solutions

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 11.1 - Matched Problem 1 Repeat Example 1 for f(x) = x2 –...Ch. 11.1 - Matches Problem 2 Find the critical numbers of f,...Ch. 11.1 - Matched Problem 3 Find the critical numbers of f,...Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 7MP Ch. 11.1 - Prob. 8MP Ch. 11.1 - Prob. 9MP Ch. 11.1 - Skills Warm-up Exercises In Problems 1-8, inspect...

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The graph of the function L = 2 x + 20000 x .

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To sketch: The graph of the function L=2x+20000x.

To find: The dimensions of the rectangle that requires the minimum average cost.

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Chapter 11 Solutions