To determine To find: The dimensions and the area of the largest garden that can be surrounded with $320 worth of fencing. Explanation Given: The homeowner owned $320 to spend for the fence building. The cost of wire fencing to cover the three sides of the rectangular garden is $2 per linear foot. The cost of wood fencing that covers only one side is $8 per linear foot. Procedure used: 1. Select the variables and find the some relations among the variables. 2. Form a mathematical model as shown below. Maximize ( or minimize ) f ( x ) on the interval I 3. Obtain the critical numbers of f ( x ) . 4. Find the absolute maximum (or minimum) of f ( x ) on the interval I and the numbers x as the absolute maximum must occur either at the critical points or at the endpoints. 5. Find the values of the left variables with the help of the solution obtained in step 4. Calculation: It is given that the garden is in rectangular shape, thus consider the length to be y and the width to be x . Then, the area becomes A = x y . Express the cost in terms of x and y as follows. C = 2 y + 2 x + 2 y + 8 x = 4 y + 10 x The total money that can be spent on fence building is $320, thus the cost 4 y + 10 x should not exceed but must be equal to the total amount $320. In order to find the largest garden that can be covered with fence, the problem that should be computed becomes, Maximize A = x y subject to 10 x + 4 y = 320 . Use the cost equation to eliminate any one variable in A and express A as a single variable expression as follows. 10 x + 4 y = 320 4 y = 320 − 10 x y = 80 − 5 2 x Then, A becomes, A = x ( 80 − 5 2 x ) = 80 x − 5 2 x 2 The dimensions of the rectangle are always greater than and equal to zero that is x ≥ 0 and y ≥ 0 . Substitute the value of y in the inequality y ≥ 0 and find the boundary of x as follows. 80 − 5 2 x ≥ 0 5 2 x ≤ 80 x ≤ 80 × 2 5 x ≤ 32 Thus, the problem becomes, Maximize A = 80 x − 5 2 x 2 subject to 0 ≤ x ≤ 32

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

13th Edition

ISBN: 9780321945518

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

Publisher: PEARSON

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Chapter 12.6, Problem 1MP

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To find: The dimensions and the area of the largest garden that can be surrounded with $320 worth of fencing.

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Chapter 12.5, Problem 80E

Chapter 12.6, Problem 2MP

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Chapter 12 Solutions

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

Ch. 12.1 - Matched Problem 1 Repeat Example 1 for f(x) = x2 –...Ch. 12.1 - Matches Problem 2 Find the critical numbers of f,...Ch. 12.1 - Matched Problem 3 Find the critical numbers of f,...Ch. 12.1 - Prob. 4MP Ch. 12.1 - Prob. 5MP Ch. 12.1 - Prob. 6MP Ch. 12.1 - Prob. 7MP Ch. 12.1 - Prob. 8MP Ch. 12.1 - Prob. 9MP Ch. 12.1 - Skills Warm-up Exercises In Problems 1-8, inspect...

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