To determine To find: The dimensions of a rectangle with an area of 200 square feet such that it has minimum perimeter. Explanation 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 a rectangle has an area of 200 square feet. Consider the length to be x and the width to be y . The equation for the area of rectangle becomes x y = 200 and the perimeter is 2 ( x + y ) . In order to find the least perimeter, the problem that should be computed becomes, Minimize P = 2 ( x + y ) subject to x y = 200 . Use the area equation to eliminate any one variable in P and express P as a single variable expression as follows. x y = 200 y = 200 x Then, P becomes, P = 2 ( x + y ) = 2 ( x + 200 x ) The dimensions of the rectangle are always greater than and equal to zero that is x ≥ 0 and y ≥ 0 . Thus, the problem becomes, Maximize P = 2 ( x + 200 x ) subject to x ≥ 0 . Set P ′ ( x ) = 0 and find the critical numbers of P as follows

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

13th Edition

ISBN: 9780321869838

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

Publisher: PEARSON

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Chapter 4.6, Problem 15E

To determine

To find: The dimensions of a rectangle with an area of 200 square feet such that it has minimum perimeter.

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Chapter 4.6, Problem 14E

Chapter 4.6, Problem 16E

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Chapter 4 Solutions

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

Ch. 4.1 - Matched Problem 1 Repeat Example 1 for f(x) = x2 ...Ch. 4.1 - Matches Problem 2 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 3 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 4 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 5 Find the critical numbers of f,...Ch. 4.1 - Prob. 6MP Ch. 4.1 - Matched Problem 7 Given f(x) = x3 9x2 + 24x 10,...Ch. 4.1 - Matched Problem 8 The graph in Figure 13...Ch. 4.1 - Matched Problem 9 The graph of the total revenue...Ch. 4.1 - Skills Warm-up Exercises In Problems 1-8, inspect...

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