Explanation Given: The owner wants to enclose a rectangular area of 12,100 square feet as in Problem 35. Procedure used: Strategy for solving Optimization Problem: 1. Introduce variables, look for relationship among the variable, and construct a mathematical model of the form, Maximize (or minimize) f ( x ) on the interval I . 2. Use the procedures developed in Section 4.5 to find the absolute maximum (or minimum) of f ( x ) on the interval I and the numbers x where this occurs. i. Check to make certain that f is continuous over [ a , b ] . ii. Find the critical numbers in the interval ( a , b ) . iii. Evaluate f at the endpoints a and b and at the critical numbers found in step (ii). iv. The absolute maximum of f on [ a , b ] is the largest value found in step (iii). v. The absolute minimum of f on [ a , b ] is the largest value found in step (iii). 3. Use this solution to the mathematical model to answer all the questions asked in in this problem. Calculation: The dimensions of the area that requires the least fencing and the total fencing required is calculated as follows. Draw the figure of the rectangular box that introduce variables, and look for the relationship among the variables is shown below in Figure 1. From Figure 1, it is observed that the length front side of the fencing is 100 + x ft and the length of the other side is y ft. Here do not know the dimensions of the fencing area, the lengths of the rectangular fencing are represented by the variables x + 100 and y . Now look for the relationships among the variables. The area rectangular fencing is, A = lenght × w i d h t = ( 100 + x ) y Given that the owner wants to enclose a rectangular area of 12,100 square feet. Hence the area of the rectangular fencing can be written as ( 100 + x ) y = 12 , 100 . The total perimeter of the fencing is, P = x + y + y + ( 100 + x ) = 2 x + 2 y + 100 Let`s assume that for enclosing the largest area will use all the fencing material available for fencing. So, the mathematical model that requires least fencing is, Minimize: P = 2 x + 2 y + 100 subject to ( 100 + x ) y = 12 , 100 . Observed that the perimeter function involving two variable, before using the calculus to find the minimum perimeter P , here must need to express P as a function of a single variable. Use the equation ( 100 + x ) y = 12 , 100 to eliminate one of the variables in the perimeter function. ( 100 + x ) y = 12 , 100 y = 12 , 100 100 + x Substitute, y = 12 , 100 100 + x in P , P = 2 x + 2 ( 12 , 100 100 + x ) + 100 = 2 x + 24 , 200 100 + x + 100 Now consider the permissible values of x . Because x is the one of the dimension of the rectangle, x must satisfy x ≥ 0 . And the equation ( 100 + x ) y = 12 , 100 is also the dimension of the rectangle, y must satisfy y ≥ 0 . Hence, the above mathematical model that minimize total fencing can be written as, Minimize : P ( x ) = 2 x + 24 , 200 100 + x + 100 for x ≥ 0 . Next, obtain the critical points if any as follows. Differentiate P ( x ) with respect to x , P ′ ( x ) = d d x ( 2 x + 24 , 200 100 + x + 100 ) = 2 − 24 , 200 ( 100 + x ) 2 + 0 = 2 − 24 , 200 ( 100 + x ) 2 For critical points P ′ ( x ) = 0

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Chapter 11.6, Problem 36E

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To find: The dimensions of the area that requires the least fencing. How many feet of fencing are required?

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Chapter 11.6, Problem 35E

Chapter 11.6, Problem 37E

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Chapter 11 Solutions

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

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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