Explanation Assume that the amount of time limited to each of the three categories of activities be represented by y 1 , y 2 and y 3 , respectively. Construct a table in order to fetch the data from the given information as follows. Walking Cycling Swimming Total Calories/min 3.5 4 8 1500 Per week 30 min Then obtain the first constraint as 3.5 y 1 + 4 y 2 + 8 y 3 ≥ 1500 since walking burns 3.5 calories per minute, cycling burns 4 calories per minute and swimming burns 8 calories per minute whereas exercises burn atleast 1500 extra calories per week. Obtain the second constraint as y 1 + y 2 ≥ 3 y 3 since she would spent the total time in walking and cycling has to be atleast 3 times as long as she spends in swimming. Obtain the third constraint as y 1 ≥ 30 since she spent atleast 30 minutes per week. Obtain the objective function which represents the minimized time is, w = y 1 + y 2 + y 3 . The linear programming problem can be stated as follows. Minimize w = y 1 + y 2 + y 3 subject to 3.5 y 1 + 4 y 2 + 8 y 3 ≥ 1500 y 1 + y 2 − 3 y 3 ≥ 0 y 1 ≥ 30 with y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 Obtain the dual of the above problem as follows. Maximize z = 1500 x 1 + 30 x 3 subject to 3 .5 x 1 + x 2 + x 3 ≤ 1 4 x 1 + x 2 ≤ 1 8 x 1 − 3 x 2 ≤ 1 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Add slack variables to the above constraints and obtain the linear programming problem as follows. Maximize z = 1500 x 1 + 30 x 3 subject to 3 .5 x 1 + x 2 + x 3 + s 1 = 1 4 x 1 + x 2 + s 2 = 1 8 x 1 − 3 x 2 + s 3 = 1 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 , s 1 ≥ 0 , s 2 ≥ 0 , s 3 ≥ 0 Rewrite the objective function as z − 1500 x 1 − 30 x 3 = 0

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 4.3, Problem 27E

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To find: The amount of time that should be spent on each activity to minimize the total exercise time per week and the minimum exercise time per week.

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Chapter 4.3, Problem 26E

Chapter 4.3, Problem 28E

Chapter 4 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 4.1 - Prob. 1YT Ch. 4.1 - Prob. 2YT Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - For Exercises 58, (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E

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The amount of time that should be spent on each activity to minimize the total exercise time per week and the minimum exercise time per week.

Solution Summary: The author explains how to find the amount of time that should be spent on each activity to minimize the total exercise time per week. The linear programming problem can be stated as follows.

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To find: The amount of time that should be spent on each activity to minimize the total exercise time per week and the minimum exercise time per week.

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Chapter 4 Solutions