(a) Explanation Method used: “Simplex method: 1. Determine the objective function. 2. Write all the necessary constraints. 3. Convert each constraint into an equation by adding a slack variable in each. 4. Set up the initial simplex tableau. 5. Locate the most negative indicator. If there are two such indicators, choose the one farther to the left. 6. Form the necessary quotients to find the pivot. Disregard any quotients with 0 or a negative number in the denominator. The smallest nonnegative quotient gives the location of the pivot. If all quotients must be disregarded, no maximum solution exists. If two quotients are both equal and smallest, choose the pivot in the row nearest the top of the matrix. 7. Use row operations to change all other numbers in the pivot column to zero by adding a suitable multiple of the pivot row to a positive multiple of each row. 8. If the indicators are all positive or 0, this is the final tableau. If not, go back to step 5 and repeat the process until a tableau with no negative indicators is obtained. 9. Read the solution from the final tableau.” Calculation: Joe daily exercise schedule and the calories burned in each exercise per hour are listed in the below table. Calisthenics Swimming Playing on Drums Maximum available hours 1 1 1 10 0 1 4 Number of calories burn 388 518 345 Let x 1 be the number of spend on calisthenics, x 2 be the number of hours spend on swimming, and x 3 be the number of hours spend to playing on drums. The first constraint is, x 1 + x 2 + x 3 ≤ 10 . Observe that Joe wants that the time spends on calisthenics and plays the drums to be at least twice as the long as he swims. That is, the second constraint of this model is − x 1 + 2 x 2 − x 3 ≤ 0 . Form the information given above the mathematical form of the linear programming problem is as follows. Maximize : z = 388 x 1 + 518 x 2 + 345 x 3 subject t o : x 1 + x 2 + x 3 ≤ 10 − x 1 + 2 x 2 − x 3 ≤ 0 x 3 ≤ 4 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0. Use the simplex algorithm and obtain the solution of the linear programming problem as follows. Add the slack variable s 1 in the first constraint, s 2 in the second constraint, s 3 in the third constraint and rewrite the original problem in equality constraints linear programming problem as follows. Maximize : z = 388 x 1 + 518 x 2 + 345 x 3 subject t o : x 1 + x 2 + x 3 + s 1 = 10 − x 1 + 2 x 2 − x 3 + s 2 = 0 x 3 + s 3 = 4 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 , s 1 ≥ 0 , s 2 ≥ 0 , s 3 ≥ 0. (1) The system (1) can be written in the system of linear equation as, x 1 + x 2 + x 3 + s 1 = 10 − x 1 + 2 x 2 − x 3 + s 2 = 0 x 3 + s 3 = 4 − 388 x 1 − 518 x 2 − 345 x 3 + z = 0 Construct the initial simplex tableau form the above linear system as follows. [ x 1 x 2 x 3 s 1 s 2 s 3 z 1 1 1 1 0 0 0 10 − 1 2 − 1 0 1 0 0 0 0 0 1 0 0 1 0 4 − 388 − 518 − 345 0 0 0 1 0 ] Look the 3 × 3 identity matrix. So the basic variables are s 1 , s 2 and s 3 . Thus, the basic solution is x 1 = 0 , x 2 = 0 , x 3 = 0 , s 1 = 10 , s 2 = 0 , s 3 = 4 and z = 0 (b) To determine To describe: The conclusion about Joe’s selection of activities.

Finite Mathematics and Calculus with Applications Books a la Carte Plus MyLab Math Package (10th Edition)

10th Edition

ISBN: 9780133935592

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

Publisher: PEARSON

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Chapter 4.2, Problem 36E

(a)

To determine

The number of hours per week should Joe spend on each exercise to maximize the number of calories he burns and then calculate the maximum number of calories he burn.

(b)

To determine

To describe: The conclusion about Joe’s selection of activities.

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Chapter 4.2, Problem 35E

Chapter 4.2, Problem 37E

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Finite Mathematics and Calculus with Applications Books a la Carte Plus MyLab Math Package (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 number of hours per week should Joe spend on each exercise to maximize the number of calories he burns and then calculate the maximum number of calories he burn.

Solution Summary: The author explains how Joe's daily exercise schedule and the calories burned per hour are listed in the table.

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The number of hours per week should Joe spend on each exercise to maximize the number of calories he burns and then calculate the maximum number of calories he burn.

To describe: The conclusion about Joe’s selection of activities.

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