(a) Explanation Given: Joe has at most 10 hours a week to exercise. He wants to do calisthenics and play the drums for at least twice as much time as he swims. His neighbors would allow him to drum for only 4 hours in the week. He burns 388 calories per hour doing calisthenics, 518 calories per hour while swimming, and 345 calories per hour while playing the drums. Explanation: Let the number of hours spent doing calisthenics, swimming, and playing drums in the week be denoted by x , y, and z, respectively. Let the number of calories he burns during the week be denoted by p : p = 388 x + 518 y + 345 z The restrictions he has can be represented by the following set of inequalities: x + y + z ≤ 10 z ≤ 4 x + z ≥ 2 y Also, the number of hours can’t be negative. This gives rise to three nonnegative constraints. x ≥ 0 , y ≥ 0 , z ≥ 0 This optimization problem can be written as Maximize p = 388 x + 518 y + 345 z Subject to x + y + z ≤ 10 z ≤ 4 − x + 2 y − z ≤ 0 with x ≥ 0 , y ≥ 0 , z ≥ 0 This can be solved using the Simplex method as follows: Add three slack variables to the three inequalities and obtain x + y + z + s 1 = 10 z + s 2 = 4 − x + 2 y − z + s 3 = 0 The objective function can be rewritten as − 388 x − 518 y − 345 z + p = 0 The initial simplex tableau is [ x y z s 1 s 2 s 3 p 1 1 1 1 0 0 0 10 0 0 1 0 1 0 0 4 − 1 2 − 1 0 0 1 0 0 − 388 − 518 − 345 0 0 0 1 0 ] _ The column corresponding to y has the highest negative value in the row and thus would enter the basis. The pivot element can be found by calculating the ratio of the elements in the extreme right column and the column corresponding to the entering variable. 10 1 = 10 0 2 = 0 The minimum ratio is 0, so s 3 leaves the basis. 1 2 R 3 → R 3 , − R 3 + R 1 → R 1 , 518 R 3 + R 4 → R 4 gives [ x y z s 1 s 2 s 3 p 1 (b) To determine A conclusion about Joe’s selection

11th Edition

ISBN: 9780321979438

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 Joe should spend on each exercise to maximize the number of calories burnt and the final maximum number of calories burnt.

(b)

To determine

A conclusion about Joe’s selection

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Chapter 4.2, Problem 35E

Chapter 4.2, Problem 37E

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Chapter 4 Solutions

Finite Mathematics (11th Edition)

Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 2E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 4E Ch. 4.1 - For Exercises 5-8. (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - Introduce slack variables as necessary, then...Ch. 4.1 - Introduce slack variables as necessary, then write...

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The number of hours Joe should spend on each exercise to maximize the number of calories burnt and the final maximum number of calories burnt.

Solution Summary: The author explains how Joe burns 388 calories per hour doing calisthenics and playing the drums, while his neighbors would allow him to drum for only 4 hours.

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The number of hours Joe should spend on each exercise to maximize the number of calories burnt and the final maximum number of calories burnt.

A conclusion about Joe’s selection

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Chapter 4 Solutions