Explanation Given: To make a large bowl 3 lb of clay and 6floz of glaze is required. To make a small bowl 2 lb of clay and 6floz of glaze is required. To make a pot 4 lb of clay and 2floz of glaze is required. She has to use more than 72 lb of clay and 108floz of glaze. She can make a large bowl in 5 hours a small bowl in 6 hours and a pot in 4 hours. Explanation: Let her make x 1 number of large bowl, x 2 number of small bowl, and x 3 numbers of pot. Then, the provided statements would convert into the following system of linear inequalities. 3 x 1 + 2 x 2 + 4 x 3 ≥ 72 6 x 1 + 6 x 2 + 2 x 3 ≥ 108 also, x 1 , x 2 , x 3 ≥ 0 wherethe objective function to be minimized is z = 5 x 1 + 6 x 2 + 4 x 3 . Use the above information to form this matrix: [ 3 2 4 72 6 6 2 108 5 6 4 0 ] _ . Transpose this to obtain a dual matrix [ 3 6 5 2 6 6 4 2 4 72 108 0 ] _ The dual problem thus is 3 y 1 + 6 y 2 ≤ 5 2 y 1 + 6 y 2 ≤ 6 4 y 1 + 2 y 2 ≤ 4 w = 72 y 1 + 108 y 2 with, y 1 , y 2 ≥ 0 . Where the maximum value of w is the minimum value of z . Introduce slack variables s 1 , s 2 , and s 3 to solve the dual problem 3 y 1 + 6 y 2 + s 1 = 5 2 y 1 + 6 y 2 + s 2 = 6 4 y 1 + 2 y 2 + s 3 = 4 w − 72 y 1 − 108 y 2 = 0 This system of linear equation gives the following tableau: [ y 1 y 2 s 1 s 2 s 3 w 3 6 1 0 0 0 5 2 6 0 1 0 0 6 4 2 0 0 1 0 4 − 72 − 108 0 0 0 1 0 ] _ This tableau gives a negative value for the last row; hence, choose the most negative number in the last row and find pivot in this column. Since, -108 is the most negative number use ratio test to find the pivot element in this column. 5 6 = 0.833 , 6 6 = 1 , 4 2 = 2 Since 6 gives the least ratio use the second element in the first row as the pivot element. Use row operation 1 6 R 1 → R 1 to change the pivot element to 1 and use R 2 − 6 R 1 → R 2 , R 3 − 2 R 1 → R 3 , R 4 + 108 R 1 → R 4 to change the remaining elements in the pivot column to 0

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ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4.3, Problem 26E

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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 amount of three items to make so that she wastes minimum time making them.

Solution Summary: The author explains how to make 16 small, 0 large, and 6 pots to minimize her time to 104 hours. The provided statements would convert into the following system of linear inequalities.

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The amount of three items to make so that she wastes minimum time making them.

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