(a) Explanation Given: A farmer has 100 acres of land for harvesting. It costs $400 to produce an acre of potatoes, $160 to produce an acre of corn, and $280 for an acre of cabbage. He can invest no more than $20,000 on harvesting. He makes a profit $120 per acre of potatoes, $40 per acre of corn, and $60 per acre of cabbage. Explanation: Let the farmer grow x 1 acres of potatoes, x 2 acres of corn, and x 3 acres of cabbage. Then the provided statements would convert into the system of linear inequalities: x 1 + x 2 + x 3 ≤ 100 400 x 1 + 160 x 2 + 280 x 3 ≤ 20 , 000 Also, x 1 , x 2 , x 3 ≥ 0 where the objective function to be maximized is z = 120 x 1 + 40 x 2 + 60 x 3 . Use the above information to form this matrix: [ 1 1 1 100 400 160 280 20 , 000 120 40 60 0 ] _ . Transpose this to obtain a dual matrix [ 1 400 120 1 160 40 1 280 60 100 20 , 000 0 (b) To determine The profit earned by the farmer if the land is cut to 90 acres, but capital increases to $21,000. (c) To determine The maximum profit earned by the farmer and the planting strategy that will produce this profit.

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Question

Chapter 4.3, Problem 21E

(a)

To determine

The dual problem.

(b)

To determine

The profit earned by the farmer if the land is cut to 90 acres, but capital increases to $21,000.

(c)

To determine

The maximum profit earned by the farmer and the planting strategy that will produce this profit.

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Chapter 4.3, Problem 20E

Chapter 4.3, Problem 22E

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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 dual problem.

Solution Summary: The author explains that the dual problem is the minimum of the objective function.

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The dual problem.

The profit earned by the farmer if the land is cut to 90 acres, but capital increases to $21,000.

The maximum profit earned by the farmer and the planting strategy that will produce this profit.

Want to see the full answer?

Chapter 4 Solutions