Explanation Given: Total of available land with the farmer is 100 acres. All the costs per acre are 100 less. He has maximum of $20,000 to spend. The taxes payable per acre for potatoes, corn and cabbage are $5, $10 and $15 respectively. The farmer does not want to pay more than $900 in taxes. He makes a profit of $120 per acre of potatoes, $40 per acre of corn and $60 per acre of cabbage. Calculation: Assume the number of acres allotted to each of the three crops be represented by x 1 , x 2 and x 3 , respectively. Then obtain the first constraint as follows. x 1 + x 2 + x 3 ≤ 100 where x 1 , x 2 and x 3 are all nonnegative. Obtain the second constraint as 300 x 1 + 60 x 2 + 180 x 3 ≤ 20 , 000 since all the costs per acre are 100 less and maximum spending on the land is $20,000, which can be further reduced to 15 x 1 + 3 x 2 + 9 x 3 ≤ 1 , 000 . Observe that the taxes are $5, $10 and $15 per acre of potatoes, corn and cabbage, respectively whereas total taxes should not exceed $900 as farmer does not want to pay more than $900 in taxes. Thus, obtain the third constraint with the help of above information about taxes as follows. 5 x 1 + 10 x 2 + 15 x 3 ≤ 900 that is x 1 + 2 x 2 + 3 x 3 ≤ 180 . Obtain the objective function which represents the profit that the farmer wants to maximize as follows. z = 120 x 1 + 40 x 2 + 60 x 3 The linear programming problem can be stated as follows. Maximize z = 120 x 1 + 40 x 2 + 60 x 3 subject to x 1 + x 2 + x 3 ≤ 100 15 x 1 + 3 x 2 + 9 x 3 ≤ 1 , 000 x 1 + 2 x 2 + 3 x 3 ≤ 180 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

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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1. A farmer has a field of 70 acres on which he plants potatoes and corn. The seed for potatoes costs $20/acre and the seed for corn costs $60/acre. The farmer has set aside $3000 to spend on seed. The profit per acre of potatoes is $150 and the profit for corn is $50 per acre. How many acres of potatoes and corn should the farmer plant to maximize his profits?

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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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To construct: The initial simplex tableau.

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