Explanation Given: The maximum number of animals raised at a time is 16. The maximum number of goats raised at a time is 10. The cost to raise one goat is $25, and the cost to raise one pig is $75. The maximum cost available to raise the animals is $900. The profit gained from one goat is $12, and the cost to raise one pig is $40. Explanation: The constraints as provided by the original problem are x + y ≤ 16 25 x + 75 y ≤ 900 x ≤ 10 x ≥ 0 y ≥ 0 The original objective function is z = 12 x + 40 y The graph of the linear inequalities as provided by the original solution is The optimal solution of the original problem is at the corner point ( 0 , 12 ) . Let the cost to raise one goat decrease from $25to $ ( 25 − b ) . The cost constraint inequality can be written as ( 25 − b ) x + 75 y ≤ 900 The corresponding boundary line is ( 25 − b ) x + 75 y = 900 y = − ( 25 − b ) 75 + 12 The slope of this boundary line is − ( 25 − b ) 75 . The cost change to raise one goat also reflects on the profit from one goat raised. The new objective function is z = ( 12 + b ) x + 40 y y = − ( 12 + b ) 40 x + z 40 The slope of the isoprofit lines for this objective function is − ( 12 + b ) 40

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

11th Edition

ISBN: 9780133886818

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

Publisher: PEARSON

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Chapter 3, Problem 3EA

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The smallest decrease in the cost of one goat raised such that the optimal solution of 0 goats and 12 pigs raised changes.

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Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

Ch. 3.1 - Graph each linear inequality. x + y 2 Ch. 3.1 - Graph each linear inequality. y x + 1 Ch. 3.1 - Graph each linear inequality. x 2 y Ch. 3.1 - Graph each linear inequality. y x 3 Ch. 3.1 - Graph each linear inequality. 4x y 6 Ch. 3.1 - Graph each linear inequality. 4y + x 6 Ch. 3.1 - Graph each linear inequality. 7. 4x + y 8 Ch. 3.1 - Graph each linear inequality. 2x y 2 Ch. 3.1 - Graph each linear inequality. x + 3y 2 Ch. 3.1 - Graph each linear inequality. 2x + 3y 6

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The smallest decrease in the cost of one goat raised such that the optimal solution of 0 goats and 12 pigs raised changes.

Solution Summary: The author determines the optimal solution of 0 goats and 12 pigs raised. The cost of one goat raised must decrease by at least 0.87.

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The smallest decrease in the cost of one goat raised such that the optimal solution of 0 goats and 12 pigs raised changes.

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