Explanation Recall: Solutions to linear programming problem: “1. If a linear programming problem has an optimal solution, then the optimal value (maximum or minimum) of an objective function occurs at a corner of the feasible region determined by the constraints. 2. When a feasible region for a linear programming problem is closed and bounded, the objective function has a maximum and a minimum value. 3. If the objective function has the same optimal value at the corners, then it also has that optimal value at any point on the boundary line segment connecting the two corners. 4. When the feasible region is not closed and bounded, the objective function may have a maximum only, a minimum only, or neither”. Given: The objective function is f = 20 x + 30 y . Subject to the constraints are, y ≤ − 1 2 x + 4 y ≤ − x + 6 y ≤ − 1 3 x + 4 x ≥ 0 , y ≥ 0 Calculation: In order to find the maximum value of the objective function, first find the feasible region of the given inequalities. Rewrite the given inequality constraints into the equality constraints by changing the inequality symbol with “equal” symbol. That is, the equations are y = − 1 2 x + 4 , y = − x + 6 and y = − 1 3 x + 4 . Now determine whether the inequality is true or false using test point. Let the test point be ( 0 , 0 ) . Substitute the test point ( 0 , 0 ) in y ≤ − 1 2 x + 4 , 0 ≤ − 1 2 ( 0 ) + 4 0 ≤ 4 It is noticed that the inequality y ≤ − 1 2 x + 4 is true. Thus, shade the half-plane that contains the test point ( 0 , 0 ) . That is, the half-plane below the line y = − 1 2 x + 4 . Substitute the test point ( 0 , 0 ) in y ≤ − x + 6 , 0 ≤ − 0 + 6 0 ≤ 6 It is noticed that the inequality y ≤ − x + 6 is true. Thus, shade the half-plane that contains the test point ( 0 , 0 ) . That is, the half-plane below the line y = − x + 6 . Substitute the test point ( 0 , 0 ) in y ≤ − 1 3 x + 4 , 0 ≤ − 1 3 ( 0 ) + 4 0 ≤ 4 It is noticed that the inequality y ≤ − 1 3 x + 4 is true

MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- for College Algebra in Context

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ISBN: 9780135909621

Author: HARSHBARGER, Ronald, YOCCO, Lisa

Publisher: PEARSON

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Chapter 8.2, Problem 9E

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To find: The maximum value of the objective function f=20x+30y.

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Chapter 8.2, Problem 8E

Chapter 8.2, Problem 10E

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Chapter 8 Solutions

MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- for College Algebra in Context

Ch. 8.1 - In Exercises 16, graph each inequality. 1. y 5x ...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - In Exercises 710, match each solution of a system...Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - In Exercises 1116, the graph of the boundary...Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - For each system of inequalities in Exercises 1722,...Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - In Exercises 2326, graph the solution of the...Ch. 8.1 - Prob. 25E Ch. 8.1 - In Exercises 2326, graph the solution of the...Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Sales Trix Auto Sales sells used cars. To promote...Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Manufacturing A company manufactures two types of...Ch. 8.1 - Prob. 34E Ch. 8.1 - Production A firm produces three different-size...Ch. 8.1 - Advertising Tire Town is developing an advertising...Ch. 8.1 - Nutrition A privately owned lake contains two...Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Manufacturing Safeco Company produces two types of...Ch. 8.1 - Manufacturing A company manufactures commercial...Ch. 8.1 - Constructing a Box An open box with a square base...Ch. 8.2 - In Exercises 1-4, use the given feasible region...Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Find the maximum possible value of f = 400x + 300y...Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Find the maximum possible value of f = 80x + 160y...Ch. 8.2 - Prob. 12E Ch. 8.2 - Find the minimum possible value of g = 40x + 30y...Ch. 8.2 - Prob. 14E Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - Manufacturing A company manufactures two types of...Ch. 8.2 - Manufacturing Evergreen Company produces two types...Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Politics A political candidate wishes to use a...Ch. 8.2 - Prob. 31E Ch. 8.2 - Manufacturing Kitchen Pride manufactures toasters...Ch. 8.3 - Prob. 1E Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - Find the first 5 terms of the sequence defined by...Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Prob. 11E Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Salaries Suppose you are offered a job with a...Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Prob. 31E Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - Prob. 35E Ch. 8.3 - Prob. 36E Ch. 8.3 - Prob. 37E Ch. 8.3 - Prob. 38E Ch. 8.3 - Prob. 39E Ch. 8.3 - Bouncing Ball A ball is dropped from 128 feet. 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The maximum value of the objective function f = 20 x + 30 y .

Solution Summary: The author explains how to find the maximum value of the objective function by rewriting the inequality constraints into the equality constraints.

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Chapter 8 Solutions