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Lesson 3-4

Lesson 3-4. Linear Programming. The inequalities are called the constraints . The intersection of the graphs is called the feasible region . The intersection of the lines are vertices . When the graph of a system of constraints is a polygonal region like the one

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Lesson 3-4

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  1. Lesson 3-4 Linear Programming

  2. The inequalities are called the constraints. The intersection of the graphs is called the feasible region. The intersection of the lines are vertices. When the graph of a system of constraints is a polygonal region like the one graphed at the right, we say that the region is bounded. The maximum or minimum value of a related function always occurs at one of the vertices. The process of finding maximum or minimum values of a function for a region defined by inequalities is called linear programming.

  3. Bounded Region • Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y) = 3x + y. x ≥ 1 y ≥ 0 2x + y ≤ 6 (x, y) 3x + y f(x, y)

  4. Unbounded Region • Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y) = 5x + 4y. 2x + y ≥ 3 3y – x ≤ 9 2x + y ≤ 10 (x, y) 5x + 4y f(x, y)

  5. Example: A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew’s schedule is set up for a maximum of 9 hours per day. Each crew is scheduled for no more than 2 pruning jobs per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the company receives per day for one of its crews. p ≤ 2 p ≥ 0 m ≥ 0 1m + 3p ≤ 9

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