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Warm-up

Warm-up. Sketch the region bounded by the system of inequalities: 1) 2). 10.8 Linear Programming. Objectives: Write an objective function modeling a quantity to be maximized or minimized. Use inequalities to model limitations in a situation Use linear programming to solve the problem.

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Warm-up

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  1. Warm-up Sketch the region bounded by the system of inequalities: 1) 2)

  2. 10.8 Linear Programming Objectives: Write an objective function modeling a quantity to be maximized or minimized. Use inequalities to model limitations in a situation Use linear programming to solve the problem.

  3. 1. Objective Function A linear objective function: Goal: Determine the values x and y that will maximize (or minimize) the value z, subject to certain constraints.

  4. 2. Writing an Objective Function Example: A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model is given: The maximumtotal weekly hours available are: 200 hrs for assembly and 108 hours for painting. The profitsper unit are $25 for model A and $15 for model B. How many of each type should be produced to maximize profit?

  5. 3. Writing Constraints The constraints are the equations that will determine the feasible region for a solution. write the constraints for the previous problem. The maximumtotal weekly hours available are: 200 hrs for assembly and 108 hours for painting.

  6. 4. Solving a Linear Programming Problem. The Linear Programming problem is: maximize : subject to: 1: Graph the feasible region 2: Determine the corner points (vertices) 3: Find the value of the objective function at each corner (make a table) 4: The largest z value is the solution.

  7. More practice p. 821 #10 maximize : subject to:

  8. Warm-up maximize : subject to:

  9. 5. Analyzing a Linear Programming problem Example: maximize : subject to: Theorem: If a Linear Programming problem has a solution, it is located at a corner point. Why? Analyze: Graph the objective function for different values of z. Why is the max at a corner point? Let z = 10: Let z = 20: Let z = 22: What happens when z > 22?

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