LINEAR PROGRAMMING A-CED.3 REPRESENT CONSTRAINTS BY EQUATIONS OR INEQUALITIES, AND BY SYSTEMS OF EQUATIONS AND/OR INEQUALITIES, AND INTERPRET SOLUTIONS AS VIABLE OR NONVIABLE OPTIONS IN A MODELING CONTEXT. FOR EXAMPLE, REPRESENT INEQUALITIES DESCRIBING NUTRITIONAL AND COST CONSTRAINTS ON COMBINATIONS OF DIFFERENT FOODS.

LINEAR PROGRAMMING Linear programming is a process of finding a maximum or minimum of a function by using coordinates of the polygon formed by the graph of the constraints Used most often to find maximum profit

VOCABULARY Constraints: limits formed by a system of inequalities Feasible region: the graph of the system of inequalities (typically a polygonal region) Objective function: models the quantity that is being minimized or maximized Bounded region: a feasible region that is closed Unbounded region: a feasible region that is open

STEPS FOR LINEAR PROGRAMMING Write the equation of the objective function Write system of linear inequalities to represent the constraints Graph the system of linear inequalities Find the vertices of the feasible region Substitute the vertices of the feasible region into the objective function to find the minimum or maximum

FINDING VERTICES OF THE FEASIBLE REGION Graph the system of inequalities on the same graph and write the coordinates of the vertices of the shaded region

FINDING MIN/MAX Substitute the coordinates of the vertices of the feasible region into the objective function EX: If profit is represented by f(x, y) = 4x + 3y, find the min. and max. values determined by the constraints:

WORD PROBLEM 1 The area of a parking lot is 600 square meters. A car requires 6 square meters. A bus requires 30 square meters. The attendant can handle only 60 vehicles. If a car is charged $2.50 and a bus $7.50, how many of each should be accepted to maximize income?

WORD PROBLEM 2 Toys-2-Go makes toys at Warehouse A and Warehouse B. Warehouse A needs to make a minimum of 1000 toy dump trucks and fire engines. Warehouse B needs to make a minimum of 800 toy dump trucks and fire engines. Warehouse A can make 10 toy dump trucks and 5 toy fire engines per hour. Warehouse B can produce 5 toy dump trucks and 15 toy fire engines per hour. It costs $30 per hour to produce toy dump trucks and $35 per hour to produce toy fire engines. How many hours should be spent on each toy in order to minimize cost? What is the minimum cost?

WORD PROBLEM 3 A lunch stand makes $.75 profit on each chef’s salad and $1.20 profit on each Caesar salad. On a typical weekday, it sells between 40 and 60 chef’s salads and between 35 and 50 Caesar salads. The total number sold has never exceeded 100 salads. How many of each type should be prepared in order t maximize profit?