2-3: Linear 2-3: Linear ProgrammingProgramming2-3: Linear 2-3: Linear

ProgrammingProgramming

Unit 2: Linear FunctionsUnit 2: Linear FunctionsEnglish CasbarroEnglish Casbarro

Definition

Linear programming is a way to figure out maxima and minima, by compressing a three-dimensional figure into two-dimensions

You will be given information about 2 independent variables (called constraints), and you will find the point that maximizes or minimizes a 3rd, dependent variablethis is called the objective function.

Steps in the Process

Graph linear inequalities according to the restrictions listed in the problem

Find out the intersection of all the areas it will be a polygon (triangle, quadrilateral)this is called the feasible region

The vertices are the points that you will need to check for maximum and minimum values.

The function where you will check these points looks something like this: P(x,y) = 2x + 3y

Turn in the following problems

1. 2.

3.

4.

5.

From the graph, you can tell that the optimum point occurs where the boundary lines (or constraints)

t= 12, and t + a = 20 intersect.

If you can’t read the intersection point from the graph, you can solve the system of equations.

The cost can be found by substituting the point (12,8) into the cost equation

d = 15(12) + 10(8)d = 180 + 80

d = 260

Remember, that the cost is in thousands, though, so that the minimum cost for the board is $260,000. This is considered the optimum (can be either

A maximum or a minimum, depending on the situation in the problem.

Turn in the following problems1.

2. Tickets to a car race cost $25 for the upper deck and $45 for the lower deck. The track may admit no more than 160,000 by order of the fire marshal.

a. If the lower deck can seat no more than 60,000 fans and the upper deck can seat no more than 120,000 fans, how many of each type of ticket should be sold to maximize profit?

b. How do the system and solution change if race officials expect to make an additional $60 per person in the upper deck and $30 per

person in the lower deck from the sale of food and merchandise?