Warm-Up1) Solve the system.

5 minutes

x y 7

2x y 8

2) Graph the solution.y 2x 3

y x 5

x 0

y 1

3.5.1 Linear Programming3.5.1 Linear Programming3.5.1 Linear Programming3.5.1 Linear ProgrammingObjectives: •Write and graph a set of constraints for a linear-programming problem

Example 1

*the inequalities contained in the problem are called the constraints

Let x = the # of afghans Let y = the # of sweaters

x y 100

10 x 60

y 20

A small company produces knitted afghans and sweaters and sells them through a chain of specialty stores. The company is to supply the stores with a total of no more than 100 afghans and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60 afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each afghan and a profit of $12 on each sweater.

*the solution to the set of constraints is called the feasible region

Example 1

*the function to be maximized or minimized is called the objective function

Write an objective function for the company’s total profit, P, from the sales of afghans and sweaters.

A small company produces knitted afghans and sweaters and sells them through a chain of specialty stores. The company is to supply the stores with a total of no more than 100 afghans and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60 afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each afghan and a profit of $12 on each sweater.

P =

10x

+ 12y

Homework

p.191 #11-17 odds

Warm-Up1) Graph the feasible region for the set of constraints below.

6 minutes

x 2y 8

2x y 10

x 0

y 0

2) Identify the vertices of the feasible region.

3.5.2 Linear Programming3.5.2 Linear Programming3.5.2 Linear Programming3.5.2 Linear ProgrammingObjectives: •Use linear programming to find the maximum or minimum value of an objective function

Example 1

a) Maximize the objective function.

A small company produces knitted afghans and sweaters and sells them through a chain of specialty stores. The company is to supply the stores with a total of no more than 100 afghans and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60 afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each afghan and a profit of $12 on each sweater.

P = 10x + 12yVertex Objective

function(10,20) P = 10(10) + 12(20) =

340

(60,20) P = 10(60) + 12(20) =

840

(60,40) P = 10(60) + 12(40) =

1080 (10,90

)P = 10(10) + 12(90) =

1180

Corner-Point Principle

In linear programming, the maximum and minimum values of the objective function each occur at one of the vertices of the feasible region.

Example 1

b) Graph the objective function, indicating the maximum profit and the feasible region.

A small company produces knitted afghans and sweaters and sells them through a chain of specialty stores. The company is to supply the stores with a total of no more than 100 afghans and sweaters per day. The stores guarantee that they will sell at least 10 and no more than 60 afghans per day and at least 20 sweaters per day. The company makes a profit of $10 on each afghan and a profit of $12 on each sweater.

P = 10x + 12y1180 = 10x +

12y 12y = –10x + 1180 5 1180

y x6 12

Example 2Find the maximum and minimum values, if they exist, of the objective function T = 3x + 2y given the set of constraints below.

x y 10

x 2y 12

4x y 13

y x 10 1

y x 62

y 4x 13

Homework

p.192 #19,21,23,27,29