1. Linear Programming
Skill 45

2. Objective
HSA-REI.12(2): Solve

3. Linear programming is method of finding a maximum or minimum value of a function that satisfies a given set of conditions called constraints.
A constraint is one of the inequalities in a linear programming problem.
The solution to the set of constraints can be graphed as a feasible region.

4. Example: Linear Programming
Lauren wants to paint no more than 70 plates for the art show. It costs her at least $50 plus $2 per item to produce red plates and $3 per item to produce gold plates. She wants to spend no more than $215. Write and graph a system of inequalities that can be used to determine the number of each plate that Lauren can make.
x represent the number of red plates
y represent the number of gold plates.

5. Example: Linear Programming
y  0
The system of inequalities is
x + y ≤ 70
50 + 2x + 3y ≤ 215

6. (x, y) 10x+15y P($) (0,0) 10(0)+15(0) (0,55) 10(0)+15(55) 850 (45,25)
Suppose Lauren sells the red plates for $10 each and the gold plates for $15 each. How many of each type of plate should she paint to maximize her profit? What is her maximum profit.
(x, y)
10x+15y
P($)
(0,0)
10(0)+15(0)
The maximum value is at the vertex (0, 55).
(0,55)
10(0)+15(55)
850
(45,25)
10(45)+15(25)
825
(70,0)
10(70)+15(0)
700
Lauren should paint 55 gold plates and 0 red plates to maximize the amount of profit, which is $850.

7. Example; Linear Programming
Leyla is selling hot dogs and spicy sausages at the fair. She has only 40 buns, so she can sell no more than a total of 40 hot dogs and spicy sausages. Each hot dog sells for $2, and each sausage sells for $2.50. Leyla needs at least $90 in sales to meet her goal. Write and graph a system of inequalities that models this situation.
d represent the number of hot dogs
s represent the number of sausages

8. Do It Yourself: Example; Application
The system of inequalities is
d + s ≤ 40
2d + 2.5s ≥ 90

9. (d, b) 2d + 2.5b P($) (0,36) 2(0)+2.5(0) (0,40) 2(0)+2.5(40) 100
How many of hotdogs (d) and sausages (s) must she sell to maximize her profit? What is her maximum profit.
(d, b)
2d + 2.5b
P($)
(0,36)
2(0)+2.5(0)
The maximum value is at the vertex (0, 40).
(0,40)
2(0)+2.5(40)
100
(20,20)
2(20)+2.5(20) 90
Leyla should sell 0 hotdogs and 40 sausages to maximize the amount of profit, which is $100.

10. #45: Linear Programming Questions Summarize Notes Homework
Google Classroom
Quiz