Foundations of Math 11
Systems of Linear Inequalities (Chapter 6)
LINEAR PROGRAMMING APPLICATIONS PART 1
RF1.1 Model a problem, using a system of linear inequalities in two variables.
RF1.5 Explain, using examples, the significance of the shaded region in the graphical solution of a system of linear
inequalities.
RF1.6 Solve an optimization problem, using linear programming.
EXAMPLE 1: PROFIT ON MANUFACTURING GOODS
A sporting goods manufacturer makes swimsuits and leotards. The manufacturer wants to find
out how many of each to make to get the greatest profit. The swimsuit needs 2 minutes on the
cutting machine and 1 minute on the stitching machine. The leotard needs 2 minutes on the
cutting machine and 4 minutes on the stitching machine. There is a maximum of 60 minutes
available on the cutting machine. There is a maximum of 60 minutes available on the stitching
machine. The profit for one swimsuit is $10 and the profit for 1 leotard is $15.
How many swimsuits and how many leotards must be made in 1 hour or less for maximum
profit? What is the maximum profit?
STEP 1 (RF1.1): Write the inequalities corresponding to the problem.
The swimsuit needs 2 minutes on the cutting machine and 1 minute on the stitching machine
.
The leotard needs 2 minutes on the cutting machine and 4 minutes on the stitching machine
There is a maximum of 60 minutes available on the cutting machine.
There is a maximum of 60 minutes available on the stitching machine.
STEP 2 (RF1.4): Graph the inequalities.
Foundations of Math 11
Systems of Linear Inequalities (Chapter 6)
STEP 3 (RF1.5): Find the vertices (the corners) for the solution (double-shaded) region.
STEP 4 (RF1.1): Write an equation for the objective function.
The profit for one swimsuit is $10 and the profit for 1 leotard is $15.
STEP 5 (RF1.6): Find the optimal solution for the objective function.
Foundations of Math 11
Systems of Linear Inequalities (Chapter 6)
DO ALL WORK ON A SEPARATE SHEET OF PAPER. SHOW ALL WORKINGS!
Include the graph.
PRACTICE 1: 1
A clothing manufacturer makes shirts and pants. The manufacturer wants to find out how many of
each to make in order to maximize the profits
The shirts need 6 minutes on the cutting machine and 12 minutes on the stitching machine. The
pants needs 5 minutes on the cutting machine and 6 minutes on the stitching machine.
There is a maximum of 370 minutes on the cutting machines available each day
There is a maximum of 540 minutes on the stitching machines available each day
The profit for one shirt is $25 and the profit for one pair of pants is $20.
How many shirts and how many pants must be made daily for maximum profit? What is the
maximum profit?
PRACTICE 2:2
A company makes two types of stereo speakers, Type X and Type Y. Each Speaker X needs 3 hours
for assembly and 2 hours for carpentry. Each Speaker Y needs 4 hours for assembly and 6 hours for
carpentry. There is a maximum of 480 hours available each week for assembly. There is a
maximum of 540 hours available each week for carpentry. The profit on one Speaker X is $50 The
profit on one Speaker Y is $80
How many of each speakers needs to be made each week for a maximum profit? What is this profit?
1
2
6S + 5P ≤ 370 12S + 6P ≤ 540 P = 25S + 20P
3X + 4Y ≤ 480 2X + 6Y ≤ 540 P = 50X + 80Y
$1500 for 20 shirts and 50 pants
$8880 for 72 of speaker X and 66 of speaker Y
Foundations of Math 11
Systems of Linear Inequalities (Chapter 6)
EXAMPLE 2: TAKING VITAMINS
A doctor advises a patient to take vitamin supplements to provide at least 55 mg of
vitamin C but no more than 3450 mg of calcium daily.
RF1.1 Write the inequalities corresponding to the problem.
Each brand A contains 5 mg of Vitamin C and 600 mg of calcium.
Each brand B contains 10 mg of Vitamin C and 150 mg of calcium.
RF1.1 Write an equation for the objective function.
Each brand A pill costs 4 cents Each brand B pill costs 9 cents
RF1.4 Graph the system of inequalities.
RF1.5 List the vertices of the solution region.
RF1.6 How many of each pill should the patient take daily in order to minimize the cost? What is
the minimum cost?
Foundations of Math 11
Systems of Linear Inequalities (Chapter 6)
PRACTICE 1: 3
A doctor advises a patient to take vitamin supplements to provide at least 54 mg of vitamin B but no
more than 800 mg of iron daily
RF1.1 Write the inequalities corresponding to the problem.
Each brand A contains 6 mg of Vitamin B and 120 mg of iron.
Each brand B contains 12 mg of Vitamin B and 100 mg of iron.
RF1.1 Write an equation for the objective function.
Each brand A pill costs 10 cents Each brand B pill costs 6 cents
RF1.4 Graph the system of inequalities.
RF1.5 List the vertices of the solution region.
RF1.6 How many of each pill should the patient take daily in order to minimize the cost? What is
the minimum cost?
3
6A + 12B ≥ 54 120A + 100B ≤ 800
C = 10A + 6B
0 pills & 4.5 pills B cost 27¢
Foundations of Math 11
Systems of Linear Inequalities (Chapter 6)
PRACTICE 2:4
A doctor advises a patient to take vitamin supplements to provide at least 45 mg of vitamin C but no
more than 800 mg of calcium daily.
E8
Write the inequalities corresponding to the problem.
Each brand A contains 10 mg of VItamin C and 150 mg of calcium.
Each brand B contains 5 mg of Vitamin C and 100 mg of calcium,
E10
Write an equation for the objective function.
Each brand A pill costs 12 cents Each brand B pill costs 10 cents
E6
Graph the system of inequalities.
E9
List the vertices of the solution region.
E11
How many of each pill should the patient take daily in order to minimize the cost? What is
the minimum cost?
4
10A + 5B ≥ 45 150A + 100B ≤ 800
C = 12A + 10B
4.5 pills A and 0 pills B cost 54¢