Warm - Up

Learning Targets

I can solve systems of inequalities by graphing.

I can determine the coordinates of the vertices of a region formed by the graph of a system of inequalities.

Quick Review of Inequalities

< MEANS LESS THAN > GREATER THAN

< LESS THAN OR EQUAL TO > GREATER THAN OR EQUAL TO

< OR > DOTTED LINE

< OR > SOLID LINE

Remember when to shade above or below the line?

Solve the system of inequalities by graphing.

solution of Regions 1 and 2

solution of Regions 2 and 3

Solve each system of inequalities by graphing.

a.

Answer:

Solve the system of inequalities by graphing.

The graphs do not overlap, so the solutions have no points in common.

Graph both inequalities.

Answer: The solution set is .

Solve the system of inequalities by graphing.

Answer:

Graph each inequality. The intersection of the graphs forms a triangle.

Answer:

The vertices of the triangle

are at (0, 1), (4, 0), and (1, 3).

Find the coordinates of the vertices of the figure formed by and

Find the coordinates of the vertices of the figure formed by and

Answer: (–1, 1), (0, 3),

and (5, –2)

Big Idea:

Explain what it means to find the vertices of a system of inequalities.

Assignment:

Practice

LESSON 3.4 ALinear Programming

Warm - Up Graph and find the vertices of the

following system of inequalities.y > -4 y < 2x + 22 x + y < 6

Homework Solutions

Learning Targets

I can determine the coordinates of the vertices of a region formed by the graph of a system of inequalities.

I can find the maximum and minimum values of a function over a region.

VocabularyConstraints:Another term for “inequalities.”

Feasible Region:The intersection of the graphs.

Bounded:The polygonal region formed by the graphing of a system of constraints.

Step 1 Find the vertices of theregion. Graph theinequalities.

The polygon formed is atriangle with vertices at

(–2, 4), (5, –3), and (5,4).

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region.

Step 2 Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function.

Answer: The vertices of the feasible region are (–2, 4), (5, –3), and

(5, 4).

The maximum value is 21 at (5, –3).

The minimum value is –14 at (–2, 4).

(x, y) 3x – 2y f(x, y)

(-2, 4)

(5, -3)

(5, 4)

Plug each of your vertices into the equation

3(-2) – 2(4) -143(5) – 2(-3) 213(5) – 2(4) 7

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region.

Answer: vertices: (1, 5), (4, 5) , (4, 2)

(x, y) 4x – 3y f(x, y)

(1 , 5)

(4 , 5)

(4 , 2)

Step 2 Use a table to find the maximum and minimum values of

f(x, y). Substitute the coordinates of the vertices into the function.

maximum: f(4, 2) = 10,

minimum: f(1, 5) = –11

Plug each of your vertices into the equation

4(1) – 3(5) -114(4) – 3(5) 1

4(4) – 3(2) 10

Vertices: (1, 0), (1, 4), (3, 0)

(x, y) 3x + y f(x, y)

(1, 0)

(1, 4)

(3, 0)

Max: Min:

Answer: vertices: (0, –3), (6, 0); maximum: f(6, 0) = 6;

no minimum

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region.

Before your assignment…

Remember the steps involved:

1. Find the vertices of the region. Graph the inequalities.

2. Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function.

Assignment

Practice Worksheet

LESSON 3.4 BLinear Programming

Homework Solutions

Learning Target:

I can solve real-world problems using linear programming.

Steps to solving linear programming problems.

Landscaping A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day.

On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the company receives per day from one of its crews.

Step 1 Define the variables.

m = the number of mowing jobs

p = the number of pruning jobs

Step 2 Write a system of inequalities.

Since the number of jobs cannot be negative, m and p must be nonnegative numbers.

m 0, p 0Mowing jobs take 1 hour. Pruning jobs take 3 hours. There are 9 hours to do the jobs.

There are no more than 2 pruning jobs a day.

p 2

Step 3 Graph the system of inequalities.

Step 4 Find the coordinates of the vertices of thefeasible region.

From the graph, the vertices are at (0, 2),

(3, 2), (9, 0), and (0, 0).

Step 5 Write the function to be maximized.

The function that describes the income is We want to find

the maximum value for this function.

Step 6 Substitute the coordinates of the vertices intothe function.

Step 7 Select the greatest amount.

(m, p) 40m + 120p f(m, p)

(0, 2) 40(0) + 120(2) 240

(3, 2) 40(3) + 120(2) 360

(9, 0) 40(9) + 120(0) 360

(0, 0) 40(0) + 120(0) 0

Answer: The maximum values are 360 at (3, 2) and 360at (9, 0). This means that the company

receives the most money with 3 mows and 2prunings or 9 mows and 0 prunings.

Step 1: Define the variables:v = # of office visitss = # of surgeries

Step 2: Write a system of inequalitiesAppointments cannot be negative, so

v > 0 and s > 0

An office visit is 20 minutes, and surgery is 40 minutes. There are 7 hours available for appointments.

20v + 40s < 420 (7 hours = 420 minutes)The veterinarian cannot do more than 6 surgeries.

s < 6

Continued… Step 3: Graph the system of inequalities Step 4: Find the vertices. Step 5: Write a function to be

maximized or minimized.

Continued…

Step 6: Substitute the coordinates of the vertices into the function.

Step 7: Select the greatest amount.Answer: When Dolores schedules 6 surgeries and 9 office visits, she earns the highest amount per day at $1245.00

(s,v) 125s + 55v f(s,v)

(0,0)

(6,0)

(6,9)

(0,21)

125(0) + 55(0) 0

125(6) + 55(0) 750

125(6) + 55(9) 1245

125(0) + 55(21) 1155

Maximum

Landscaping A landscaping company has crews who rake leaves and mulch. The company schedules 2 hours for mulching jobs and 4 hours for raking jobs. Each crew is scheduled for no more than2 raking jobs per day. Each crew’sschedule is set up for a maximum of 8 hours per day. On the average, the charge forraking a lawn is $50 and the charge for mulching is$30. Find a combination of raking leaves andmulching that will maximize the income the company receives per day from one of its crews.

List out the steps…

1. Define the variables

2. Write a system of inequalities.

3. Graph the system of inequalities.

4. Find the coordinates of the vertices of the feasible region.

5. Write a function to be maximized or minimized.

6. Substitute the coordinates of the vertices into the function.

7. Select the greatest or least result. Answer the problem.

Answer:

0 raking jobs and 4 mulching jobs

Assignment

3.4B Linear Programming Worksheet

Linear Programming Practice # 1 1. A test consists of short answer and essay problems. Short answer problems are

worth six points, and essay problems are worth 10 points.

(c) Write the max points equation: f(x, y) =

It takes 2 minutes to answer the short answer problems and 4 minutes to answer the essay problems. You have 40 minutes for the test and you may choose no more than 12 problems to answer.

Assuming you answer all of the problems you attempted correctly, how many of each type of problem should you answer to get the highest score?

4042 yx

12 yx

0x0y You should answer ____short answer and

__ essay problems.

2. Jimmy is baking cookies for a bake sale. He is making chocolate chip cookies and oatmeal raisin cookies. He gets $0.25 for each chocolate chip cookie and $0.30 for each oatmeal raisin cookie.

(c) Write the profit equation: f(x, y) =

He cannot make more than 500 cookies of each kind, and he cannot make more than 800 cookies total.

How many of each kind of cookie should he make to get the most money?

500x

500y

800 yx

0x0y Jimmy should make ch. chip and

___oatmeal-raisin cookies.

3) Jayne assembles two types of cameras: the Excellent and the Premier. Her cost of assembling these cameras is $21 for the Excellent and $42 for the Premier. The total funds available for her to use are $1,260.

It takes 5 hours to assemble an Excellent and 2 hours to assemble a Premier. She can work for no more than 100 hours total.

If the profit on the two cameras is $8 for the Excellent and $10 for the Premier,

determine how many cameras of each kind should Jayne assemble to maximize profit?

12604221 yx

10025 yx

0x0y You should answer ____excellent and

__ premier.

4) A studio sells photographs and prints. It cost $20 to purchase each photograph and it takes 2 hours to frame it. It costs $25 to purchase each print and it takes 5 hours to frame it. The store has at most $400 to spend and at most 60 hours to frame.

It makes $30 profit on each photograph and $50 profit on each print.

Find the number of each that the studio should purchase to maximize profits.

4002520 yx

6052 yx

0x0y You should answer ____photographs and

__ prints.

5. A tent-maker makes a Standard model and an Expedition model. Each standard tent requires one hour of labor from the cutting department and 3 hours of labor from the assembly department. Each Expedition tent requires 2 hours of labor from the cutting department and 4 hours of labor from the assembly department. The maximum labor hours available per week in the cutting department is 32 and in the assembly department 84.

If the company makes $50 profit on each Standard tent and $80 profit on each Expedition tent,

how many tents of each type should be made to maximize weekly profit?

3221 yx

8443 yx

0x0y You should answer ____Standard and

__ Expedition.

21

16

28 32

yxyxf 8050),(

?

6. A carpenter makes bookcases in two sizes, large and small. It takes 6 hours to make a large bookcase and 2 hours to make a small one. The carpenter can spend only 24 hours per week making bookcases and must make at least two of each size per week.

The profit on a large bookcase is $50 and the profit on a small bookcase is $20.

How many of each size must be made per week in order to provide maximum profit?

2426 yx

2x

2y

You should answer ____large and __ small.

12

42

2

yxyxf 2050),(