f(x,y) = 2x+3y make a table to evaluate the objective function: … 13... · escalades. it takes...

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Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities

Lesson 13: Linear Programming

Example 1: Bounded Region

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

inequalities.

x≤ 3

x-3y ≥ -12 or y ≤ 1/3x +4 or x int: -12, y int: 4

4x+3y ≥12 or y ≥ -4/3x +4 or x int: 3, y int: 4

Vertices:

**The vertices represent the maximum or

minimum values of a related function.

Objective Function:

F(x,y) = 2x+3y

Make a table to evaluate the Objective Function:

(x,y) 2x+3y F(x,y)

What are the maximum and minimum values for the objective function?



Example 2: Unbounded Region

Step 1: Graph the following system of inequalties.

y ≥ 2x – 8

2x – y ≥ -4

Y ≤ 1/3x +2

Name the vertices:

Find the maximum and minimum

Values of the function:

f(x,y) = 4x+3y

Make a table to evaluate the Objective Function:

What are the maximum and minimum values for the objective function?



Example 3

Betty is baking cakes and pies for

her church bake sale. A pie will

take a half hour to make and a cake

will take one hour to make. She

cannot bake for more than 20 hours

this week and she does not want to

make more than 30 pies. She plans

to charge $10 per pie and $12 per

cake. Find a combination of cakes

and pies that will maximize her

profits for the sale.

Step 1: Define your variables and write a system for the constraints.

Step 2: Write the objective function.

Step 3: Graph.

Step 4: Find the coordinates of the feasible region. (Vertices)

Step 5. Create a table with the objective function to determine the maximum values.

Step 6: Answer the question.



Lesson 7: Linear Programming

Example 4

An assembler of battery operated child-

sized cars manufactures a mustang

version and an escalade version. The

mustang has a profit margin of $100 and

the escalade has a profit margin of

$160. The company can assemble no

more than 60 mustangs and 40

escalades. It takes 150 hours of labor to

assemble the mustang and 200 hours to

assemble the escalade. The company

has up to 12,000 hours per month for

assembly of both vehicles. Find the

number of each model that the company

can manufacture in ordered to maximize

their monthly profit.



Step 3: Graph.

Step 4: Find the coordinates of the feasible region.





Lesson 13: Linear Programming – Practice Problems

Part 1: Skill Practice – Graph each system of inequalities. Identify the vertices of the feasible

region. Find the maximum and minimum values of the given function for this region.

1. y ≥ 2

x ≤ 7

y ≤ 2x+1

f(x,y) = 2x+y

Step 1: Graph the system.

Step 2: Identify the vertices:

Step 3: Create a table to find the maximum

and minimum values for the objective

function.

Step 4: Identify the maximum and minimum

values.



2. y ≤ x+6

y +2x ≥ 6

3 ≤ x ≤ 6

f(x,y) = x -2y





function.


values.



3. y ≥ 4

1 ≤ x ≤ 10

x- 2y ≥ -4

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





function.


values.



Part 2: Apply your knowledge to real world problems.

.



Step 3: Graph.




4. A school based theatre program is putting

on a production. According to fire safety

procedures, no more than 100 student tickets

can be sold and no more than 200 general

admission tickets can be sold. It costs $0.50

per ticket to advertise to students and $1 per

ticket to advertise to the general public. They

have an advertising budget of $200. Find the

maximum profit the program can make if they

sell student tickets for $3 and general

admission tickets for $5.





Step 3: Graph.




5. A receptionist for a pediatric doctor

schedules appointments. She allots 15

minutes for a sick visit and 40 minutes for a

well visit. The pediatrician cannot have more

than 5 well visits per day. The office has 7

hours available for appointments. A sick visit

costs $50 and a well visit costs $85. Find a

combination of sick and well visits that will

maximize the income of the pediatrician each

day.





Step 3: Graph.




5. A charter airline company sells first class

and coach class seats. To charter a plane, at

least 5 first class seats and at least 10 coach

class seats must be sold. The plane does

not hold more than 30 passengers. The

company makes a $60 profit for each first

class seat and a $40 profit for each coach

class seat sold. In order to maximize profits,

how many coach and first class seats should

they sell?



Step 1: Define your variables and write a system for the constraints. (3 points)

Step 2: Write the objective function. (1 point)

Step 3: Graph. (3 points)


(Vertices) (2 points)

Step 5. Create a table with the objective function to determine the maximum values. (3 points)

Step 6: Answer the question. (1 point)

A theatre company is selling tickets to their most recent production. At

least 200 general tickets must be sold and at least 50 balcony tickets

can be sold. There are only 70 balcony seats in the theatre. The

theatre holds a total of 350 seats. The company makes a profit of $6

for each general ticket and $8 for each balcony seat sold. How many

of each type of ticket should they sell in order to maximize their profit?



Lesson 13: Linear Programming – Practice Problems – Answer Key

Part 1: Skill Practice – Graph each system of inequalities. Identify the vertices of the feasible

region. Find the maximum and minimum values of the given function for this region.

1. y ≥ 2

x ≤ 7

y ≤ 2x+1

f(x,y) = 2x+y



The vertices are: (1/2,2) (7,2) (7,15)

Blue/red lines: Red and green lines: Blue and green lines Y = 2x+1(blue) and y = 2 (red) We know x = 7 from constraints We know x = 7 2 = 2x+1 Substitute 2 for y We know y = 2 from problem Blue line: y = 2x+1 2-1 = 2x+1-1 Subtract 1 (7,2) y = 2(7)+1 Substitute 7 1 = 2x y = 15 ½ = 2x/2 Divide by 2 (7,15) ½ = x We know that y = 2 (1/2, 2)



function.

(x,y) 2x+y f(x,y)

(1/2,2) 2(1/2)+2 3

(7,2) 2(7)+2 16

(7,15) 2(7)+15 29


values.

The minimum value is 3 at (1/2,2)

The maximum value is 29 at (7,15)



2. y ≤ x+6

y +2x ≥ 6

3 ≤ x ≤ 6 x≤ 6 and x≥ 3

f(x,y) = x -2y



Vertex 1: intersection of yellow and red.

Yellow: x = 3 red: y = x+6

Intersection: y = 3+6 y = 9 (3,9)

Vertex 2: intersection of blue and red.

Blue: x = 6 red: y = x+6

Intersection: y = 6+6 y = 12 (6,12)

Vertex 3: intersection of green and yellow

(3,0)

Vertex 4: Intersection of green and blue

Blue: x = 6 green: y +2x = 6 or y=-2x+6

Intersection: y = -2(6)+6 y = -6 (6,-6)



function.

(x,y) x-2y f(x,y)

(3,9) 3 – 2(9) -15

(6,12) 6 – 2(12) -18

(3,0) 3-2(0) 3

(6,-6) 6-2(-6) 18


values.

The minimum value is -18 at (6,12).

The maximum value is 18 at (6,-6)

1

2

3

4



3. y ≥ 4

1 ≤ x ≤ 10 x ≤ 10 and x ≥1

x- 2y ≥ -4 or y ≤ 1/2x+2 (yellow)

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



Vertex 1: yellow and green line.

Green: x = 10 yellow: y = 1/2x+2

Y = ½(10)+2 y = 7 (10,7)

Vertex 2: yellow and red line.

Red: y = 4 yellow: y = 1/2x +2

4 = 1/2x +2 x = 4 (4,4)

Vertex 3: red and green lines.

Green: x = 10 red: y = 4 (10,4)



function.

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

(10,7) -10+3(7) 11

(4,4) -4+3(4) 8

(10,4) -10 +3(4) 2


values.

The minimum value is 2 at (10,4)

The maximum value is 11 at (10,7)

3

2

1



Part 2: Apply your knowledge to real world problems.

.


Let x = the number of student tickets Let y = the number of general admission tickets

x ≥ 0 and x≤ 100

y ≥ 0 and y ≤ 200

.50x +1y ≤ 200 (advertising budget)


P = 3x+5y

f(x,y) = 3x+5y

Step 3: Graph.

For .50x +y ≤ 200 x and y intercepts: x = 400 y = 200


Vertex 1: (0,0)

Vertex 2: (0, 200)

Vertex 3: blue and green line. Green: x = 100 blue: y = -.5x+200

y = -.5(100) +200 y = 150 (100, 150)

Vertex 4: (100,0)


4. A school based theatre program is putting

on a production. According to fire safety

procedures, no more than 100 student tickets

can be sold and no more than 200 general

admission tickets can be sold. It costs $0.50

per ticket to advertise to students and $1 per

ticket to advertise to the general public. They

have an advertising budget of $200. Find the

maximum profit the program can make if they

sell student tickets for $3 and general

admission tickets for $5.

2

1

3

4

(x,y) 3x+5y f(x,y) (0,0) 3(0)+5(0) 0 (0,200) 3(0)+5(200) 1000 (100,150) 3(100)+5(150) 1050 (100,0) 3(100)+5(0) 300

The maximum profit that

can be made is $1050.

They can make this profit

by selling 100 student

tickets and 150 general

admission tickets.




Let x = the number of sick visits Let y = the number of well visits

15x+40y ≤ 420 (number of visits per day) (420 minutes) = 7 hours (60 *7)

x≥ 0

y ≥ 0 and y ≤ 5


P = 50x+85y

f(x,y) = 50x+85y

Step 3: Graph.

15x +40y ≤ 420 find the x and y intercepts: x = 28 y = 10.5


Vertex 1: (0,0)

Vertex 2: (0,5)

Vertex 3: intersection of blue and red line.

Blue: 15x+40y = 420 red: y = 5

15x +40(5) = 420 15x+200=420

15x +200-200=420-200

15x = 220

X = 14.67 (14.67, 5)

Vertex 4: (28,0)


5. A receptionist for a pediatric doctor

schedules appointments. She allots 15

minutes for a sick visit and 40 minutes for a

well visit. The pediatrician cannot have more

than 5 well visits per day. The office has 7

hours available for appointments. A sick visit

costs $50 and a well visit costs $85. Find a

combination of sick and well visits that will

maximize the income of the pediatrician each

day. 2

1

3

4

(x,y) 50x+85y f(x,y) (0,0) 50(0)+85(0) 0 (0,5) 50(0)+85(5) 425 (14.67,5) 50(14)+85(5) 1125 (28,0) 50(28)+85(0) 1400


can be made is $1400. This

profit could be made by

scheduling 28 sick visits

and no well visits.




Let x = the number of first class seats Let y = the number of coach class seats

x ≥ 5 (the problem tells us that they must have more than 5 first class seats sold)

y ≥ 10 (the problem tells us that they must have more than 10 coach class seats sold)

x+y ≤ 30 (plane only holds 30 seats)


40x+50y = P

f(x,y) = 40x+50y

Step 3: Graph.

x+y ≤ 30 find the x and y intercepts: x = 30 y = 30


Vertex 1: (5,10)

Vertex 2: intersection of blue and red

Blue x= 5 red x +y = 30

5 +y = 30 y = 25 (5,25)

Vertex 3: intersection of green and red

Green y = 10 red: x +y = 30

X+10 = 30 x = 20 (20,10)


5. A charter airline company sells first class

and coach class seats. To charter a plane, at

least 5 first class seats and at least 10 coach

class seats must be sold. The plane does

not hold more than 30 passengers. The

company makes a $60 profit for each first

class seat and a $40 profit for each coach

class seat sold. In order to maximize profits,

how many coach and first class seats should

they sell?

3

2

1

(x,y) 60x+40y f(x,y) (5,10) 60(5)+40(10) 700 (5,25) 60(5)+40(25) 1300 (20,10) 60(20)+40(10) 1600




selling 20 first class seats

and 10 coach seats.



Step 1: Define your variables and write a system for the constraints. (3 points)

Let x = the number of general tickets

Let y = the number of balcony tickets

X ≥ 200

Y ≥ 50 and y ≤ 70

x+y ≤ 350 (only 350 seats in theatre)

Step 2: Write the objective function. (1 point)

P = 6x+4y

f(x,y) = 6x+8y

Step 3: Graph. (3 points)

x+y ≤ 350 find the x and y intercepts

x intercept = 350 y intercept = 350


(Vertices) (2 points)

Vertex 1: (200,50)

Vertex 2: (200, 70)

Vertex 3: Intersection of yellow and red line

Yellow: y = 70 red: x+y = 350

X+70 =350 x = 280 (280,70)

Vertex 4: Intersection of green and red line

Green: y = 50 red: x+y = 350

X+50=350 x = 300 (300, 50)

Step 5. Create a table with the objective function to determine the maximum values. (3 points)

Step 6: Answer the question. (1 point)

A theatre company is selling tickets to their most recent production. At

least 200 general tickets must be sold and at least 50 balcony tickets

can be sold. There are only 70 balcony seats in the theatre. The

theatre holds a total of 350 seats. The company makes a profit of $6

for each general ticket and $8 for each balcony seat sold. How many

of each type of ticket should they sell in order to maximize their profit?

1

2 3

4

(x,y) 6x+8y f(x,y) (200,50) 6(200)+8(50) 1600 (200,70) 6(200)+8(70) 1760 280,70) 6(280)+8(70) 2240 (300,50) 6(300)+8(50) 2200




selling 280 general tickets

and 70 balcony tickets.

f(x,y) = 2x+3y make a table to evaluate the objective function: … 13... · escalades. it takes...

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