f(x,y) = 2x+3y make a table to evaluate the objective function: … 13... · escalades. it takes...
TRANSCRIPT
Copyright© 2009 - 2013 Algebra-class.com
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?
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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?
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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 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.
Step 5. Create a table with the objective function to determine the maximum values.
Step 6: Answer the question.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
2. y ≤ x+6
y +2x ≥ 6
3 ≤ x ≤ 6
f(x,y) = x -2y
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.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
3. y ≥ 4
1 ≤ x ≤ 10
x- 2y ≥ -4
f(x,y) = -x+3y
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.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
Part 2: Apply your knowledge to real world problems.
.
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.
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.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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.
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.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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.
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?
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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)
Step 4: Find the coordinates of the feasible region.
(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?
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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
Step 1: Graph the system.
Step 2: Identify the vertices:
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)
Step 3: Create a table to find the maximum
and minimum values for the objective
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
Step 4: Identify the maximum and minimum
values.
The minimum value is 3 at (1/2,2)
The maximum value is 29 at (7,15)
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
2. y ≤ x+6
y +2x ≥ 6
3 ≤ x ≤ 6 x≤ 6 and x≥ 3
f(x,y) = x -2y
Step 1: Graph the system.
Step 2: Identify the vertices:
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)
Step 3: Create a table to find the maximum
and minimum values for the objective
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
Step 4: Identify the maximum and minimum
values.
The minimum value is -18 at (6,12).
The maximum value is 18 at (6,-6)
1
2
3
4
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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
Step 1: Graph the system.
Step 2: Identify the vertices:
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)
Step 3: Create a table to find the maximum
and minimum values for the objective
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
Step 4: Identify the maximum and minimum
values.
The minimum value is 2 at (10,4)
The maximum value is 11 at (10,7)
3
2
1
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
Part 2: Apply your knowledge to real world problems.
.
Step 1: Define your variables and write a system for the constraints.
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)
Step 2: Write the objective function.
P = 3x+5y
f(x,y) = 3x+5y
Step 3: Graph.
For .50x +y ≤ 200 x and y intercepts: x = 400 y = 200
Step 4: Find the coordinates of the feasible region. (Vertices)
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)
Step 5. Create a table with the objective function to determine the maximum values.
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.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
Step 1: Define your variables and write a system for the constraints.
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
Step 2: Write the objective function.
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
Step 4: Find the coordinates of the feasible region. (Vertices)
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)
Step 5. Create a table with the objective function to determine the maximum values.
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
The maximum profit that
can be made is $1400. This
profit could be made by
scheduling 28 sick visits
and no well visits.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
Step 1: Define your variables and write a system for the constraints.
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)
Step 2: Write the objective function.
40x+50y = P
f(x,y) = 40x+50y
Step 3: Graph.
x+y ≤ 30 find the x and y intercepts: x = 30 y = 30
Step 4: Find the coordinates of the feasible region. (Vertices)
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)
Step 5. Create a table with the objective function to determine the maximum values.
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
The maximum profit that
can be made is $1600. This
profit could be made by
selling 20 first class seats
and 10 coach seats.
Copyright© 2009 - 2013 Algebra-class.com
Algebra 1Algebra 1Algebra 1Algebra 1: Inequalities: Inequalities: Inequalities: Inequalities
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
Step 4: Find the coordinates of the feasible region.
(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
The maximum profit that
can be made is $2240. This
profit could be made by
selling 280 general tickets
and 70 balcony tickets.