Culminating Experience 2 - Linear programming
3 Weeks
Course Level Expectations
1.1.9
Develop, compare and apply functions using a variety of technologies (i.e. graphing calculators,
spreadsheets, and on-line resources).
1.2.1
Develop and apply linear equations and inequalities that model real-world situations.
1.2.2
Represent functions (including linear and nonlinear functions such as square, square root, and
piecewise defined) with tables, graphs, words and symbolic rules; translate one representation of a
function into another representation.
1.2.4
Explain how changes in the parameters m and b affect the graph of a linear function.
1.2.5
Recognize and explain the meaning and practical significance of the slope and the x- and y-intercepts
as they relate to a context, graph, table or equation.
1.3.1
Simplify and solve equations and inequalities.
1.3.5
Solve systems of linear equations that model real world situations using both graphical and algebraic
methods.
2.1.1
Compare, locate, label and order integers, rational numbers and real numbers on number lines, scales
and graphs.
2.2.2
Use technological tools such as spreadsheets, probes, algebra systems and graphing utilities to
organize, analyze and evaluate large amounts of numerical information.
2.2.3
Choose from among a variety of strategies to estimate and find values of formulas, functions and
roots.
2.2.4
Judge the reasonableness of estimations, computations, and predictions.
4.1.1
Collect real data and create meaningful graphical representations of the data with and without
technology.
Extended Course Level Expectations
1.2.7 Solve systems of linear equations that model real world situations using both graphical and algebraic
methods. - is extended to systems of linear inequalities.
Overview
The Linear Programming Culminating Experience builds on the mathematics learned in this course and extends
students’ understanding of algebra and graphing techniques to the solution of optimization problems. Students
will graph the solution set of a linear inequality in two variables and solve systems of inequalities graphically by
hand, and with a graphing calculator. They will identify the boundary lines, half-planes, feasible region and
vertices of a feasible region and determine the objective function for a real-world problem. Students will apply
the Fundamental Principle of Linear Programming (the maximum/minimum solution occurs at a vertex of the
feasible region) and determine the optimal solution to real-world problems. The students’ experience solving
optimization problems will result in an understanding of the historical applications and practical efficacy of
linear programming and its importance in present day decision-making.
Assessment Activities
Evidence of Success: What students will be able to do.
Students will use linear programming to identify optimal solutions to practical problems.
Assessment Strategies: How will they show what they know.
Homework, research questions, student presentations and a differentiated Linear Programming Problem.
Launch Notes:
Closure Notes:
This activity provides three end-of-unit
Real Life Context –
assessments – two collaborative performance tasks
Linear programming provides a mathematical way to
and a culminating problem done individually.
identify optimal conditions. Linear programming was
developed out of necessity during World War II, and with
In the first performance task students identify a
the invention of computers, made advances into the 21st
practical application and create a corresponding
century. Linear programming was used, beginning with
World War II and then other conflicts, to optimize the use problem. Or, to differentiate, you may select an
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of resources for the military: food, vehicles, ships,
personnel, etc. (View Video Clip
http://www.videospider.tv/Videos/Detail/681116719.aspx
or http://video.yahoo.com/watch/10072/871993).
In the 1960’s Linear programming was used in the
launch of the first rocket that carried an astronaut into
space. In industry, linear programming is used to
determine optimal solutions to many real life
situations. Linear programming is a process or
algorithm that determines a maximum or minimum
value (optimal solution) for a linear function of more
than one variable where the independent variables are
subject to linear constraints. The process may be
represented graphical or algebraically. In this activity
we will examine the graphing approach to finding the
optimal solution.
application and give the student a problem to
solve. Students work in small groups to apply the
seven-step linear programming algorithm. They
will present the solution to the class orally. The
students will also submit a detailed written
solution.
For the second performance task, students will
investigate the role of linear programming. They
may research the history of linear programming,
identify mathematicians who have had or may
continue to have a major role in the area, or
identify linear programming’s role in
contemporary decision-making (See Activity Sheet
1.4b). Students will write a report about their
research so that all members of the class learn
more about linear programming’s development
and benefits. Skits, videos, power point
presentations, blogs and other creative presentation
modes may be used. Interdisciplinary planning and
sharing are encouraged. Students will be assessed
individually through the solution of an application
problem. Form A is more challenging then Form
B.
Important to Note: Vocabulary, connections, common mistakes, typical misconceptions
Algorithm, inequality, constraint, equivalent equation and inequality, half-plane, shading,
boundary line, system of equations, system of inequalities, test point, feasible region, objective
function, maximum/minimum, optimization
Learning Strategies
Learning Activities
Differentiated Instruction
This investigation provides an opportunity to complete
Week 1
some interdisciplinary work with the history, computer
1.1 During the first two days, provide an overview of
linear programming. The rest of the week, students science, and/or English department teachers. The
will hone their abilities to apply components of the history of linear programming may be researched and
used in conjunction with applications and techniques
seven-step algorithm using practical linear
used in computer science. English department teachers
programming contexts. To launch the project,
may use this nonfiction project as an opportunity for
students may watch the two- minute video clip
students to write a research report. English teachers
about army planning and the movement of goods.
http://www.videospider.tv/Videos/Detail/6811167 may assist students in setting reading and writing
goals. The extensive new vocabulary may be
19.aspx).
supported by diagrams, graphic organizers, and a
student-constructed glossary.
Give students a real world problem involving
transportation of military vehicles to investigate in
small groups, possibly with teacher guidance. See 1.1 Activity Sheets 1.1b and 1.1c Homework is
provided so that students come to class on the
Activity Sheets 1.1a Defining Variables and
second day with solutions and non-solutions and a
Writing Constraints. Challenge the groups to
graph (that has the same scaling as the one used in
work together for 20 minutes and find a “best”
class for the weight constraint) of the ordered pairs
solution in the allotted time. Students then share
for the area constraint. Two forms of the
and explore all the different solutions and agree on
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the one that appears to be the “best”. Then pose
the question “How do we know if we have the
very best solution?”
Introduce the idea that there is a mathematical way
to judge the solution. Have students try the first
two steps of the linear programming algorithm:
defining the variables and writing a constraint as
well as determining solutions and non-solutions
for the weight constraint. Have each group graph
these on a transparency. Collect each group’s
transparency.
(Note: Prior to the next class, take the groups’
transparencies with the ordered pairs from the
weight constraint and plot all points on one master
transparency for use the next day.)
homework sheet are provided. Each one has
different ordered pairs so that there will be
sufficient points to draw a reasonable conclusion
regarding how to quickly graph the solutions of an
inequality with respect to the boundary line.
Teacher may also want to provide some homework
examples for students who need to review x- and yintercepts of a linear function and solve some linear
systems. Students will need to find x- and yintercepts and solve systems on day two to obtain
the corner points of a feasible region.
1.2 Display the master transparency showing all
ordered pair solutions and non-solutions to the
weight constraint that the groups tested the day
before. Guide students to draw the conclusion that
all the solutions are on one side of the boundary
line. Using Activity Sheets 1.2a Graphing the
Constraints and Determining the Objective
Function and student homework, each group will
make a transparency for the area constraint. You
may then overlay all the area constraint
transparencies. Again pose a question, “Where are
all the solutions to the inequality?” After students
agree on the answer to that question, overlay the
master weight constraint transparency onto the
area constraint transparencies so students may
“see” a feasible region. Then continue with the
activity to complete the 7-step process. This will
provide the optimal solution to the military
transport problem. Have students compare it to the
earlier suggested “best” solution. Then assign
homework Activity Sheet 1.2b. Work from it will
be used to launch the next lesson.
1.2 Keep the transparencies produced today so that
future work for the feasible region can be
demonstrated. Homework is provided (Activity
Sheet 1.2b) where students graph the objective
function for several force values is provided so
students can come to the next class and see that for
changing values of c in ax + by = c , it generates
parallel lines. Thus, we only need to look at edges
of the feasible region and ultimately just the
corners of the feasible region. For students
needing more practice finding x- or y-intercepts or
solving systems of equations, additional
homework practice may be provided.
1.3 Using Activity Sheets 1.3 The Rationale Behind
Only Checking Corner Points, students will gain
an understanding of the relationship of the
objective function to the corner points of the
feasible region; that is, that the optimal solution to
a linear programming problem will be found at a
corner point (vertex). Students will then apply the
7-step algorithm to a new problem, the Stop World
Hunger Fundraiser.
1.3 Emphasize the need to check the final answer with
the actual problem to be sure it is indeed the
solution.
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Time may be short for completing the Stop the
World Hunger problem in section 3 of Activity
Sheet 1.3. You may want to prepare a solution to
this problem ahead of time and walk the students
through the process up to the graph of the feasible
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The URL below dynamically demonstrates
objective function lines moving in parallel
fashion. It may be used here or with Activity 1.5
below
http://people.richland.edu/james/ictcm/2006/slope.
html
1.4 Students may watch a video about the manufacture
of Belgian chocolates. You may use the video at
www.metalproject.co.uk/METAL/Resources/Film
s/linear_programming/index.html. (produced by
Mathematics to Enhance Economics: Enhancing
Teaching and Learning). You may pause the video
and have students do the associated linear
programming problem and then restart the clip to
"see" the solution. Or you can just let the teacher
in the video talk students through the problem-- it
is done slowly and with clarity. The problem has a
fractional solution, which is fine since the
company can produce a fraction of a batch of
chocolate.
For the next problem of the day, Activity Sheets
1.4a Eduardo’s Jobs, is provided.
At this point it is time to describe the research
activity and the research questions. Students
should take a few days to select a topic of interest.
Activity Sheets 1.4b Suggested Research
Question List gives some ideas.
region (steps 1 – 4). Then, you can have the
students carry out steps 5, 6 and 7 to gain practice
with the final steps of the linear programming
algorithm.
Many students are overwhelmed with the amount
of reading in a linear programming problem. You
may prompt students needing assistance by
verbally (or with highlighter) highlighting the
sentences that contain the variable definitions or
constraints. For example, “The first constraint
uses the statement ‘No more than 100 shirts can be
made’.”
In order to quickly sketch the boundary line,
students should be reminded that the intercepts are
easy to find because the boundary equations are
generally in standard form.
Emphasize the use of the “test point”; i.e. graph
the boundary line and then use one “test” point. If
the inequality is satisfied, that is the side that
contains the entire solution set. If the inequality is
not satisfied then the other side of the line contains
the solutions. The boundary line is included since
we are graphing ax + by ≤ c or ax + by ≥ c. Often
the origin can be used as the test point. The use of
a test point extends to solving a system of two
constraints where one needs to examine a test
point in each of the four regions created by 2
intersecting lines.
1.4 Students may begin their research of LP in
general, or learn about George Dantzig and his
contributions to linear programming (see Activity
Sheet 1.4b). Again, you may want to emphasize
that linear programming is used daily and recent
advancements have been made in the field. Linear
programming software is used to aid in finding
optimal solutions in many industries. The
following website may be used to give insight into
who is using it and why:
http://www.ilog.com/products/optimization/.
See the end of the unit plan for some research
questions.
The teacher may continue to assist those students
challenged by the reading by verbally highlighting
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or physically highlighting important key phrases
such as “Combined he will work a total of 12
hours a week at the two jobs.”
1.5 Make sure that students are considering the many
topics they can research so that by early in the
second week in this unit, actual research can begin.
See the choice of videos and descriptions is in the
teacher notes at the right.
Another problem with the 7-step support template
has been provided (see Activity Sheets 1.5: The
Farmer). This feasible region does not have
corners at the origin or on the x or y axis. By next
class students can provide their own “template” for
assigned problems.
However, see teacher note at right. This day may
be needed to reinforce the idea that the process
used to solve this problem “fits the same mold” as
those presented thus far.
1.5 You may emphasize that linear programming
problems (now often called operations research)
are seen in many real world contexts where a
maximum or minimum solution is sought:
transportation, manufacturing, agriculture,
business, health care, advertising, military,
telecommunications, financial services, energy
and utilities, marketing and sales. See
www.hsor.org for a very rich site: the
Mathematics for Decision Making in Industry
and Government, the High School Operations
Research site. It has modules from Bethlehem
Steel, the Meat Industry, LL Bean, and Special
Education School Buses. Videos are available
too. Most materials are for free. There are some
that can be purchased. This site was a link from
the NCTM illuminations site
Activity Sheet 1.5, The Farmer, is a challenging
problem in which only one of the constraints is in
the form ax  by  c . The second constraint
is y  x . Furthermore, instead of the nonnegative constraints x  0 , y  0 , The Farmer
requires the constraints x  100 , y  100 .
Therefore, if during the first 4 days of the unit,
time has not permitted the class to investigate all
problems listed in 1.3 and 1.4, you may instead
decide to use this day to catch up with those
problems.
Or, you may decide to replace The Farmer with
the first supplemental problem listed at the end of
the unit plan. (See Activity Sheets 1.5b). This
problem, about an artist, has a style much like
those addressed so far in the unit.
The URL below (also cited in 1.3 above)
reinforces the relationship between the objective
function and the corner points of the feasible
region and it opens up the discussion of what
happens when the slope of the graph of an
objective function is equal to the slope of an edge.
In week 2 activities, one of the recommended
problems has an edge parallel to the objective
function
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http://people.richland.edu/james/ictcm/2006/slope.html
Week 2
1.6 Sometime during this 3-day period, it is hoped that
students can be provided with some library time
and/or computer time to work on their research
question. Students should also be given a copy of
the research rubric, Activity Sheet 1.4c that will
be used to assess their project.
Students should begin Activity Sheets Farm
Subsidies 1.6a in class. The teacher may want to
check the students’ constraints before they delve
too deeply into the problem. The same set of
constraints applies to A1, A2 and B. A2 may be
assigned for homework and B may be assigned for
homework for students ready for a challenge. Or
part B might be done in class. Problem B
illustrates that when two adjacent corners both
produce the optimum result, then all points along
the entire edge do also. If the video listed above in
Activity 1.5 was shown in class, students will have
been forewarned of this possibility.
Research questions should be selected and
assigned by the beginning of week two and some
class time may be scheduled for computer lab or
library work in this three-day period. Students may
begin to work in groups on a research question.
Following the activity about the farm subsidies,
you may share the lesson about the tomato farmer
from the video at
www.metalproject.co.uk/METAL/Resources/Film
s/linear_programming/index.html.
A supplemental problem, Activity Sheet 1.6b
Natasha’s Cat, requires students to minimize an
objective function. The coordinates of one of the
corner points are fractions, but make sense since
one can feed a cat a half serving of cat food.
Sources for more problems to try during this threeday period are provided at the right. And a few
problems are provided on Activity Sheet 1.5b.It is
recommended that students provide their own
template for these.
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1.6 You may need to monitor groups closely on the
construction of the constraints in Activity 1.6. The
students should be alerted to look for three
constraints in addition to the non-negative
constraints. And, when students are creating their
graphs, you may need to remind students to label
their lines clearly so that they know which line
matches each constraint. This will help in
formulating the equations that are used to find the
intercepts.
During Activity 1.6, you might ask students to look
up farm subsidies and ask if farmers in other
countries get subsidies, what crops are subsidized,
and if any farmers in Connecticut get one. See a
farm subsidy database at
http://farm.ewg.org/farm/region.php?fips=09003.
The tomato farmer video contains three constraints
in addition to non-negative constraints. Select 4.04
and 4.05 at
www.metalproject.co.uk/METAL/Resources/Films
/linear_programming/index.html
You may just show the greenhouses and situation
(4 minutes) and do the problem in class or you can
let the economics teacher in the video do the
problem (12 – 16 minutes). The ultimate solution is
to not introduce lettuce farming, similar to problem
A1 in Activity Sheet 1.6a where the optimal
solution is to plant only oats.
Natasha’s cat uses the AAFCO guidelines for
felines and is based on dry matter values. There
may be a need to do some review work with
percents to determine which cat food really does
provide the most protein (or other nutrient).
Students ready for a challenge may be provided
with problems that have more than two constraints
in addition to the non-negative constraints or may
be directed to www.hsor.org mentioned in 1.5 or to
the dirt bike activity cited in the resources at the
end of this plan http://illuminations.nctm.org (Go
under grade 9 – 12 algebra and select Dirt Bike
Dilemma.)
See Activity Sheets 1.5b Supplemental Problem
List for alternative problem ideas.
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Now is a good time to administer, score and share
solutions to the Unit Test problems. To
differentiate the assessment, two versions are
provided.
1.7 In preparation for the final presentations on the
research questions, students should be encouraged
to create skits, videos, power point presentations,
etc. In addition to the time that students are
spending on their research question, students
should also be encouraged to create their own
linear programming problem individually or in
pairs. Then ideally have the two students or pairs
of students join to form a group. They will select
and solve one of the two problems and share the
problem and its solution with the class. A written
solution should also be collected from each group
See the Activity Sheet 1.7a, Create Your Own
Linear Programming Problem for ideas. A
rubric, Activity Sheets 1.7b, for the solution
should be given to the students and used as a selfassessment and reflection of progress.
1.8 All groups report out on the final research
questions. Any student-created skits, videos etc.
are shared with the class. Written reports are due.
To motivate students while they continue their
study of linear programming, you may wish to use
the following video on Operational Research http://www.bnet.com/2422-13950_23-178846.html
- the individual in this video worked with George
Dantzig at Stanford University.
1.7 Given that time has been allocated to research
already in Activity 1.6 and that four additional
days are provided here in Activity 1.7, it is
recommended that, in addition to the research
question, all students work in small groups (2-3
students – perhaps their research teams) to create
their own linear programming problem.
If students struggle in creating their own problem,
the teacher may have them choose from Activity
Sheets 1.5b Supplemental Linear Programming
Problems list at the end of the unit plan.
1.8 Focus should be on sharing the work that the
students researched and studied in groups. The
goal is to have students leave with new knowledge
and new questions, and some questions which
might remain unanswered. Leave time for
discussion and posing, “What if’s?” Guests may
be invited to share interesting business and
industry applications and see the work your
students have accomplished.
For the presentations related to the linear
programming problems that were created,
students should explain each of the steps used in
solving their problem, i.e. follow the algorithm.
See Activity Sheets 1.4d for the rubric for the
research presentation and Activity Sheets 1.7b
for the rubric for the “create your own linear
programming problem” presentation.
Resources:

http://www.videospider.tv/Videos/Detail/681116719.aspx (1.1)

http://people.richland.edu/james/ictcm/2006/slope.html (1.3, 1.5)

Belgian Chocolates and Tomato Farmers videos (produced by Mathematics to Enhance Economics:
Enhancing Teaching and Learning – at
www.metalproject.co.uk/METAL/Resources/Films/linear_programming/index.html (1.4, 1.6)

The following website may be used to give insight into who is using linear programming and why:
http://www.ilog.com/products/optimization/. (1.4)

See www.hsor.org for many high school linear programming problems and information on linear
programming (1.5)

Contemporary College Algebra: Data, Functions, Modeling, 6th ed., McGraw – Hill Primis Custom
Publishing . Don Small author (Farm Subsidy problem, 1.6)
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







Farm subsidy data base at http://farm.ewg.
org/farm/region.php?fips=09003 (1.6)
video on Operational Research use in 1.6 or later- http://www.bnet.com/2422-13950_23-178846.html
TI Activity Exchange for Design your Own Can at http://education.ti.com/
educationportal/activityexchange/
Activity.do?aId=8272&cid=US (supplemental problem #2)
http://illuminations.nctm.org (supplemental problem #3)
http://www.wikihow.com/Make-a-Duct-Tape-Wallet
http://www.ducttapefashion.com/products/prod01.htm (supplemental problem #4)
TI Numb3rs Linear Programming Activity at
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=7508
(supplemental problem #6)
Bakers Choice published by Key Curriculum Press (supplemental problem #7)
Information relevant to George Dantzig: http://www.lionhrtpub.com/orms/orms-8-05/dantzig.html and
http://www.lionhrtpub.com/orms/orms-6-05/dantzig.html (Suggested research question #4)
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Unit 8: Culminating Experience 2: Linear Programming
Activity 1.1a, p. 1 of 4
Defining Variables and Writing Constraints
Name: _________________________________________ Date: _________________
Section 1: The Peacekeeping Problem
Read the following problem. Your team will have approximately 20 minutes to explore
the following problem and offer a possible “best” solution.
The UN needs to move a number of units of Hummers (one unit has 5 Hummer
vehicles), and Apache helicopters, for air support, to a region in the world for
peacekeeping purposes. A ship is used to transport these vehicles and can
accommodate a total equipment weight of 150 tons. Each Hummer unit weighs 15
tons and each Apache helicopter weighs 6 tons. Each Hummer unit uses an area
of 1,200 square feet and each helicopter uses an area of 1,800 square feet. The
ship has a total of 25,200 square feet of area to use for transport. What
combination of Hummer units and Apache helicopters will produce a maximum
force (force = the sum of Hummer units and helicopters) of Hummer units and
Apache helicopters for the UN?
After working in your group, please submit this sheet with the information below
completed and all calculations (there is room on the back):
Group Member Names:
Number of Hummer units to transport:
Number of Apache helicopters to transport:
Is the weight of the Hummer units and Apache helicopters (using the number of
units you claim) give a maximum force less than or equal to 150 tons? ____ Show
your calculation.
Is the total area (square feet) for the number of Hummer units and Apache
helicopters (using the number of units you claim) give a maximum force less than
or equal to 25200 square feet? _____ Show your calculation:
Why do you think you have the best possible combination of vehicles that
maximizes the force for the UN?
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Unit 8, Culminating Experience 2
Activity 1.1a, p. 2 of 4
Section 2: Defining Variables and Writing the First Constraint--Weight
After examining each team’s “best” solution, we have now identified the “best” solution
put forth by this class. But how do we know that this is the best possible solution (i.e. the
“optimal” solution)? While we may not be able to answer this question in the remainder
of this class period, there is indeed a mathematical technique called linear programming
that will allow us to find the optimal solution. There are seven steps in this process.
Today we will examine the first two.
The first step in the algorithm for solving a linear programming problem is to define
variables for the unknowns in the problem. For example,
There are two kinds of vehicles that need to be transported: units of Hummers and
Apache helicopters.
Let x represent the number of Hummer units to be transported.
Let y represent the number of Apache helicopters to be transported.
The second step is to use these defined variables to write the constraints for the problem.
A ship is used to transport these vehicles and can accommodate a total equipment
weight of 150 tons. Each Hummer unit weighs 15 tons and each Apache
helicopter weighs 6 tons.
Weight Constraint: 15x + 6y < 150
1. It may be helpful to build a table and explore combinations of vehicles to find
possible solutions. A possible solution is a pair of values for x and y which satisfy
the constraint. Continue filling in the table, finding solutions to the constraint for
this problem. Each member of the team should find their own three pairs of
values.
Number of Hummer
units to be
transported, x
Number of Apache
helicopter to be
transported, y
Weight Constraint
15x + 6y < 150
Solution to
the
Constraint?
4
10
8
20
15 (4) + 6(8) ≤ 150
15 (10) + 6(20) ≤ 150
Yes
No
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Unit 8, Culminating Experience 2
Activity 1.1a, p. 3 of 4
2. Since the solutions to the problem must satisfy all conditions in the problem, and
there cannot there be a negative number of units of Hummers nor can there be a
negative number of helicopters, this means that even though (-1, -2) satisfies 15x
+ 6y ≤ 150, we would not consider it to be a solution to our peacekeeping
problem nor would we graph it as a solution of the problem.
Write inequalities that state (1) that the number of units of Hummers cannot be
negative and (2) that the number of Apache helicopters cannot be negative.
_____________________________________________
3. Using the graph paper below, label the axes on the graph and plot the points from
your table. Note: that the graph is not a graph of the entire plane because we do
not want points that have one or both coordinates negative. Use green to plot each
ordered pair that represents a solution to the constraint. Use red to plot each
ordered pair that did not result in a solution to the constraint.
Graph the equation 15x + 6y = 150. The line you get is called a boundary line.
Make a conjecture about the location (with respect to the boundary line) of the
points on the graph that represent solutions to the weight constraint,
15x + 6y ≤ 150.
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Unit 8, Culminating Experience 2
Activity 1.1a, p. 4 of 4
Section 3: More team work
1. Within each team, pool the team’s data (from member’s tables and the work
getting a best solution) using green dots for points whose coordinates are
solutions and red dots for points whose coordinates are “not possible.” Make sure
you are plotting the points on your individual graphs while the team recorder
makes a graph of the team data on a team transparency. You will need the graphs
and transparency next class.
2. Graph the boundary line for the weight constraint if you have not already.
3. You may not have been able to make a conjecture with just your own data, but
with all the team members’ data, you should be able to make a conjecture about
the location (with respect to the boundary line) of the points on the graph that
represent solutions to the weight constraint, 15x + 6y ≤ 150.
4. Is it necessary to find a lot of solutions of 15x + 6y < 150 in order to make the
graph of this inequality? How many solutions do we need to find, assuming we
have plotted the boundary line?
5. There is a second constraint for this problem. Using the sentence, “Each Hummer
unit uses an area of 1,200 square feet and each helicopter uses an area of 1,800
square feet. The ship has a total of 25,200 square feet of area to use for transport.”
Write an inequality to define this area constraint.
_______________________________________
For homework you will need to make a table and graph for the second constraint. Bring it
to class tomorrow so we can find that “best” solution. A table and graph sheet will be
distributed.
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Unit 8, Culminating Experience 2: Linear Programming
Homework 1.1b, p. 1 of 2
Defining Variables and Writing Constraints
Name: __________________________________________ Date: __________________
In class you defined the variables for the Peacekeeping Problem.
Let x represent the number of Hummer units to be transported.
Let y represent the number of Apache helicopters to be transported.
Near the end of class you wrote the constraint 1200x + 1800 y ≤ 25200 using the
sentence, “Each Hummer unit uses an area of 1,200 square feet and each helicopter uses
an area of 1,800 square feet. The ship has a total of 25,200 square feet of area to use for
transport.”
1. It may be helpful to build a table to explore combinations of vehicles and
possible solutions as you did in class for the weight constraint. A possible
solution is a pair of values for x and y which satisfy the constraint. Continue
filling in the table, finding solutions to the constraint for this problem. Select
two points of your choice for the last two rows.
Number of Hummer
units, x
Number of Apache
helicopters, y
Area Constraint
1200x + 1800y < 25200
Solution?
4
10
2
6
4
9
8
20
2
8
6
5
1200 (4) + 1800(8) ≤ 25200
1200 (10) + 1800(20) ≤ 25200
Yes
No
2. Using the graph paper below, label the axes on the graph and plot the points
from your table. Use green to plot each ordered pair that represents a solution
to the constraint. Use red to plot each ordered pair that is not a solution to the
constraint. Next, graph the equation 1200x + 1800y = 25200. (or the
equivalent equation 2x + 3y = 42) The resulting line is called a boundary
line. Make a conjecture about the location (with respect to the boundary line)
of the points on the graph that represent solutions to the area constraint 1200x
+ 1800y ≤ 25200.
_____________________________________________________________________
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Unit 8, Culminating Experience 2
Homework 1.1b, p. 2 of 2
______________________________
______________________________
______________________________
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Unit 8: Culminating Experience 2
Homework 1.1c, p. 1 of 2
Defining Variables and Writing Constraints
Name: ____________________________________ Date: ____________________
In class you defined the variables for the Peacekeeping Problem.
Let x represent the number of Hummer units to be transported.
Let y represent the number of Apache helicopters to be transported.
Near the end of class you wrote the constraint 1200x + 1800 y ≤ 25200 using the
sentence, “Each Hummer unit uses an area of 1,200 square feet and each helicopter uses
an area of 1,800 square feet. The ship has a total of 25,200 square feet of area to use for
transport.”
1 It may be helpful to build a table to explore combinations of vehicles to find possible
solutions as you did in class for the weight constraint. A possible solution is a pair of
values for x and y which satisfy the constraint. Continue filling in the table, finding
solutions to the constraint for this problem. Select 2 points of your choice for the last
2 rows.
Number of Hummer
units, x
Number of Apache
helicopters, y
Area Constraint
1200x + 1800y < 25200
Solution?
4
10
1
6
3
10
8
20
2
12
15
8
1200 (4) + 1800(8) ≤ 25200
1200 (10) + 1800(20) ≤ 25200
Yes
No
2 Using the graph paper below, label the axes on the graph and plot the points from
your table. Use green to plot each ordered pair that represents a solution to the
constraint. Use red to plot each ordered pair that is not a solution to the constraint.
Next, graph the equation 1200x + 1800y = 25200 (or the equivalent equation 2x + 3y
= 42). The resulting line is called a boundary line. Make a conjecture about the
location (with respect to the boundary line) of the points on the graph that represent
solutions to the area constraint 1200x + 1800y ≤ 25200.
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Unit 8, Culminating Experience 2
Homework 1.1c, p. 2 of 2
______________________________
______________________________
______________________________
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Unit 8: Culminating Experience 2
Activity 1.2a, p. 1 of 5:
Graphing the Constraints and Determining the Objective Function
Name: _______________________________________ Date: _____________________
Section 1: A class graph
Last class we found the best team solution to the UN peacekeeping problem by the guess,
check and revise method. Let us continue our study of linear programming so we can find
the best solution. (We may have found it already, but we do not know we have.)
Each team recorder made a transparency of the points their team used for the weight
constraint. The team recorders also drew in the boundary line of the weight constraint.
Let us overlay all the transparencies.
1
You should see a pattern with regard to the location of the solutions and the
boundary line emerging. What is it?
Section 2: The Second Constraint--Area
The UN needs to move a number of units of Hummers (one unit has 5 Hummer
vehicles), and Apache helicopters, for air support, to a region in the world for
peacekeeping purposes. A ship is used to transport these vehicles and can
accommodate a total equipment weight of 150 tons. Each Hummer unit weighs 15
tons and each Apache helicopter weighs 6 tons. Each Hummer unit uses an area
of 1,200 square feet and each helicopter uses an area of 1,800 square feet. The
ship has a total of 25,200 square feet of area to use for transport. What
combination of Hummer units and Apache helicopters will produce a maximum
force of Hummer units and Apache helicopters for the UN?
Let x represent the number of Hummer units to be transported.
Let y represent the number of Apache helicopters to be transported.
Weight constraint: 15x + 6y < 150
Area constraint: 1200x + 1800y ≤ 25200
Implied constraints x ≥ 0, y ≥ 0
Last night you completed a table, looking for solutions to our area constraint,
1200x + 1800y ≤ 25200. You also made a graph.
1. Using a new transparency, make a team graph for the points your team
members tested last night for the weight constraint. When plotting coordinates
from your tables, plot points whose coordinates are solutions in green and
“not possible” points in red. Also be sure to draw the graph of the boundary
line, 1200x + 1800y = 25200.
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Unit 8, Culminating Experience 2
Activity 1.2a, p. 2 of 5
2. Now, let’s pool the class data by overlaying the team transparencies. You
should see a pattern with regard to the solutions and the boundary line
developing. What is it?
3. Is it necessary to find a lot of solutions of 1200x + 1800y < 25200 in order to
make the graph of this inequality? How many do we need to find, assuming
we have plotted the boundary line?
Now that you can graph a constraint, you have almost mastered the third
requirement to be able to solve a linear programming problem graphically.
Section 3: Graphing All Constraints Simultaneously For the Peacekeeping Problem
We are going to continue to develop our understanding of constraints and their graphical
representations. The third step of the linear programming algorithm requires that we
graph all constraints on the same axes. The region containing just the points that satisfy
all the constraints simultaneously (hence the solutions to the system of inequalities) is
called the feasible region.
Let x represent the number of Hummer units to be transported.
Let y represent the number of Apache helicopters to be transported.
Weight constraint: 15 x + 6y < 150
Area constraint: 1200x + 1800 y ≤ 25200
Implied constraints x ≥ 0, y ≥ 0
Last night, your teacher made a master graph of all points plotted for the weight
constraint using each team’s transparency from the prior day’s lesson. Your teacher will
now place the master transparency for the weight constraint on top of the team
transparencies for the area constraint. You should see a region in the shape of a
quadrilateral that appears to have only points that are green. Some parts of the first
quadrant will have only red points; others will have red and green points.
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Unit 8, Culminating Experience 2
Activity 1.2a, p. 3 of 5
1. Our feasible region is the quadrilateral region that has only green points.
Why?
Below is a graph of the first quadrant with the boundary lines of the two
inequalities. Region B contains all the possible solutions of our peacekeeping
problem. You have now completed step three because you have identified the
feasible region. You now have a graph of all the solutions of x ≥0, y ≥ 0,
15x + 6y ≤ 150, and 1200x + 1800y ≤ 25200. Usually we would lightly shade in
region B. Do so now.
Section 4: Corner points:
So far we have completed the following steps
1. Clearly define each variable to be used in the constraints.
2. Develop all constraints. Recall that many real world problems must include the
constraints of x > 0 and y > 0.
3. Graph the system of inequalities to obtain the feasible region. Be sure to label the
graph and shade the feasible region.
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Unit 8, Culminating Experience 2
Activity 1.2a, p. 4 of 5
4. The four vertices (corner points) of region B are very important. Finding their
coordinates is our fourth step. Next class we will explore in greater detail just
why. But today we would like to just get an overview of the procedure so we can
find out if any team got the optimal solution. As you find the intersections,
remember you can use the equivalent system: area constraint line, 2x + 3y = 42
and weight constraint line, 5x + 2y = 50. Why?
Region B is bounded by sections of the four boundary lines of our constraints.
1. The boundary lines x = 0 and y = 0 intersect at the origin whose coordinates are:
_________________
2. The area constraint line 1200 x + 1800 y = 25200 (equivalently 2x + 3y = 42) and
x = 0 meet at the y-intercept of the area constraint line. That y –intercept is
_________________
3. The weight constraint line with equation 15x + 6y = 150 (equivalently 5x + 2y =
50) and x = 0 meet and the weight constraint’s x-intercept which is _____
4. Lastly the weight constraint line with equation 15x + 6y = 150 and the area
constraint line whose equation is 1200x + 1800y = 25200 meet at __________.
Finding this corner point will require a little work. You will have to solve a
system of equations. The algebra will be simpler if you use the equivalent system
given above.
5. Label the points with their coordinates on the graph on page 3.
Section 5: The Objective Function and the Linear Programming Model
Even though we have identified all the ordered pairs whose coordinates satisfy all the
constraints, we now need to be able to select the best (optimal) solution. When you were
working as teams last class, you found there was one sentence in the problem that helped
you to determine whether one solution you obtained was better than another one. This
leads us to our fifth step.
For the fifth step we need to reread the problem looking for an “objective function,” a
quantity that needs to be maximized or minimized. For the UN peacekeeping problem,
our objective is to maximize the force of Hummer units and Apache helicopters that we
can transport. We need to write an equation for that which we are maximizing (in our
case, force) in terms of the variables that we used to define the constraints (in our case,
number of Hummer units (x) and number of Apache helicopters (y)). When we have a
linear function that must be optimized (in other words minimized or maximized) subject
to linear constraints (our linear inequalities), we have a linear programming model. Our
peacekeeping problem meets these conditions.
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Unit 8, Culminating Experience 2
Activity 1.2a, p. 5 of 5
Let us define an objective function. It is linear. Reread the problem. The important
sentence for the definition of the objective function is:
“What combination of Hummer units and Apache helicopters will produce a
maximum force of Hummer units and Apache helicopters for the military?” Note:
The “force” is sum of the number of Hummer units and the number of Apache
helicopters.
Force =
Section 6: The Solution to our Peacekeeping Problem
Tomorrow we will demonstrate why of all the many points in the feasible region, the only
ones we need to examine to find the maximum function value or minimum function value
are the ones at the corners. If we evaluate our objective function at each of the corners
and take the coordinates that give us the largest value, we will get the maximum force for
our peacekeeping problem, the sixth step in this algorithm.
1 Now evaluate the objective function Force = x + y at (0,0), (0, 14), (10, 0) and (6, 10).
What is the maximum force?
Section 7: Sixth of the seven linear programming steps
So far we have completed the following steps:
1. Clearly define each variable to be used in the constraints.
2. Develop all constraints. Recall that many real world problems must include the
constraints of x > 0 and y > 0.
3. Graph the system of inequalities to obtain the feasible region. Be sure to label
your graph and shade the feasible region.
4. On the graph, find and label the corner points with their coordinates.
5. Define the objective function.
6. Now evaluate the objective function f = ax +by at all corner points. Make a table
or use the table feature of your grapher. What is the maximum or minimum
value?
Next class we will examine the rationale behind step 6 and add the important step 7.
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Unit 8, Culminating Experience 2
Homework 1.2b, p. 1 of 1
Name ____________________________ Date _________________________
We determined in class that the objective function for the peacekeeping problem is:
F= x+y
1. Let F = 1 and graph x + y = 1 on the graph below.
2. Let F = 3 and graph x + y = 3 on the graph below.
3. Select 3 more values for F and graph each resulting equation. Show your work
and equations below.
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Unit 8: Culminating Experience 2
Activity 1.3, p. 1 of 6
The Rational Behind Only Checking Corner Points
Name: _____________________________________ Date: __________________
Today we will revisit our peacekeeping problem. We will first examine the rationale
behind testing just the corner points for a largest or smallest function value, and we will
then make sure our answer does indeed make sense for the problem (the seventh and final
step in this linear programming algorithm). Lastly as we apply all our steps to a new
problem, we will determine an efficient method for locating the feasible region.
Section 1: Why Only Evaluate at the Corner Points?
Last night you made a careful graph of the peacekeeping problem’s constraints
and each of you was asked to draw three Force lines on your graphs.
Our goal is to define a method that will always identify the optimal solution
within the feasible region. The key to the answer is the objective function. In our
case, the objective function is Force = x + y where x represents the number of
Hummer units and y represents the number of Apache helicopters.
While making a table of the coordinates of some of the points in the feasible
region and evaluating the objective function at those points provides insight into
the problem, this process will not let us know if we have found the maximum or
minimum value of the objective function. However, it does provide the insight
we need to understand why we only need test the corner points.
# Hummer units, x
0
2
4
6
# helicopters, y
6
4
6
8
Force = x + y
Force = 0 + 6 = 6
Force = 2 + 4 = 6
Force = 4 + 6 = 10
Force = 6 + 8 = 14
Each of the points whose coordinates are in the table belongs to a particular Force
line. In the table above, both (0, 6) and (2, 4) are on the line x + y = 6. Since the
coordinates of all points on each line are always going to produce the same Force,
note that our line intersects points on the edges of the feasible region as well as
interior points, so the interior points can be ignored and we can just use the
coordinates of an edge (a boundary point). For our example, there is no need to
test (2, 4), an interior point because it is on the same line as (0, 6). See the graph
on the next page.
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Unit 8, Culminating Experience 2
Activity 1.3: p. 2 of 6
In the same manner (7, 0) and (1, 6) both deliver the same Force and are on the
same Force line x + y = 7. Again, we can just use the boundary point (7, 0).
So now we know we only need to examine all points on the edges of our polygon.
However that is still too many (an infinite number to be exact).
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Unit 8, Culminating Experience 2
Activity 1.3, p. 3 of 6
Each of you graphed three Force lines for homework. Compare your lines with
those of a neighbor. Notice all the Force lines are parallel to each other. Since the
coordinates of any point on a given line are going to always produce the same
Force, the greatest Force will occur on the line that intersects the feasible region
with the largest Force. Using the edge of a ruler as a model of a force line,
continue to “move” the ruler in parallel fashion (and so the F amounts get larger)
until it only touches the feasible region at one point, and so that if you continued
to move it, it would no longer intersect the feasible region. What is special about
the one point you got?
See the graph below.
For our peacekeeping problem, the point with coordinates (6, 10) gives us
the largest value, Force = 16. We can see graphically that this is where the
line x + y = 16 will intersect the feasible region in exactly one point. For
values less than 16 but greater than or equal to zero the Force lines will
intersect the region in an infinite number of points. And for values greater
than 16, the force lines will not intersect the feasible region at all.
We do not usually graph the objective function f. Because we know that a
maximum or minimum for the function must occur at a corner point, we
merely find the coordinates of all corner points and evaluate the objective
function at each point. If there are a lot of corner points (there would need
to be more constraints), we can use our calculator’s table feature to
evaluate the objective function.
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Unit 8, Culminating Experience 2
Activity 1.3, p. 4 of 6
Lastly, as with all real world problems, we step back and make sure that the
solution we are proposing makes sense for the problem we are solving. It is
always wise to be sure the coordinates of the ordered pair do indeed satisfy the
constraints, that we have evaluated the objective function correctly, and that we
have included appropriate units in our answer.
Section 2: The Seven Steps of the linear programming algorithm
We have now completed the steps in the linear programming process:
1. Clearly define each variable to be used in the constraints.
2. Define all constraints. Recall that many real world problems must include the
constraints of x > 0 and y > 0.
3. Graph the system of inequalities and shade the feasible region.
4. On the graph, determine and label the corner points with their coordinates.
5. Define the objective function
6. Now evaluate the objective function f = ax + by at all corner points. Make a
table or use the table feature of your grapher. What is the maximum or minimum
value?
7. Step back and make sure your solution makes sense. Then state your solution and
make sure to use appropriate units.
Section 3: The “Stop World Hunger Fundraiser” Problem
Let us apply our new linear programming algorithm to a new problem. Read the
problem below and identify the variables.
The Say No to World Hunger committee has decided to have a fund raiser next
month. The idea for the fund raiser is to make and sell t-shirts with tie-dyed
designs. There are two designs being offered: a child’s face and a custom design.
No more than 100 shirts can be made and production may not cost over $560. It
costs $5 to make a child face design tie-dye and $7 to make a custom designed
tie-dye. The profit for a child’s face design t-shirt is $3.50 per shirt. The profit for
a custom design t-shirt is $5.00 per shirt. Find the maximum possible profit from
the fund raiser.
1. The variables are:
x is
y is
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Unit 8, Culminating Experience 2
Activity 1.3: p. 5 of 6
2. Now write the four inequalities (do not forget the non-negativity constraints) that
constrain the problem.
3.
Graph the constraints. Apply the “test” point principle by identifying four points,
each one being clearly in the region you wish to test. Use the origin for one of the
test points. Write down your four test points and determine if its coordinates
satisfy BOTH constraints. There is only one region for which that will happen.
Shade the feasible region.
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Unit 8, Culminating Experience 2
Activity 1.3, p. 6 of 6
4. Find the coordinates of the corner points. Some will be intercepts of the constraint
lines. For one corner you will need to solve a system of equations. Show your
work here.
5. Write down your objective function.
6. Now make a table of values for the corner points and select the corner that gives
you a maximum.
x
y
Profit =
7. Write down the solution after you have checked to be sure it does make sense.
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Unit 8: Culminating Experience 2:
Activity 1.4a, p. 1 of 3
Eduardo’s Jobs
Name: ____________________________ Date: _________________________
Eduardo is a high school student who is going to work two jobs this summer. He will be a
painter for one job and will mow lawns for the second job. Combined he will work a total
of at most 24 hours a week at the two jobs. For his first job, painting, he has found that he
needs 1.5 hours of preparation time for every hour he works. He is not paid for the prep
time and so does not count this in his hours of work. For his second job, mowing, he
needs 0.5 hour of preparation time for every hour he works. He wants his total
preparation time to be at most 18 hours per week. If Eduardo makes 12 dollars an hour
painting and 9 dollars an hour mowing lawns, what is the best combination of hours of
painting and mowing (he wants to make the most money) for him to schedule each week?
1. The variables are:
x is the
y is the
2. Now write the four constraints.
3. Graph the constraints. Last class you found that the x- and y-intercepts were
helpful. Find them first. You will need to shade the feasible region. Often the
origin can be used as the test point. Write down your four test points and
determine if its coordinates satisfy ALL constraints. There is only one region, the
feasible region, for which that will happen.
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Unit 8, Culminating Experience 2
Activity 1.4a, p. 2 of 3
4. Find the coordinates of the corner points. Some will be intercepts of the
constraint lines. For one corner you will need to solve a system of equations.
Show your work here.
5. Write down your objective function.
6. Now make a table of values for the corner points and select the corner that gives
you a maximum.
x
y
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Profit =
Page 30 of 58
Unit 8, Culminating Experience 2
Activity 1.4a, p. 3 of 3
7. Write down the solution after you have checked to be sure it does make sense.
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Unit 8, Culminating Experience 2
Activity 1.4b, p. 1 of 3
Suggested Research Question List
Name: ______________________________________ Date: _____________________
1. Research how linear programming came to being. How did it get its name? What role
did World War II and computers play in its development?
2. Have there been any recent developments in the field of linear programming,
sometimes called Operations Research (OR)? How, if at all, does linear programming
or OR affect our daily life and how we live?
3. Joseph Fourier proved the Linear Programming Theorem in 1826. It states: If a
feasible region in a linear programming problem is convex and bounded, then the
maximum or minimum quantity for the linear objective function is determined at one
(or more) of the vertices of the region. Like many mathematical discoveries and
proofs, it waited many years before being widely used, in this case more than 100
years. Who is Fourier? What is his role in the field of mathematics?
4. In 1947 George Danzig, Leonld Hurwitz and T. C Koopmans invented the simplex
algorithm or method used for solving “big” linear programming problems. President
Gerald Ford in 1975 gave George Danzig a Presidential Award for his contribution.
Koopmans received a Nobel Prize in 1982. Research the lives of these men and their
contribution to the simplex method.
5. In 1947 John von Neumann developed the theory of the duality. Who is von
Neumann and what is meant by the duality? Demonstrate its usefulness. ( For
students needing challenge)
6. Study the simplex method and solve a small linear programming problem using it. (
For students needing challenge)
7. In 1945, George Stiegler worked on the problem of determining a least expensive, yet
healthy diet. He received a Nobel Prize for his work. Find out more about what he did
and his diet.
8. In 1979 the Soviet mathematician L. G. Kachian developed the ellipsoid method,
which has not proved as useful as the simplex method, but it is noteworthy in this
developing field of Operations Research. Research Kachian and his contributions.
9. In 1984, 29-year old Neredra Karmarkar of ATT Bell Laboratory announced his
algorithm. Is it better than the simplex method? Research Karmarkar and the state of
his linear programming algorithm.
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Unit 8, Culminating Experience 2
Activity 1.4b, p. 2 of 3
Student can research large linear programming problems. See below for some sources
Large Scale Linear Programming Application Areas
Problem areas where large linear programming problems arise are:

Pacific Basin facility planning for AT&T
The problem is to determine where undersea cables and satellite circuits should be
installed, when they will be needed, the number of circuits needed, cable technology, call
routing, etc over a 19 year planning horizon (an linear programming with 28,000
constraints, 77,000 variables).

Military officer personnel planning
The problem is to plan US Army officer promotions (to Lieutenant, Captain, Major,
Lieutenant Colonel and Colonel), taking into account the people entering and leaving the
Army and training requirements by skill categories to meet the overall Army force
structure requirements (an linear programming with 21,000 constraints and 43,000
variables).

Military patient evacuation
The US Air Force Military Airlift Command (MAC) has a patient evacuation problem
that can be modeled as a linear programming problem. They use this model to determine
the flow of patients moved by air from an area of conflict to bases and hospitals in the
continental United States. The objective is to minimize the time that patients are in the air
transport system. The constraints are:
All patients that need transporting must be transported; and
Limits on the size and composition of hospitals, staging areas and air fleet must
be observed.
1.
2.
MAC have generated a series of problems based on the number of time periods (days). A
50 day problem consists of an linear programming with 79,000 constraints and 267,000
variables (solved in 10 hours).

Military logistics planning
The US Department of Defense Joint Chiefs of Staff have a logistics planning problem
that models the feasibility of supporting military operations during a crisis.
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Unit 8, Culminating Experience 2
Activity 1.4b, p. 3 of 3
The problem is to determine if different materials (called movement requirements) can be
transported overseas within strict time windows.
The linear programming includes capacities at embarkation and debarkation ports,
capacities of the various aircraft and ships that carry the movement requirements and
penalties for missing delivery dates.
One problem (using simulated data) that has been solved had 15 time periods, 12 ports of
embarkation, 7 ports of debarkation and 9 different types of vehicle for 20,000 movement
requirements. This resulted in an linear programming with 20,500 constraints and
520,000 variables (solved in 75 minutes).
Source: http://people.brunel.ac.uk/~mastjjb/jeb/or/solvelp.html
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Unit 8, Culminating Experience 2
Activity Rubric 1.4c, p. 1 of 3
Linear Programming Research Presentation Rubric
Component
0 = Missing
1. The research
question or topic
is clearly
identified.
Students fail to
identify a research
question/topic
during the
presentation.
2. Insightful
information is
provided based
on the research.
Students provide
no information
relevant to the
research question
or topic.
3. Appropriate and
informative
graphics/visual
aids are included
to enhance the
presentation.
The group does
not provide visual
aids.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
1 = Needs
Improvement
Students define a
research topic or
question, but the
question is not
“interesting” or
relevant.
2 = Proficient
Students define the
research question or
topic, but it is not
communicated
clearly or comes
too late in the
presentation.
Students provide a
Students provide a
very limited quantity reasonable quantity
of research about
of research, but
their topic or
only a limited
question.
amount of
insightful
information about
their topic or
question.
The group shares
The group shares
visual aids during
visual aids during
the presentation, but the presentation
the information
that are clearly seen
cannot be easily seen by the audience.
by the audience. Or, However, the
aids do not provide
number is limited
relevant information. or only some
Page 35 of 58
3 =Advanced
Students clearly
define the research
question or topic
early in the
presentation.
Students provide a
sufficient quantity
of research, most
or all of which is
insightful,
informative and
relevant.
The group shares
numerous visual
aids that enhance
the presentation
with relevant and
interesting
information.
Student SelfReflection
Pts.
Earned
Component
0 = Missing
1 = Needs
Improvement
4. Teamwork is
demonstrated
during the
presentation.
Only one member
of the group
presents the
solution.
One team member
dominates
presentation while
others play minor
roles.
5. Presentation
follows a logical
sequence and
conclusion.
A logical or
progressive
sequence is
completely absent.
The presentation
jumps around,
showing a lack of
organization.
2 = Proficient
provide relevant
information.
Most team
members have an
important part of
the presentation,
but one team
member plays a
very minor role.
The presentation is
organized, but some
topics are presented
out or order. Or, a
conclusion is not
clearly stated.
6. Students answer Students are
related questions, unable to address
showing a strong questions.
understanding of
the topic.
Students attempt to
answer questions,
but they show a very
poor understanding
of the research.
Students are able to
answer questions,
but responses are
not as insightful as
would be expected
following intense
research on their
topic.
7. The written
report clearly
captures the
essence of the
The written report
omits key
components of the
oral presentation
The written report
follows the
sequence of the oral
presentation, but it
Students submit no
written report to
accompany the
oral presentation.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 36 of 58
3 =Advanced
All team members
have an important
role in the
presentation of the
project.
The presentation is
well organized,
presented in a
logical sequence,
and has a clearly
defined
conclusion.
Students answer
questions showing
a strong
understanding of
the research on
their topic.
The written report
follows the
sequence of the
oral presentation
Student SelfReflection
Pts.
Earned
Component
0 = Missing
oral presentation.
8. The written
report cites three
or more
resources in
sufficient detail.
Resources are not
cited.
1 = Needs
Improvement
and/or does not
follow the sequence
of the presentation.
Only one or two
resources are cited.
2 = Proficient
Student SelfReflection
Pts.
Earned
omits a key
and includes a
component of the
detailed account
oral presentation or consistent with all
fails to fully collate components of the
all material.
presentation.
Three or more
Three or more
resources are cited, resources are cited
but not in sufficient in sufficient detail.
detail. For
No important
example, internet
references are
links do not lead to omitted.
the precise
webpage that
contains the
information cited.
Total Points Earned = _____ out of 24 possible points = ______
Comments:
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
3 =Advanced
Page 37 of 58
Unit 8: Investigation 1
Activity 1.5a, p. 1 of 3
The Farmer
Name: ____________________________ Date: _________________________
A Connecticut farmer grows corn and apples on his farm. He ships both items in the same
size box and his truck can carry at most 500 boxes per trip. Past records indicate that each
shipment should contain at least 100 boxes of each product. Also, the number of boxes of
apples should not exceed the number of boxes of corn. If this Connecticut farmer receives
a profit of $6 for each box of apples and $4 for each box of corn, how many boxes of
each should he load on his truck to maximize his profit?
1. The variables are:
x is the
y is the
2. Now write the 4 constraints.
3. Graph the inequalities. Shade the feasible region. Write down your test points and
determine if the coordinates of each satisfy all constraints.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 38 of 58
Unit 8, Investigation 1
Activity 1.5a, p. 2 of 3
4. Find the coordinates of the corner points. Show your work here.
5. Write down your objective function.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 39 of 58
Unit 8, Investigation 1
Activity 1.5a, p. 3 of 3
6. Now make a table of values for the corner points and select the corner that gives
you a maximum.
x
y
Profit =
7. Write down the solution after you have checked to be sure it does make sense.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 40 of 58
Unit 8, Investigation 1
Activity 1.5b, p. 1 of 2
Supplemental Linear Programming Problems
1. A local artist working at the town fair makes caricatures and portraits. The
caricatures cost $4 to make while the portraits cost $10 to produce. It takes 10
minutes to make a caricature and 30 minutes to make a portrait. The artist has
$500 for supplies and a total of 22.5 hours at the fair. She would like to charge
$10 for a caricature and $20 for a portrait. She is going to donate all proceeds of
the weekend fair to a local charity. What number of each type of work should she
make in order to maximize the donation for the charity?
2. At the TI Activities Exchange
http://education.ti.com/educationportal/activityexchange/Activity.do?aId=8272&c
id=US. “Design a Better Drink Can” can be used for students who may be having
trouble coming up with their own linear programming problem to research or as
an additional classroom problem because it relates geometry and algebra. Activity
is designed to come up with optimal dimensions for different can sizes. It explores
the relationships between dimensions of a can, its volume and its surface area.
3. At http://illuminations.nctm.org under grade 9 – 12 algebra select “Dirt Bike
Dilemma”. You may just use the problem or parts of the lesson and activity
sheets.
4. A student is making two types of wallets out of duct tape. She is going to make a
tri-fold wallet and a bi-fold wallet. The tri-fold wallet takes 1 hour to finish and
the bi-fold wallet takes ½ hour to finish. The tri-fold wallet costs $1 of duct tape
to make and the bi-fold costs $0.75 of duct tape to make. The tri-fold wallet costs
$10 and the bi-fold wallet cost $8. If the student has $24 to purchase tape and 20
hours throughout the week to make wallets, what combination of tri-fold and bifold wallets should she make to maximize profit?
5. For Valentine’s Day the student council is going to sell single roses and chocolate
hearts. The council must first buy roses and hearts to sell. Each rose costs $0.75
while each chocolate heart costs $0.50. The council has at most $270 to spend on
this fundraiser. On average each rose takes 2 minutes to wrap with a red ribbon
and write the student’s name on a card and the chocolate heart takes about one
minute to write the student’s name on a card. The student council has at most 10
hours to wrap the ribbons and write student’s names on cards. If the student
council will make a profit of $0.75 on each rose arrangement and $0.50 on each
chocolate heart, what combination of roses and chocolate hearts should they try to
sell?
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 41 of 58
Unit 8, Investigation 1
Activity 1.5b, p. 2 of 2
6. Numb3rs Season 3 “The Mole - Branch and Bound” provided on the Activities
Exchange at
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId
=7508. This problem is designed to answer how linear programming can be used
to find the most probable outcome of a situation. It will study the branching and
bounding algorithm that can be used when integer solutions are desired but not
obtained. It is written for Algebra 1 students.
7. Bakers Choice published by Key Curriculum Press has a real life problem
involving a Bakery that wants to maximize their profits. It is a supplemental
module for Interactive Mathematics.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 42 of 58
Unit 8: Investigation 1:
Activity 1.6a, p. 1 of 5
Farm Subsidies
Name: ________________________________ Date: _________________________
Farming can be a very difficult way of life, as sometimes “Mother Nature” does not
cooperate. A drought can wipe out entire crops. A spring season with too much rain can
delay planting and make it difficult for seeds to sprout before they rot. On occasion,
insects can damage entire fields of crops. Without healthy crops, there is nothing for the
farmer to sell and therefore very little income in those difficult years. Without help, many
farmers would sell their land and find another job.
But, our country needs the farmers and the crops they grow. As a result, the US
government helps our nation’s farmers by providing them with subsidies. A subsidy is
extra money paid directly to the farmers to help them and their families earn enough
money to be able to live comfortably and continue farming, especially in those difficult
years. If the farmer accepts a subsidy, the government determines which crops are needed
and then tells the farmer what to grow on his or her land. The farmer must follow these
instructions to receive the subsidy.
A farmer does not have to accept a subsidy. He or she may try to earn a living without the
government’s help. Some farmers feel they can earn more money by making their own
decisions. Others do not like to take directions from the government.
Suppose a farmer is trying to decide whether or not to accept a subsidy from the
government. Based on the following questions, how does getting a subsidy affect this one
farmer’s decision on what to plant and how much?
A1. Suppose a farmer has 200 acres of land on which the family can plant any
combination of corn and oats. Each acre of corn that is planted requires 2 workerdays of labor and costs the farmer $10. Each acre of oats that is planted requires ½
worker-day of labor and costs the farmer $5. Suppose the farmer gets $30 in
revenue for each acre of corn planted and $20 in revenue for each acre of oats
planted. If the farmer has $1,100 to spend on planting and 160 worker-days of labor
available for the year, how many acres of corn and how many of oats should the
farmer plant to maximize his revenue?
A2. A corn subsidy is available and increases the corn revenue from $30 to $50 per
acre. Since the farmer had planned on planting corn anyway, he is considering
taking the subsidy. If he takes the subsidy, should the farmer change the number of
acres he plants with corn? If so, how many acres should he now plant with corn and
what will be his new maximum revenue?
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 43 of 58
Unit 8, Investigation 1
Activity 1.6a: p. 2 of 5
Part A1
1. The variables are:
x is the
y is the
2. Now write the 3 constraints in addition to the non negativity constraints.
3. Graph and shade the feasible region. Write down your test points and determine if the
coordinates of each satisfy all the constraints. There is only one region for which that
will happen. Your teacher will give you a separate piece of graph paper.
4. Find the coordinates of the corner points. Show your work here.
5. Write down your objective function.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 44 of 58
Unit 8, Investigation 1
Activity 1.6a, p. 3 of 5
6. Now make a table of values for the corner points and select the corner that gives
you a maximum.
x
y
Revenue =
7. Write down the solution after you have checked to be sure it does make sense.
Part A2
8. Write down your new objective function.
9. Now make a table of values for the corner points (the corner points have not
changed) and select the corner that gives you a maximum.
x
y
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Revenue =
Page 45 of 58
Unit 8, Investigation 1
Activity 1.6a, p. 4 of 5
10. Write down the solution after you have checked to be sure it does make sense. Do
you think the farmer should accept the subsidy? Explain.
Part B Instead suppose the subsidy increases the corn revenue from $30 to $40 for an
acre of corn. How will this affect the decision on planting?
11. Write down your new objective function.
12. Now make a table of values for the corner points (the corner points have not
changed) and select the corner that gives you a maximum.
x
y
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
revenue =
Page 46 of 58
Unit 8, Investigation 1
Activity 1.6a, p. 5 of 5
13. Write down the solution (something should be different here) after you have
checked to be sure it does make sense. Make sure you answered the question that
was posed in part B.
14. Write a summary paragraph about how subsidies might affect this farmer’s decisions.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 47 of 58
Unit 8: Investigation 1
Activity 1.6b, p. 1 of 1
Natasha’s Cat
Name: ___________________________________ Date: _________________________
Use your linear programming algorithm to solve the problem below. The seven steps are
repeated here for your convenience.
1. Clearly define each variable to be used in the constraints.
2. Define all constraints. Recall that many real world problems must include the
constraints of x > 0 and y > 0.
3. Graph the system of inequalities and shade the feasible region.
4. On the graph, determine and label the corner points with their coordinates.
5. Define the objective function
6. Now evaluate the objective function f = ax + by at all corner points. Make a
table or use the table feature of your grapher. What is the maximum or minimum
value?
7. Step back and make sure your solution makes sense. Then state your solution and
make sure to use appropriate units.
Natasha’s cat, Dancer, will eat both dry food and wet (canned) food. Natasha wants
Dancer to eat meals that are nutritious, but would also like to save some money on her
cat’s food bill. Dancer usually eats wet food. However, the wet food is more expensive.
In order to save money, Natasha is trying to decide if she can switch her cat to a diet with
some wet and some dry food, or perhaps all dry food. However, she still wants to meet
the nutritional requirements of her cat.
Protein and fat are two important components of a cat’s diet. Proteins serve as building
blocks for bones and muscles, while fat provides energy and helps with the absorption of
vitamins. Dancer needs at least 60 grams of protein per day and at least 12 grams of fat
per day. The wet food provides about 10 grams of protein per serving while the dry food
provides about 30 grams of protein per serving. Both the wet and dry food contain about
4 gram of fat per serving. The wet food costs 80 cents per serving and the dry food costs
66 cents per serving. How many servings of wet and dry food will Dancer need each day
to meet her nutritional requirements while keeping Natasha’s pet food bill a minimum?
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 48 of 58
Unit 8, Investigation 1
Activity 1.7a, P. 1 of 1
Create Your Own Linear Programming Problem
Name: _______________________________________ Date: ____________________
Here are some ideas to follow:
1. The oil industry uses linear programming to maximize their profit. Research the
use of linear programming within the oil industry and how it has improved their
profits. How did they make decisions before linear programming was developed?
How do they use it in their product and distribution?
2. In a group, members will construct a company which makes two different
products. Members will build realistic constraints for their products and assign a
profit or cost to their products. Once the question has been developed, it should be
solved and presented to the class.
3. Telephone companies have to schedule millions and millions of phone calls each
day. How is linear programming incorporated into their delivery system? How
does this affect their product costs and profits?
4. UPS and FedEx must ship millions of packages a day, all over the world. How do
they incorporate linear programming into their shipping schedules? How are
computer models developed and implemented? How has the development of
different technologies aided in their success or failures?
5. A local tire company may be contacted to determine what the manufacturing costs
are for two or three different tires, and the profit and the time needed to produce a
specific tire.
6. A local restaurant can be contacted to determine how menu items are selected and
how they set their prices.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 49 of 58
Unit 8, Investigation 1
Activity 1.7b, p. 1 of 4
Rubric for Create Your Own Problem
Component
0 = Missing
1. Definition of
variables
Students do not
define the
variables.
1 = Needs
Improvement
Students define
variables, but one or
more is incorrect.
2. Construction of
problem
constraints
Students do not
construct
constraints.
Students construct
constraints, but one
or more is incorrect.
3. Graph of feasible
region
Students do not
provide a graph.
Students provide a
graph, but lines do
not match equations
or lines appear
correct but none of
the following are
shown:
 Labeled axes
and correct
scale
 Clearly
labeled
intercepts
 Correct
region
shaded
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
2 = Proficient
3 =Advanced
Students define
variables correctly
but do not use clear
statements.
Students construct
constraints
correctly but do not
list the nonnegativity
constraints.
Students provide a
graph with
correctly drawn
lines. However, one
or two of the
following is
missing:
 Labeled
axes and
correct scale
 Clearly
labeled
intercepts
 Correct
region
shaded
Students define
variables correctly
and clearly.
Page 50 of 58
Students construct
constraints
correctly,
including the nonnegativity
constraints.
Students provide a
graph and define
the feasible region
correctly and all
three bulleted
items are included:



Labeled
axes and
correct
scale
Clearly
labeled
intercepts
Correct
region
shaded
Student SelfReflection
Pts.
Earned
Component
0 = Missing
4. Identification of
the objective
function
Students do not
identify an
objective function.
5. Calculation of
the intersection
point of the
boundary lines
Students do not
identify the
intersection point
of the boundary
lines of the nontrivial constraints.
6. Evaluation of the
objective
function at each
vertex and
identification of
the optimal
solution
Students do not
evaluate objective
function at the
vertices.
7. Contextual
understanding of
solution.
Students do not
Students present an
attempt to put the
incorrect
solution in context. interpretation of
each of the
coordinates of the
optimal solution in
context, or omit one
coordinate.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
1 = Needs
Improvement
Students define an
objective function,
but are optimizing
the wrong quantity.
Students list an
incorrect ordered
pair for the
intersection of the
non-trivial
constraints and show
little, if any,
algebraic work.
Students evaluate
the objective
function at the
vertices, but half or
more of the
calculations are
incorrect or missing.
2 = Proficient
3 =Advanced
Students define an
objective function,
but use incorrect
coefficients or
reverse coefficients
of x and y.
Students identify
the correct point of
intersection for the
non-trivial
constraints but
show insufficient
algebraic
justification.
Students evaluate
the objective
function at the
vertices, and the
majority of the
outputs are correct.
Students identify
the optimal solution
based on their
calculations.
Students define a
correct and
appropriate
objective function.
Students correctly
identify one of the
coordinates in the
optimal solution in
context, but err in
the interpretation of
the other.
Students present
correct
understanding of
optimal solution in
context.
Page 51 of 58
Students identify
the correct point of
intersection for the
non-trivial
constraints and
show a complete
algebraic
justification.
Students evaluate
the objective
function correctly
at each of the
vertices and
clearly identify the
optimal solution.
Student SelfReflection
Pts.
Earned
Component
0 = Missing
8. Quality of the
created problem:
Students create
an interesting
and relevant
problem with
reasonable
constraints and
optimal solution
in context.
9. Teamwork
The problem is not
coherent.
Only one member
of the group
presents the
solution.
One team member
dominates the
presentation while
others play minor
roles.
10. Visual display
Group does not
provide visual
aids.
Group shares graphs
and tables during the
presentation, but the
information cannot
be easily seen by the
audience.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
1 = Needs
Improvement
The problem is little
more than a clone of
a prior problem in
the unit.
2 = Proficient
3 =Advanced
The problem is
interesting and
relevant. However,
the constraint
equation and/or
optimal solution do
not provide
reasonable values
in context.
The problem is
interesting and
relevant. The
constraint
equations and
optimal solution
are reasonable in
the context.
Most team
members have an
important part of
the presentation,
but one team
member plays a
very minor role.
Group shares
graphs and tables
during the
presentation that
are clearly seen by
the audience.
All team members
have an important
role in the
presentation of the
project.
Student SelfReflection
Pts.
Earned
Group shares
graphs and tables
and at least one
other creative
visual aid that
enhance the
presentation.
Total Points Earned = ______ out of 30 possible points = ______
Page 52 of 58
UNIT 8
LINEAR PROGRAMMING – CULMINATING PROBLEM
PART A
Name: _______________________________________________ Date: ________________
Jewelry Business
1
Hector is starting his own business making jewelry for a large discount store. Long-term
projections indicate an expected demand of at least 100 pairs of silver hoop earrings and 80
beaded bracelets a week. Because of limitations on production capacity, no more than 200
hoop earrings and 170 beaded bracelets can be made weekly. To satisfy the contract, a total
of at least 200 pieces of jewelry must be shipped each week.
If each pair of hoop earrings sells at a $2.00 loss, but each beaded bracelet produces a $5.00
profit, how many of each type should he make weekly to maximize net profit?
a. Choose your variables:
x=
y=
b. Write inequalities for the constraints on the variables.
c. Write an equation of the Objective Function.
PLEASE GIVE YOUR TEACHER THIS SHEET. YOUR TEACHER WILL GIVE YOU
PART B OF THE TEST.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 53 of 58
UNIT 8
LINEAR PROGRAMMING TEST – CULMINATING PROBLEM
PART B
Name: ___________________________________________ Date: ______________
Jewelry Business
1. Hector is starting his own business making jewelry for a large discount store. Long-term
projections indicate an expected demand of at least 100 pairs of silver loop earrings and 80
beaded bracelets a week. Because of limitations on production capacity, no more than 200
hoop earrings and 170 beaded bracelets can be made weekly. To satisfy the contract, a total
of at least 200 pieces of jewelry must be shipped each week.
If each pair of hoop earrings sells at a $2.00 loss, but each beaded bracelet produces a $5.00
profit, how many of each type should he make weekly to maximize net profit?
The constraints for the variables are:
100  x  200
80  y  170
x  y  200
The Objective Function is:
P = -2x + 5y
Use the constraints and objective function above to complete the problem.
a.
Use your calculator to graph the constraint inequalities.
b. Use the grid below to draw your graph and indicate the feasible region. Label your
graph and indicate scales on your axes.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 54 of 58
c. Use your calculator to determine the corner points (vertices) of the feasible region.
d. For each corner point, determine the value of the Objective Function. Show your work.
e. Explain how you determined the number of silver hoop earrings and beaded bracelets to
make to maximize your profit. Is the profit reasonable for Hector to start his business?
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 55 of 58
UNIT 8
LINEAR PROGRAMMING TEST – CULMINATING PROBLEM
PART A
Name: ___________________________________________ Date: ______________
Greenhouse Business
1
Marlene builds greenhouses. Her greenhouses are popular because many people are
interested in growing their own vegetables. She uses ten small glass panes and fifteen large
glass panes for small greenhouses. She uses fifteen small glass panes and forty-five large
glass panes for a large greenhouse. She has available sixty small glass panes and one hundred
thirty-five large glass panes.
If Marlene makes $390 profit on a small greenhouse and $520 on a large greenhouse, how
many of each type should she build to maximize profit?
a. Choose your variables:
a. x =
b. y =
b. Write inequalities for the constraints on the variables.
c. Write an equation of the Objective Function.
PLEASE GIVE YOUR TEACHER THIS SHEET. YOUR TEACHER WILL GIVE YOU
PART B OF THE TEST.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 56 of 58
UNIT 8
LINEAR PROGRAMMING TEST – CULMINATING PROBLEM
PART B
Name: ___________________________________________ Date: ______________
Greenhouse Business
1. Marlene builds greenhouses. Her greenhouses are popular because many people are
interested in growing their own vegetables. She uses ten small glass panes and fifteen large
glass panes for small greenhouses. She uses fifteen small glass panes and forty-five large
glass panes for a large greenhouse. She has available sixty small glass panes and one hundred
thirty-five large glass panes.
If Marlene makes a $390 profit on a small greenhouse and $520 on a large greenhouse, how
many of each type should she build to maximize profit?
The constraints for the variables are:
x0 y0
10x + 15y  60
15x + 45y  135
The Objective Function is:
P = 390x + 520y
Use the constraints and objective function above to complete the problem.
a. Use your calculator to graph the constraint inequalities.
b. Use the grid below to draw your graph and indicate the feasible region. Label your graph
and indicate scales on your axes.
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 57 of 58
c. Use your calculator to determine the corner points (vertices) of the feasible region.
d. For each corner point, determine the value of the Objective Function. Show your work.
e. Explain how you determined the number of small greenhouses and large greenhouse to
maximize your profit. Is the profit reasonable for Marlene’s business?
CT Algebra One for All
Unit 8, Culminating Experience 8 13 09
Page 58 of 58