Algebra 2
Name: ___________________________
Linear Programming
Period: ______ Date: _______________
Worked Example
The Juice Mixing Company has 240 quarts of cranberry juice and 120 quarts of apple juice. One (1) gallon of
cranapple juice is made from 3 quarts cranberry juice and 1 quart of apple juice. One gallon of appleberry is
made from 2 quarts of apple juice and 2 quarts of cranberry juice. The company can make a profit of 20 cents
on a gallon of cranapple and 50 cents on a gallon of appleberry. How many gallons of cranapple and how
many gallons of appleberry should be made to obtain the highest profit? What will this highest profit be?
Step 1: Assign variables x, y, and p
x=
y=
p=
Step 2: Make the mixture table
Step 3: Write the resource constraints & Situational Constraints as algebraic inequalities
Step 4: Graph the feasible region
Neatly and completely graph the linear inequalities from step 3. Make sure that you shade the appropriate
regions to correctly identify the feasible region. Locate and label all intersections and intercepts. Check
your work if necessary.
Step 5: Apply the Corner Point Principle to maximize/minimize the situation
a.
Write an equation relating total profit to the amount of resources used.
b.
Compute the profit for each corner point of the feasible region.
c.
Identify the point that maximizes/minimizes the scenario appropriately.
d.
Interpret the meaning of the corner point.
e.
Demonstrate (with arithmetic) that the resources available are adequate to make the optimal
production policy you have stated in answer c.
Avery 2009
Algebra 2
Name: ___________________________
Linear Programming
Period: ______ Date: _______________
Provide a neat and complete solution showing all work, steps, charts, graphs, labels, etc!
1. The famous high priced law firm of Dewey, Cheetim, and Howe wants to get as much money as
it can. DCH has two basic services: “Initial consultations” for a profit of $300 and “representing
you” for a profit of $600. “Initial consultations” require 1 hour of paralegal time and 1 hour of
lawyer time, while “representing you” requires 4 hours of paralegal time and 1 hour of lawyer
time. The firm has 20 hours of paralegal time available and 8 hours of lawyer time available daily.
How many of each type of service should be handled daily to maximize profit?
What is the maximum profit per day?
2. A toy manufacturer makes bikes and wagons. It requires 2 hours of machine time and 4 hours of
painting time to produce a bike. It requires 3 hours of machine time and 2 hours of painting time
to produce a wagon. There are 12 hours of machine time and 16 hours of painting time available
per day. The profit on bikes is $12 and the profit on wagons is $10.
How many bikes and wagons should be produced per day to maximize profit?
What is the maximum profit per day?
3. Websites-R-US maintains two types of web sites: “hot” and “cold”. “Hot” sites change their layout
frequently, but keep their content for a long time. “Cold” sites keep their layout consistent, but
often change their content. Daily maintenance of a “Hot” site requires 1.5 hours of layout time
and 1 hour for content changes. Daily maintenance of a “Cold” site requires 1 hour of layout time
and 2 hours for content changes. Each day the company has 12 hours for layout changes and 16
hours for content changes. Net profit is $50 for a set of changes on a “hot” site and $250 for a set
of changes on a “cold” site.
How many of each type of website should the company maintain daily to maximize profit?
What is the maximum profit per day?
Avery 2009