Linear Programming
June 14, 2012
Notes
1. Pet Food Cost A pet owner is buying two brands of food, X and Y, for his animals. Each
serving of the mixture of foods should contain at least 60 grams of protein and 40 grams of
fat. Brand X costs 75 cents per box and Brand Y costs 50 cents per box. Each box of
Brand X contains 20 grams of protein and 10 grams of fat, whereas each box of Brand Y
contains 10 grams of protein and 10 grams of fat. Determine how many boxes of each
brand should be purchased to obtain a minimum cost per serving.
a. Define the variables.
b. Write the objective function.
c. Write the inequalities that describe the situation.
d. Graph and shade the feasible region. Find all corner points and label the coordinates on
the graph.
Pet Food
e. Determine how much of each brand should be bought to obtain a minimum cost per
serving. Show all mathematical work.
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Linear Programming
June 14, 2012
2. Carmella and Walt produce handmade shawls and afghans. They spin the yarn, dye it, and
then weave it. A shawl requires 1 hour of spinning, 1 hour of dyeing, and 1 hour of weaving.
An afghan needs 2 hours of spinning, 1 hour of dyeing, and 4 hours of weaving. Together,
they spend at most 8 hours spinning, 6 hours dyeing, and 14 hours weaving daily. They
make $10 profit per shawl and $20 profit per afghan. Determine the combination of afghans
and shawls that will maximize their profit. Below is a graph that may be used to answer this
question. Show all work.
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Linear Programming
June 14, 2012
3. Airlines An airline with two types of airplanes, P1 and P2, has contracted with a tour group
to provide accommodations for a minimum of 2000 first-class passengers, 1500 tourist
passengers, and 2400 economy-class passengers. Each airplane P1 costs $12,000 to
operate and can accommodate 40 first-class passengers, 40 tourist, and 120 economyclass passengers, whereas each airplane P2 costs $10,000 to operate and can
accommodate 80 first-class, 30 tourist, and 40 economy-class passengers. How many of
each type of airplane should be used to minimize the operating cost?
a. Define the variables.
b. Write the objective function for this situation.
c. Write the inequalities that describe the situation.
d. Shade the feasible region and circle all of the corner points (ON the graph). Solve for all
corner points that are not labeled and label them on the graph.
P2
(0,60)
Economy
(0,50)
Tourist
(0,25)
(20,0)
(37.5,0)
(50,0)
First Class
P1
e. Determine the combination of planes will minimize costs. Show your work.
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Linear Programming
June 14, 2012
4. Maximizing storage An office manager wants to buy filing cabinets. Cabinet X costs
$100, requires 6 square feet of floor space, and has the storage capacity of 8 cubic feet.
Cabinet Y costs $200, requires 8 square feet of floor space, and has the storage capacity of
12 cubic feet. No more than $1400 can be spent, and the office has room for no more than
72 square feet of cabinets. The office manager wants the maximum storage capacity within
the limits imposed by funds and space. How many of each type of cabinet should be
bought?
a. Define the variables.
b. Write the objective function.
c. Write the inequalities that describe the situation.
d. Graph and shade the solution set (feasible region).
Cabinet X and Cabinet Y
y
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e. Find all corner points and label the coordinates on the graph.
f.
Write a statement indicating how many of each type of cabinet should be bought to
maximize the storage capacity?.
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