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Packet of linear programming word problems
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Name:___________________________
LINEAR PROGRAMMING
NOTES AND HOMEWORK
EXAMPLE 1: The Firm
It takes a tailoring firm 2 hours of cutting and 4 hours of sewing to make a knit suit. To make a wool suit, it takes 4 hours
of cutting and 2 hours of sewing. At most, 20 hours per day are available for cutting and 16 hours per day for sewing.
The profit on a knit suit is $34 and on a wool suit is $31. How many of each type of suit should be made to maximize
profit and what will that maximum profit be?
HELPFUL TOOL: The Box
Max or Min circle
Knit (x) Wool (y) Symbol Total
Cutting
Sewing
Profit
Objective Function:
The variables are at the top
of the box & fill down
Constraints:
*Hidden Constraints:
Knit (x)
Wo
ol
(y)
Answer:
They should make
_______ knit suits & ________ wool suits
For a maximum profit of $___________.
Feasible Region:
Vertices:
EXAMPLE 2: Papa Johns
John is hosting dinner for his large family. He has decided to buy pizzas. He needs at least 4 cheese pizzas and
6 pepperoni pizzas but needs at least 14 pizzas total. Each cheese pizza costs $8.00 and each pepperoni is $2 more. How
many pizzas of each type should he buy to minimize the cost of dinner?
Max or Min circle
Linear Programming Steps:
1. Identify Variables
2. Write constraints
(Remember the hidden ones)
and fill in box
3. Solve for y and graph
4. Shade feasible region
5. Find vertices of feasible
region
6. Substitute vertices into
objective function
7. Find max or min of objective
function
Cheese (x) Pepperoni (y) Symbol
Total Pizzas
Cost
Other Inequalities:
Cheese (x)
Pep
per
on
i (y
)
Answer:
John should order
_______ cheese &
________ pepperoni pizzas
For a minimum cost of
$___________.
PROBLEM 1: Its Snow Problem
A water ski manufacturing company makes two types of skis: trick and slalom. Trick skis require 6 hours in the
fabricating department and slalom skis require 4 hours. Each type of ski requires one hour in the finishing department.
There are a total of 108 hours available for fabricating and 24 hours for finishing. The profit on the trick skis is $40 and
on slalom is $30. How many of each type of ski should be produced to maximize profit?
Max or Min
circle
Trick Slalom Symbol Total
Trick Skis (x)
Answer:
They should make
_______ trick &
________ slalom skis
For a maximum profit of
$___________.
Sla
lom
Sk
is (
y)
Profit
Fabricate
Finish
PROBLEM 2: Keep Fishin
A private fishing resort has bass and trout in its lake. The owner provides two types of food, A and B, for these fish.
Each week, the bass require 2 units of food A and 4 units of food B. The trout require 5 units of food A and 2 units of
food B. The owner can only supply 800 units of each type of food on a weekly basis. What is the maximum number of
each type of fish the lake can support under these conditions?
Max or Min
circle
Bass Trout Symbol Total
Answer:
The lake can
support
_______ Bass
&
_______ Trout.
For _______ total
fish.
Tro
ut
# of Fish
Food A
Food B
Bass 20
20
400
400
PROBLEM 3: Dont Keep It Lowe
A tool company manufactures two types of drills, one cordless and the other corded. The cord-type drill requires 2 labor
hours to make and the cordless type requires 3. The company only has 600 labor hours available each day and the
packaging department can package no more than 250 drills each day. The cordless drill sells for a profit of $60 and the
corded for $45. How many of each type of drill should be made to maximize profit?
Max or Min
circle
Answer:
They should make
_______ corded
&
_______ cordless.
For a max profit of
$______________
Profit
Packaging
10
10
200
100
Corded Symbol Total
200
PROBLEM 4: Fly Away
An airline with two types of airplanes, X and Y, has contracted with a tour group to provide accommodations for a
minimum of 2000 first-class passengers, 1500 tourist-class passengers, and 2400 economy class passengers. Airplane X
costs $12,000 per plan to operate ad can accommodate 40 first-class, 40 tourist-class, and 120 economy class passengers.
Airplane Y can accommodate 80 first-class, 30 tourist-class and 40 economy-class passengers and costs $10,000 per plane
to operate. How many of each type of airplane should be used to minimize cost?
Max or Min
circle
Answer:
They should use
_______ type- X airplanes
&
_______ type Y- airplanes
For a minimum cost of
$_____________________
Cost
Economy
10
200
Symbol Total
Tourist
First-Class
5
5
Problem 5: Will You Choose the Red or Blue?
Neo is ill and decides to take vitamins to help speed up his recovery. Each day he must have at least 16 mg of vitamin A,
at least 5 mg of vitamin B, and at least 20 mg of vitamin C. He can choose between red pills that cost $0.10 each and
contain 8 mg of A, 1 mg of B, and 2 mg of C; or blue pills that contain 2 mg of A, 1 mg of B and 7 mg of C but cost $0.20
each. How many of each pill should Neo buy to satisfy the minimum daily requirements at minimum cost?
Max or Min
circle
10
Answer:
Robin should take
_______ Red vitamin pills
&
_______ Blue vitamin pills
For a minimum cost of
$_____________________
For a profit of
Cost
Vitamin C
10
200
Red Pill Blue Pill Symbol Total
Vitamin B
Vitamin A
1
1
Red Pill (x)
Blue Pill
Symbol
Total
Problem 6: So Need an Answer
Sony makes two types of television sets. It produces an LCD set that gives a profit of $100 or a plasma set
which sells for $150 profit. On the assembly line the LCD set requires 3 hours while the plasma set requires 5 hours. The
electronics department spends one hour on the LCD and 3 hours on the plasma. Both sets require 2 hours for testing and
packaging. On a particular production run, the company has 3900 work hours available for the assembly line, 2100 hours
in electronics and 2200 hours in the testing and packaging area. How many of each type of set should be made to
maximize profit?
Max or Min
circle
Answer:
They should produce
_______ LCD TVs
&
_______ Plasma TVs
For a maximum profit of
$____________________
For a profit of
$______________
Profit
10
200
50
50
LCD (x)
Blue Pill
Symbol
Total
Problem 7: McMe Some Money
A fast-food chain plans to expand by opening several new restaurants. The chain operates two types of
restaurants, drive thru and full-service. A drive-thru restaurant costs $100,000 to construct, requires 5 employees and has
an expected annual revenue of $200,000. A full-service restaurant costs $150,000 to construct, requires 15 employees and
has an expected annual revenue of $500,000. The chain has $2,400,000 available for construction costs. Labor contracts
require that they hire no more than 210 employees and licensing restrictions require that they open no more than 20 new
restaurants. How many restaurants of which type should the chain open in order to maximize their expected revenue?
Max or Min
circle
Answer:
They should open
_______ Drive-Thrus
&
_______ Full-Service Restaurants
For a maximum profit of
$_____________________.
They will use
$_______________ in capital and
Will hire ________ employees and
Open _________ restaurants.
For a profit of
Profit
10
200
Drive-Thru Full-Service Symbol Total
1
1
Problem 8: Soy Maize
A farmer has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, her
expenses are $50 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of
storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90. If the
total amount of storage space available is 19,200 bushels and the farmer has only $20,000 on hand, how many acres of
each crop should she plant in order to maximize her profit? What will her profit be if she follows this strategy?
Max or Min
circle
Answer:
He should plant
_______ acres of corn
&
_______ acres of soybeans
For a maximum profit of
$____________________
Profit
200
Hint: You may need
to solve for the
intersection as at
least one wont be
easy to tell from
the graph.
Problem 9: Its Electric
A plant makes aluminum and copper wire. Each pound of aluminum wire requires 5 kwh of electricity and 1/4 hr. of
labor. Each pound of copper wire requires 2 kwh of electricity and hr. of labor. Production of copper wire is restricted
by the fact that raw materials are available to produce at most 60 lbs./day. Electricity is limited to 500 kwh/day and labor
to 40 personhrs./day. If the profit from aluminum wire is $.25/lb. and the profit from copper is $.40/lb., how much of
each should be produced to maximize profit and what is the maximum profit?
Max or Min
circle
10
Answer:
They should product
_______ lbs. of aluminum
&
_______ lbs. of copper
For a maximum profit of
$____________________
200
Other constraint:
Problem 10: Burning for an Answer
TAE Electronics manufactures portable tape players and CD players. The manufacturing plant has the capacity to
manufacture at most 750 tape players and 500 CD players in one month. Combined, they can only manufacture 900
products. It takes 2 hours to make a tape player and 5 hours to make a CD player. The company can spend no more than
3000 hours manufacturing these products. TAE Electronics makes $4 profit on tape players and $7 profit on CD players.
To maximize profits, how many tape players and how many CD players should they make? What is the maximum profit?
Max or Min
circle
Answer:
They should make
_______ tape players
&
_______ CD players
For a maximum profit of
$____________________
200
Other:
Problem 11: Soarin
Charles is chief mathematician for Fly-By-Night Aircraft Corp. He is responsible for mathematical analysis of
the manufacturing of the company's two models of planes, the Eagle and the Hippo. The production department can make
no more than 7 Hippos and 11 Eagles. The shipping department can move no more than 12 planes total per day. The sales
department can sell no more than twice the number of Hippos than the number of Eagles. The personnel department must
use more than 800 man-hours of labor per day (it takes 100 man-hours to manufacture each Eagle and 200 for each
Hippo.) If one Eagle makes $300 in profit and one Hippo makes $200 in profit, how many of each should they make to
maximize profit?
Max or Min
circle
Answer:
They should make
_______ Eagles
&
_______ Hippos
For a maximum profit of
$____________________
200
Note: 3 of the constraints may
not be able to be placed in the
box. Write an inequality for it
here.
Eagles (x)
Blue Pill
Symbol
B ONUS LINEAR PROGRAMMING PROBLEM
Problem 12: Dont Gas
The manufacturing process requires that oil refineries must manufacture at least 2 gal of gasoline for every gallon of fuel
oil. To meet the winter demand for fuel oil, at least 3 million gal a day must be produced. The demand for gasoline is no
more than 6.4 million gal per day. It takes .25 hour to ship each million gal of gasoline and 1 hour to ship each million gal
of fuel oil out of the warehouse. No more than 4.65 hours are available for shipping. If the refinery sells gasoline for $1.25
per gal and fuel oil for $1 per gal, how much of each should be produced to maximize revenue? Find the maximum
revenue.
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