Applications of Linear
Programming
Section 3-6
(a.k.a. STORY PROBLEMS)
Review
• Graph a system of inequalities.
• Find the vertices of the feasible region.
• Determine which lines are intersecting.
• Solve that system of equations.
• Plug the vertices into the objective
function to find the maximum and minimum
values.
How will the applications
be different?
• Your problems will be
in paragraph form.
• You will have to create
your own system of
inequalities.
• You also get to create
your own function.
Not to worry, I have
a plan!
Step 1
• Define your variables.
• Look at the last sentence in your
problem.
• Write “x =“ and “y =“ as part of
your work.
Step 2
• Organize the given information in a
chart.
Step 3
• Write a system of inequalities to
represent all of the limitations.
• Can the items in my problem be
negative?
• Does the chart show me
anything?
Step 4
• Graph the system of inequalities and
find the vertices of the feasible
region.
Step 5
• Write a function to be maximized or
minimized.
• Does the chart tell you anything?
• Often the function represents
cost, revenue, or profit.
Step 6
• Test your points and answer the
question.
• SENTENCES!!
Sounds easy, ready to
try?
The AC Telephone Company manufactures two
styles of cordless telephones, deluxe and
standard. Each deluxe phone nets the company
$9 in profit and each standard phone nets $6.
Machines A and B are used to make both styles
of telephones. Each deluxe model requires 3
hours of machine A time and 1 hour on B. Each
standard phone requires 2 hours on both
machines. If the company has 12 hours
available on machine A and 8 hours available on
B, determine the mix of phones that will
maximize the company’s profit.
“determine the mix of
phones that will maximize
the company’s profit”
• Let x = # of deluxe phones.
• Let y = # of standard phones.
Make a chart.
Profit
# Deluxe
x
#
Standard
y
$9
Machine A Machine B
Time
Time
3
1
$6
2
12
2
8
System of Inequalities
• Can I make negative phones?
• Does my table help?
x≥0
y≥0
3x + 2y ≤ 12
x + 2y ≤ 8
Graph and find the
vertices.
“maximize the company’s
profit”
P(x, y) = 9x + 6y
P(0, 0) = 9(0) + 6(0) = 0
P(0, 4) = 9(0) + 6(4) = 24
P(2, 3) = 9(2) + 6(3) = 36
P(4, 0) = 9(4) + 6(0) = 36
The company should either
make 2 deluxe phones and 3
standard phones or 4 deluxe
phones and 0 standard phones.
If you are the CEO, which
option would you choose?
Why?