3.4 Linear Programming
optimization: finding the maximum or minimum value of some quantity
linear programming: the process of maximizing or minimizing a linear objective function
subject to constraints that are linear inequalities
constraints: graphing restrictions
feasible region: the graph of the system of constraints
Ex: Find the minimum value and the maximum value of the objective
function C = 5x – 2y subject to the following constraints.
x !0
y !0
2x + y " 8
x + 3y " 9
minimum value: ____________
maximum value: ____________
Ex: F U ND RA IS ER Your class plans to raise money by selling T-shirts and baseball caps.
The plan is to buy the T-shirts for $8 and sell them for $12 and to buy the caps for $4
and sell them for $7. The planning committee estimates that you will not sell more than
120 items. Your class can afford to spend as much as $800 to buy the articles. The
constraints on your fun-raising activity are given by the system of inequalities below.
Your class can only sell combinations of T-shirts and caps indicated by points that are
solutions to the system.
c + t ! 120
4c + 8t ! 800
c "0
t "0
a) Your club wants to maximize profit. Write the profit functions p in terms of c and t.
b) At which point do you think the maximum value of p will occur?
c) Which combination of baseball caps and T-shirts maximizes profit? What is the
maximum profit?
Ex: PIN A TA S Piñatas are made to sell at a craft fair. It takes 2 hours to make a mini
piñata and 3 hours to make a regular-sized piñata. The owner of the craft booth will
make a profit of $12 for each mini piñata sold and $24 for each regular-sized piñata
sold. If the craft booth owner has no more than 30 hours available to make piñatas
and wants to have at least 12 piñatas to sell, how many of each size piñata should be
made to maximize profit?