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Linear Programming
Ex1) Use the system of constraints below to maximize the objective function z = -0.4
x + 3.2
y .
 x  0





 y  0 x  5 x  y  7 x  2 y  4 y  x  5
Corner Points

Linear Programming
Ex1) Use the system of constraints below to maximize the objective function z = -0.4
x + 3.2
y .
( x , y ) z = -0.4
x + 3.2
y
(0,2)
(0,5)
(1,6)
(4,0)
(5,0)
(5,2)


Linear Programming
Ex2) Use the system of constraints below to minimize the objective function z = 2 x + 3 y .
 x  0
 y  0
 x  6
 y  5
 x  y  2
Corner Points

Linear Programming
Ex2) Use the system of constraints below to minimize the objective function z = 2 x + 3 y .
( x , y )
(0,2)
(0,5)
(2,0)
(6,0)
(6,5) z = 2 x + 3 y

Linear Programming
Ex3) A calculator company makes scientific and graphing calculators. There is a demand for at least 100 scientific and
80 graphing calculators a day; however, due to materials, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, at least 200 calculators must be shipped each day. If each scientific calculators results in a loss of $2, but each graphing calculator gains a profit of $5, how many of each type should be made to maximize profit?
a) Identify the system of constraints and the objective function.


 x = sci calcs y = graph calcs



Linear Programming b) Graph the system of constraints to find the corner points
Corner Points
 x  100
 y  80
 x  200
 y  170
 x  y  200

Linear Programming c) Plug in the corner points to determine what amount of each type of calculator would maximize profit.
( x , y )
(100,100)
(100,170)
(120,80)
(200,80)
(200,170)
P = -2 x + 5 y

Linear Programming
Ex4) 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?