1. LINEARPROGRAMMING
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2. Example 1 (-5, 3) (4, 3) (-5, -1) (4, -1) 5/23/2018 11:13 AM
3.5 Linear Programming 2 2 2

3. Definitions Optimization is finding the minimum and maximum value
For the most part, optimization involves point, P
Steps in Linear Programming
1. Find the vertices by graphing
2. Plug the vertices into the P equation, which is given
3. Find the minimum and maximum optimization values of P
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3.4 Linear Programming 3

4. Linear Programming is a method of finding a maximum or minimum value of a function that satisfies a set of conditions called constraints
A constraint is one of the inequalities in a linear programming problem.
The solution to the set of constraints can be graphed as a feasible region.

5. Optimization
A Haunted House is opened from 7pm to 4am. Look at this graph and determine the maximization and minimization of this business.
MAXIMIZATION
MINIMIZATION
MINIMIZATION
7p p p p p a a a a a
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6. Example 1 Given Find the minimum and maximum for equation, Step 1:
Find the vertices by graphing
(-5, 3)
(4, 3)
(-5, -1)
(4, -1)
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7. Example 1 vertices P = –2x + y profit (-5, 3) (4, 3) (4, -1) (-5, -1)
Given Find the minimum and maximum for equation,
Step 2: Plug the vertices into the P equation, which is given
vertices
P = –2x + y
profit
(-5, 3)
(4, 3)
(4, -1)
(-5, -1)
P = -2(-5) + (3)
P = 13
P = -2(4) + (3)
P = –5
P = -2(4) + (-1)
P = –9
P = -2(-5) + (-1)
P = 9
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8. Example 1 Minimum: –9 @ (4,-1) 13 @ (-5,3) Maximum:
Given Find the minimum and maximum for equation,
Step 3: Find the minimum and maximum optimization values of P
vertices
P = -2x + y
Profit
(-5, 3)
(4, 3)
(4, -1)
(-5, -1)
P = -2(-5) + (3)
P = 13
P = -2(4) + (3)
P = –5
P = -2(4) + (-1)
P = –9
P = -2(-5) + (-1)
P = 9
Minimum:
(4,-1)
Maximum:
(-5,3)
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5/23/ :13 AM
3.5 Linear Programming

9. Given Find the minimum and maximum optimization for equation,
Example 2
Given Find the minimum and
maximum optimization for equation,
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10. Example 2 39 10 Minimum: 10 @ (2,1) Maximum: 39 @ (5,6)
Given Find the minimum and maximum for
equation,
Maximum:
(5,6)
Vertices
P = 3x+4y
Profit
(2, 6)
P = 3(2) + 4(6) 30
(5, 6)
P = 3(5) + 4(6) 39
(2, 1)
P = 3(2) + 4(1) 10
(5, 1)
P = 3(5) + 4(1) 19
(2, 6)
(5, 6)
(2, 1)
(5, 1)
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3.5 Linear Programming 10

11. Example 3 Given Find the minimum and maximum for equation, Vertices:
y ≥ 1.5
2.5x + 5y ≤ 20
3x + 2y ≤ 12
Given Find the minimum and maximum
for equation,
(0, 4)
(2, 3)
Vertices:
(0, 4), (0, 1.5), (2, 3), and (3, 1.5)
(0, 1.5)
(3, 1.5)
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12. Example 3 45 140 Given Find the minimum and maximum for equation,
y ≥ 1.5
2.5x + 5y ≤ 20
3x + 2y ≤ 12
Given Find the minimum and maximum
for equation,
(x, y)
25x + 30y
P($)
(0, 4)
25(0) + 30(4)
120
(0, 1.5)
25(0) + 30(1.5) 45
(2, 3)
25(2) + 30(3)
140
(3, 1.5)
25(3) + 30(1.5)
(0, 4)
(2, 3)
(0, 3/2)
(3, 3/2)
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13. Your Turn Given Find the minimum and maximum for equation, Step 1:
(0, 2)
Step 1:
Find the vertices by graphing
(0, 0)
(2, 0)
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14. Your Turn vertices P = x + 2y profit (0, 2) (0, 0) (2, 0)
Given Find the minimum and maximum for equation,
Step 2: Plug the vertices into the P equation, which is given
vertices
P = x + 2y
profit
(0, 2)
(0, 0)
(2, 0)
P = (0) + 2(2)
P = 4
P = (0) + 2(0)
P = 0
P = (2) + 2(0)
P = 2
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15. Example 4 24 4 (0, 8) (0, 2) 6 (2, 0) (4, 0) 8 (0, 8) P = 2(0) + 3(8)
Given Find the minimum and maximum for equation,
(0, 8)
vertices
P = 2x + 3y
profit
(0, 8)
P = 2(0) + 3(8) 24
(0, 2)
P = 2(0) + 3(2) 6
(2, 0)
P = 2(2) + 3(0) 4
(4, 0)
P = 2(4) + 3(0) 8
(0, 2)
(2, 0)
(4, 0)
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16. Example 5
A charity is selling T-shirts in order to raise money. The cost of a T-shirt is $15 for adults and $10 for students. The charity needs to raise at least $3000 and has only 250 T-shirts. Write and graph a system of inequalities that can be used to determine the number of adult and student T-shirts the charity must sell.
Let a = adult t-shirts
Let b = student t-shirts

17. Warm-up
Sue manages a soccer club and must decide how many members to send to soccer camp.
It costs $75 for each advanced player and $50 for each intermediate player.
Sue can spend no more than $13,250.
Sue must send at least 60 more advanced than intermediate players and a minimum of 80 advanced players.
Find the number of each type of player Sue can send to camp to maximize the number of players at camp.
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18. Example 6 MAKE a TABLE to show your work for the objective function
x = the number of advanced players,
y = the number of intermediate players.
x ≥ 80
The number of advanced players is at least 80.
The number of intermediate players cannot be negative.
y ≥ 0
There are at least 60 more advanced players than intermediate players.
x – y ≥ 60
The total cost must be no more than $13,250.
75x + 50y ≤ 13,250
Let P = the number of players sent to camp.
The objective function is P = x + y.
MAKE a TABLE to show your work for the objective function
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19. Example 6 P(80, 0) = (80) + (0) = 80 P(80, 20) = (80) + (20) = 100
Graph the feasible region, and identify the vertices. Evaluate the objective function at each vertex.
P(80, 0) = (80) + (0) = 80
P(80, 20) = (80) + (20) = 100
P(176, 0) = (176) + (0) = 176
P(130,70) = (130) + (70) = 200
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20. Example 6 Check the values (130, 70) in the constraints.  x ≥ 80
y ≥ 0
130 ≥ 80 
70 ≥ 0 
x – y ≥ 60
75x + 50y ≤ 13,250
(130) – (70) ≥ 60
75(130) + 50(70) ≤ 13,250
60 ≥ 60 
13,250 ≤ 13,250 
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21. Assignment
Pg 202: odd, 20, 29, 31 (no need to identify the shape from 16-19)
Pg 209: 9-21 odd
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