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Do This Problem Right Now. Given Find the minimum and maximum for equation,. (0, 8). (0, 2). (2, 0). (4, 0). Section 3.4, Revised 2011. LINEAR PROGRAMMING Day 2. Steps for solving Real Life Linear Programming Problems. Solve List all of your restraints - PowerPoint PPT Presentation
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Do This Problem Right Now
Given Find the minimum and maximum for equation,
0
0
2 8
2 2 4
x
y
x y
x y
2 3 .C x y
(0, 8)
(4, 0)
(0, 2)
(2, 0)
vertices C = 2x + 3y Min/Max
(0, 8) C = 2(0) + 3(8) 24
(0, 2) C = 2(0) + 3(2)
6
(2, 0) C = 2(2) + 3(0) 4
(4, 0) C = 2(4) + 3(0)
8
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LINEARPROGRAMMINGDay 2
Section 3.4, Revised 2011Section 3.4, Revised 2011
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Steps for solving Real Life Linear Programming Problems
1. Solvea) List all of your restraintsb) Determine your Objective Equation (usually dealing with
Profit)c) Find the x-intercept (y=0)
and the y-intercept (x =0) Use Cover-up method to determine the intercepts
d) Use Elimination/Substitution to determine the intersection points of the 2 equations
2. Check
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A grocer buys cases of almonds and walnuts. Almonds are packaged 20 bags per case. The grocer pays $30 per case of almonds and makes a profit of $17 per case. Walnuts are packaged 24 bags per case. The grocer pays $26 per case of walnuts and makes a profit of $15 per case. He orders no more than 300 bags of almonds and walnuts together at a maximum cost of $400. Use x for cases of almonds and y for cases of walnuts.
Example 1
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Example 1
A grocer buys cases of almonds and walnuts. Almonds are packaged 20 bags per case. The grocer pays $30 per case of almonds and makes a profit of $17 per case. Walnuts are packaged 24 bags per case. The grocer pays $26 per case of walnuts and makes a profit of $15 per case. He orders no more than 300 bags of almonds and walnuts together at a maximum cost of $400. Use x for cases of almonds and y for cases of walnuts.
X = Cases of AlmondsY = Cases of Walnuts
0x
0
0
x
y
0
0
30 26 400
x
y
x y
0
0
30 26 400
20 24 300
x
y
x y
x y
17 15P x y
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Example 1
A grocer buys cases of almonds and walnuts. Almonds are packaged 20 bags per case. The grocer pays $30 per case of almonds and makes a profit of $17 per case. Walnuts are packaged 24 bags per case. The grocer pays $26 per case of walnuts and makes a profit of $15 per case. He orders no more than 300 bags of almonds and walnuts together at a maximum cost of $400. Use x for cases of almonds and y for cases of walnuts.
X = Cases of AlmondsY = Cases of Walnuts
0
0
30 26 400
20 24 300
x
y
x y
x y
17 15C x y (0, 0)
(0, 12.5)Using Cover Up
(13.3, 0) Using Cover Up
(9, 5) Using Elimination
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A grocer buys cases of almonds and walnuts. Almonds are packaged 20 bags per case. The grocer pays $30 per case of almonds and makes a profit of $17 per case. Walnuts are packaged 24 bags per case. The grocer pays $26 per case of walnuts and makes a profit of $15 per case. He orders no more than 300 bags of almonds and walnuts together at a maximum cost of $400. Use x for cases of almonds and y for cases of walnuts.
Example 1
17 15P x y
vertices P= 17x + 15y Profit
(0, 0) P = 17(0) + 15(0) P = 0
(0, 12.5) P = 17(0) + 15(12.5) P = $187.50
(13.3, 0) P = 17(13.3) + 15(0) P = $226.10
(9, 5) P = 17(9) + 15(5) P = $228
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X = Cases of AlmondsY = Cases of Walnuts
A grocer buys cases of almonds and walnuts. Almonds are packaged 20 bags per case. The grocer pays $30 per case of almonds and makes a profit of $17 per case. Walnuts are packaged 24 bags per case. The grocer pays $26 per case of walnuts and makes a profit of $15 per case. He orders no more than 300 bags of almonds and walnuts together at a maximum cost of $400. Use x for cases of almonds and y for cases of walnuts.
Example 1
How many cases of almonds and walnuts maximize the grocer’s profit?
The grocer should buy 9 cases of almonds and 5 cases of walnuts to have a maximum profit of $228.
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X = Cases of AlmondsY = Cases of Walnuts
A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rental cost for a large bus is $800 and $600 for the small bus. Calculate how many buses of each type should be used for the trip for the least possible cost.
Example 2
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Example 2A school is preparing a trip for 400 students. The company who is providing the transportation has 8 small buses of 40 seats each and 10 big buses of 50 seats each but only has 9 drivers available. The rental cost is for $600 for the small bus and $800 for a large bus . Calculate how many buses of each type should be used for the trip for the least possible cost.
X = Small Buses
Y = Big Buses
Small Buses
Big
Bu
se
s
1010
0x0y
9x y 40 50 400x y
600 800C x y
Example 2A school is preparing a trip for 400 students. The company who is providing the transportation has 8 small buses of 40 seats each and 10 big buses of 50 seats each but only has 9 drivers available. The rental cost is for $600 for the small bus and $800 for a large bus . Calculate how many buses of each type should be used for the trip for the least possible cost.
X = Small BusesY = Big Buses
Small Buses
Big
Bu
se
s
1111
(0, 9)Using Cover Up
(5, 4) Using Elimination(0, 8)
Using Cover Up
0x0y
9x y 40 50 400x y
600 800C x y
Example 2A school is preparing a trip for 400 students. The company who is providing the transportation has 8 small buses of 40 seats each and 10 big buses of 50 seats each but only has 9 drivers available. The rental cost is for $600 for the small bus and $800 for a large bus . Calculate how many buses of each type should be used for the trip for the least possible cost.
X = Small BusesY = Big Buses
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Vertices C = 600x + 800y Max/Min
(0, 8)
(0, 9)
(5, 4)
Vertices C = 600x + 800y Max/Min
(0, 8) C = 600(0) + 800(8)
(0, 9) C = 600(0) + 800(9)
(5, 4) C = 600(5) + 800(4)
Vertices C = 600x + 800y Max/Min
(0, 8) C = 600(0) + 800(8) $6,400
(0, 9) C = 600(0) + 800(9) $7,200
(5, 4) C = 600(5) + 800(4) $6,200
The school should rent 4 large buses and
5 small buses for the least possible cost of $6,200.
1212
0
0
9
40 50 400
600 800
x
y
x y
x y
C x y