1. LINEARPROGRAMMING Day 2
Section 3.4
LINEARPROGRAMMING Day 2
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2. Steps of Problem Solving
Understand the problem
Translate the problem
Solve
List all of your restraints
Determine your Objective Equation (usually dealing with Profit)
Use Cover-up to determine the intercepts
Use Elimination/Substitution to determine the intersection points
Check
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3. 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.
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4. Example 1 X = Cases of Almonds Y = 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.
X = Cases of Almonds
Y = Cases of Walnuts
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5. Example 1 X = Cases of Almonds Y = 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.
X = Cases of Almonds
Y = Cases of Walnuts
(0, 12.5) Using Cover Up
(9, 5) Using Elimination
(0, 0)
(13.3, 0) Using Cover Up
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6. Example 1 X = Cases of Almonds Y = Cases of Walnuts vertices
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 Almonds
Y = Cases of Walnuts
vertices
C = 17x + 15y
Profit
(0, 0)
C = 17(0) + 15(0)
C = 0
(0, 12.5)
C = 17(0) + 15(12.5)
C = $187.5
(13.3, 0)
C = 17(13.3) + 15(0)
C = $226.1
(9, 5)
C = 17(9) + 15(5)
C = $228
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7. Example 1 X = Cases of Almonds Y = 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.
X = Cases of Almonds
Y = Cases of Walnuts
How many cases of almonds and walnuts maximize the grocer’s profit?
9 cases of almonds and 5 cases of walnuts help maximize the grocer’s profit.
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8. Example 2
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.
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9. Example 2 X = Small Buses Y = Big Buses
A 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
Big Buses
(0,9)
(9,0)
(0,8)
(10,0)
Small Buses
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10. Example 2 X = Small Buses Y = Big Buses
A 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
(0, 9) Using Cover Up
Y = Big Buses
Big Buses
(0, 8) Using Cover Up
(5, 4) Using Elimination
Small Buses
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11. Example 2 X = Small Buses Y = Big Buses Vertices C = 600x + 800y
A 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
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)
(0, 9)
C = 600(0) + 800(9)
(5, 4)
C = 600(5) + 800(4)
Vertices
C = 600x + 800y
Max/Min
(0, 8)
(0, 9)
(5, 4)
$6400
$7200
$6,200
The school should rent 4 large buses and 5 small buses for the least possible cost of $6200
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