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