Do This Problem Right Now

x  0
y  0

Given 
Find the minimum and maximum for equation, C
2 x  y  8

2 x  2 y  4
 2 x  3 y.
(0, 8)
vertices
C = 2x + 3y
Min/Max
(0, 8)
C = 2(0) +
3(8)
24
(0, 2)
C = 2(0) +
3(2)
6
(0, 2)
(2, 0)
C = 2(2) +
3(0)
4
(4, 0)
C = 2(4) +
3(0)
8
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(2, 0)
(4, 0)
1
Section 3.4, Revised 2011
LINEARPROGRAMMING
Day 2
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2
Steps for solving Real Life Linear
Programming Problems
1. Solve
a)
b)
c)
d)
List all of your restraints
Determine your Objective Equation (usually dealing with
Profit)
Find the x-intercept (y=0)
and the y-intercept (x =0)
Use Cover-up method to determine the intercepts
Use Elimination/Substitution to determine the
intersection points of the 2 equations
2. 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
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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 x  0 P  17 x  15 y
y  0


 30 x  26 y  400

20 x  24 y  300
5
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 Almonds
Y = Cases of Walnuts
x  0
(0, 12.5)
Using Cover Up
y  0


30 x  26 y  400

20 x  24 y  300
C  17 x  15 y
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(0, 0)
(9, 5) Using Elimination
(13.3, 0) Using Cover Up
6
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 Almonds
Y = Cases of Walnuts P  17 x  15 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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7
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 Almonds
Y = Cases of Walnuts
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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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
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
x0
y0

x y 9

 40 x  50 y  400
C  600 x  800 y
Big Buses
Y = Big Buses
Small Buses
10
Example 2
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
x  0
y0

8)
 x  y  9 Using(0,Cover
Up

 40 x  50 y  400
C  600 x  800 y
Big Buses
Y = Big Buses
(0, 9)
Using Cover Up
(5, 4) Using Elimination
Small Buses
11
Example 2
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
The school should rent 4 large buses and
5 small buses for the least possible cost of $6, 200.
x  0
y  0


x  y  9

40 x  50 y  400
C  600 x  800 y
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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
12