Using Mathematics to Solve Real World Problems
Transcript of Using Mathematics to Solve Real World Problems
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Using Mathematics to Solve Real World Problems
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Creating a mathematical model:
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Creating a mathematical model:
• We are given a word problem
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Creating a mathematical model:
• We are given a word problem
• Determine what question we are to answer
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Creating a mathematical model:
• We are given a word problem
• Determine what question we are to answer
• Assign variables to quantities in the problem so that you can answer the question using these variables
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Creating a mathematical model:
• We are given a word problem
• Determine what question we are to answer
• Assign variables to quantities in the problem so that you can answer the question using these variables
• Derive mathematical equations containing these variables
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Creating a mathematical model:
• We are given a word problem
• Determine what question we are to answer
• Assign variables to quantities in the problem so that you can answer the question using these variables
• Derive mathematical equations containing these variables
• Use these equations to find the values of these variables
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Creating a mathematical model:
• We are given a word problem
• Determine what question we are to answer
• Assign variables to quantities in the problem so that you can answer the question using these variables
• Derive mathematical equations containing these variables
• Use these equations to find the values of these variables
• State the answer to the problem
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Creating a mathematical model:
• We are given a word problem
• Determine what question we are to answer
• Assign variables to quantities in the problem so that you can answer the question using these variables
• Derive mathematical equations containing these variables
• Use these equations to find the values of these variables
• State the answer to the problem
Today we will do this using straight lines as our equations, and we will solve the problem by drawing these lines (graphing).
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Creating a mathematical model:
• We are given a word problem
• Determine what question we are to answer
• Assign variables to quantities in the problem so that you can answer the question using these variables
• Derive mathematical equations containing these variables
• Use these equations to find the values of these variables
• State the answer to the problem
Today we will do this using straight lines as our equations, and we will solve the problem by drawing these lines (graphing).
This process is called “Linear Programming” and is one of the most powerful mathematical methods used by businesses and companies to solve problems and help them make the best decisions.
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Creating a mathematical model:
• We are given a word problem
• Determine what question we are to answer
• Assign variables to quantities in the problem so that you can answer the question using these variables
• Derive mathematical equations containing these variables
• Use these equations to find the values of these variables
• State the answer to the problem
Today we will do this using straight lines as our equations, and we will solve the problem by drawing these lines (graphing).
This process is called “Linear Programming” and is one of the most powerful mathematical methods used by businesses and companies to solve problems and help them make the best decisions.
“Operations Research” is the profession that applies mathematical methods like this to problems arising in industry, healthcare, finance, etc.
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A problem:
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A problem:
A furniture manufacturer produces two sizes of boxes (large, small) that are used to make either a table or a chair.
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A problem:
A furniture manufacturer produces two sizes of boxes (large, small) that are used to make either a table or a chair.
large block
small block
table
chair
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A problem:
A furniture manufacturer produces two sizes of boxes (large, small) that are used to make either a table or a chair.
A table makes $3 profit and a chair makes $5 profit.
large block
small block
table
chair
$3
$5
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A problem:
A furniture manufacturer produces two sizes of boxes (large, small) that are used to make either a table or a chair.
A table makes $3 profit and a chair makes $5 profit.
If M small blocks and N large blocks are produced, how many tables and chairs should the manufacturer make in order to obtain the greatest profit?
large block
small block
table
chair
$3
$5M
N
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Large block
Small block
Table1 large block1 small block
Chair1 large block2 small blocks
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Table Chair
Profit: $3 $5
Problem: Given M small blocks and N large blocks, how many tables and chairs should we make to obtain the most profit?
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Example: 12 small blocks
and 12 large blocks 12
12
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We can make 4 tables and 4 chairs:
Example: 12 small blocks
and 12 large blocks 12
12
4 4
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We can make 4 tables and 4 chairs:
4 large4 small
4 large8 small
Example: 12 small blocks
and 12 large blocks
8 large12 small
12
12
4 4
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We can make 4 tables and 4 chairs:
4 large4 small
4 large8 small
Example: 12 small blocks
and 12 large blocks
8 large12 small
Profit = ($3) x 4 + ($5) x 4 = $32
12
12
4 4
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4 tables and 4 chairs:
12 small blocks
12 large blocks 12
12
4 4Used: 8 large blocks12 small blocks(4 large blocks left)
$3 $5
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4 tables and 4 chairs:
12 small blocks
12 large blocks
Profit = ($3) x 6 + ($5) x 3 = $33
12
12
4 4Used: 8 large blocks12 small blocks(4 large blocks left)
I can make 2 more tables if I make 1 less chair; 3 chairs and 6 tablesincrease my profit! (1 chair 2 tables, profit goes up by $1)
$3 $5
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6 tables and 3 chairs:
12 small blocks
12 large blocks 12
12
6 3Used: 9 large blocks12 small blocks(3 large blocks left)
$3 $5
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6 tables and 3 chairs:
12 small blocks
12 large blocks
Profit = ($3) x 8 + ($5) x 2 = $34
12
12
6 3Used: 9 large blocks12 small blocks(3 large blocks left)
I can do it again; change one chair into 2 tables; 8 tables and 2 chairs
$3 $5
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8 tables and 2 chairs:
12 small blocks
12 large blocks 12
12
8 2Used: 10 large blocks12 small blocks(2 large blocks left)
$3 $5
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8 tables and 2 chairs:
12 small blocks
12 large blocks
Profit = ($3) x 10 + ($5) x 1 = $35
12
12
8 2Used: 10 large blocks12 small blocks(2 large blocks left)
I can do it again; change one chair into 2 tables; 10 tables and 1 chair
$3 $5
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10 tables and 1 chair:
12 small blocks
12 large blocks 12
12
10 1Used: 11 large blocks12 small blocks(1 large block left)
$3 $5
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10 tables and 1 chair:
12 small blocks
12 large blocks
Profit = ($3) x 12 + ($5) x 0 = $36
12
12
10 1Used: 11 large blocks12 small blocks(1 large block left)
I can do it again; change one chair into 2 tables; 12 tables and 0 chairs
$3 $5
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12 tables and 0 chairs:
12 small blocks
12 large blocks
Profit = $36
12
12
12Used: 12 large blocks12 small blocks(no blocks left)
$3
Is this the best?
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How many tables and chairs?
20 small blocks
12 large blocks12
20
$3 $5
Now you try:
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How many tables and chairs?
25 small blocks
12 large blocks12
20
$3 $5
Another:
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tables 0 2 3 4 6 8 10 11 12
chairs 6 5 4 4 3 2 1 0 0
profit 30 31 29 32 33 34 35 33 36
small = 12large = 12
tables 2 4 5 6 7 8 10 11 12
chairs 9 8 7 6 5 4 2 1 0
profit 51 52 50 48 46 44 40 38 36
small = 20large = 12
tables 0 4 5 6 7 8 10 11 12
chairs 12 8 7 6 5 4 2 1 0
profit 60 52 50 48 46 44 40 38 36
small = 25large = 12
Not many small blocks
Not many small blocks
Many small blocks
table chair$3 $5
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Summary M = # small blocks
N = # large blocks
Table
Chair
N
M
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Two cases: M = # small blocks
N = # large blocks
Table
Chair
N
M
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Two cases:
1. Many small blocks: M >= 2N
M = # small blocks
N = # large blocks
Table
Chair
N
M
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Two cases:
1. Many small blocks: M >= 2N
make N chairs, 0 tables
M = # small blocks
N = # large blocks
Table
Chair
N
M
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Two cases:
1. Many small blocks: M >= 2N
make N chairs, 0 tables
M = # small blocks
N = # large blocks
2. Not many small blocks: M < 2N
Table
Chair
N
M
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Two cases:
1. Many small blocks: M >= 2N
make N chairs, 0 tables
M = # small blocks
N = # large blocks
2. Not many small blocks: M < 2N
mixture of tables and chairs . . . .
Table
Chair
N
M
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Two cases:
1. Many small blocks: M >= 2N
make N chairs, 0 tables
M = # small blocks
N = # large blocks
2. Not many small blocks: M < 2N
mixture of tables and chairs . . . .
What is the magic number of tables and chairs??
Table
Chair
N
M
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Two cases:
1. Many small blocks: M >= 2N
make N chairs, 0 tables
M = # small blocks
N = # large blocks
2. Not many small blocks: M < 2N
mixture of tables and chairs . . . .
What is the magic number of tables and chairs??
Table
Chair
N
M
Let’s make a mathematical model to find out
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M = # small blocks
N = # large blocks
Table
Chair
N
M
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M = # small blocks
N = # large blocks
Table
Chair
N
MX = # tables builtY = # chairs built
Our variables
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M = # small blocks
N = # large blocks
Table
Chair
N
MX = # tables builtY = # chairs built
How many can I build?
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M = # small blocks
N = # large blocks
Table
Chair
N
MX = # tables builtY = # chairs built
How many can I build?
X + 2Y <= M; I have only M small blocks
X + Y <= N; I have only N large blocks
Our equations
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M = # small blocks
N = # large blocks
Table
Chair
N
MX = # tables builtY = # chairs built
How many can I build?
X + 2Y <= M; I have only M small blocks
X + Y <= N; I have only N large blocks
Our equations
(Note: these are inequalities, not equalities!)
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M = # small blocks
N = # large blocks
Table
Chair
N
MX = # tables builtY = # chairs built
How many can I build?
X + 2Y <= M; I have only M small blocks
X + Y <= N; I have only N large blocks
Let’s plot all possible choices for X and Y for a given M and N,and then we’ll pick the X,Y that gives the greatest profit.
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M = # small blocks
N = # large blocks
Table
Chair
N
MX = # tables builtY = # chairs built
How many can I build?
X + 2Y <= M; I have only M small blocks
X + Y <= N; I have only N large blocks
Let’s plot all possible choices for X and Y for a given M and N,and then we’ll pick the X,Y that gives the greatest profit.
Solving our equations . . .
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Intermission: A primer on linear equations
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Intermission: A primer on linear equations
A linear equation: aX + bY = c, or Y = mX + b, a,b,c,m numbers, X,Y variables
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Intermission: A primer on linear equations
A linear equation: aX + cY = d, or Y = mX + b, a,b,c,d,m numbers, X,Y variables
If we plot all the X,Y that satisfy a linear equation, it forms a line:
Y
X
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Intermission: A primer on linear equations
A linear equation: aX + cY = d, or Y = mX + b, a,b,c,d,m numbers, X,Y variables
If we plot all the X,Y that satisfy a linear equation, it forms a line:
Y
X
Slope of a line: riserun
rise
run
= m
b
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Intermission: A primer on linear equations
A linear equation: aX + cY = d, or Y = mX + b, a,b,c,d,m numbers, X,Y variables
If we plot all the X,Y that satisfy a linear equation, it forms a line:
Y
X
Slope of a line: riserun
rise
run
= mm=0
m=1m=-1
m=2
m=1/2
m=-2
m=-1/2
b
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Back to our problem . . .
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Let’s say M = 12, N = 12
X + 2Y <= 12 Y = -(1/2) X + 6 slope = -1/2
X + Y <= 12 Y = -(1) X + 12 slope = -1
X = # tablesY = # chairsM = # small blocksN = # large blocks
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Let’s say M = 12, N = 12
X + 2Y <= 12 Y = -(1/2) X + 6 slope = -1/2
X + Y <= 12 Y = -(1) X + 12 slope = -1
We want to find all possible X and Y that satisfy these two equations.
X = # tablesY = # chairsM = # small blocksN = # large blocks
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Let’s say M = 12, N = 12
X + 2Y <= 12 Y = -(1/2) X + 6 slope = -1/2
X + Y <= 12 Y = -(1) X + 12 slope = -1
We want to find all possible X and Y that satisfy these two equations. First draw the equality lines;
Y
X
12
12
6
X + Y = 12
X + 2Y = 12
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2
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Now the inequalities:
Y
X
12
12
6
X + Y = 12
X + Y <= 12 which side of the line is this region?
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Now the inequalities:
Y
X
12
12
6
X + Y = 12
X + Y <= 12 which side of the line is this region?
Let’s check one point: Is (0,0) in this region?
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Now the inequalities:
Y
X
12
12
6
X + Y = 12
X + Y <= 12 which side of the line is this region?
Let’s check one point: Is (0,0) in this region?Yes: When X=0 and Y=0 then 0 + 0 <= 12
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Now the inequalities:
Y
X
12
12
6
X + Y = 12
X + Y <= 12 which side of the line is this region?
Let’s check one point: Is (0,0) in this region?Yes: When X=0 and Y=0 then 0 + 0 <= 12
So the allowed values of X and Y are below this line(feasible points)
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Y
X
12
12
6 X + 2Y = 12
X + 2Y <= 12 which side of the line is this region?
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Y
X
12
12
6 X + 2Y = 12
X + 2Y <= 12 which side of the line is this region?
Region is also below the line
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So the allowed region for both inequalities is the common region (the intersection)
Y
X
12
12
6 X + Y <= 12 AND X + 2Y <= 12
Only X and Y in this region are allowed (for M=12, N=12)
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Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6 X + Y <= 12 AND X + 2Y <= 12
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5
![Page 67: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/67.jpg)
Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6 X + Y <= 12 AND X + 2Y <= 12
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5
Let’s plot the profit line P = 3X + 5Y
![Page 68: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/68.jpg)
Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6Slope = -1/2
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5, Y intercept = P/5
Slope = -1
Slope = -3/5
Slope = -1/2
Along the profit line, all (X,Y) give the same profit P
Slope = -3/5All X and Y along this line give the same profit P
P/5
![Page 69: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/69.jpg)
Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5, Y intercept = P/5
Slope = -1
Slope = -3/5
Slope = -1/2
Along the profit line, all (X,Y) give the same profit P
lower profit
higher profit
![Page 70: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/70.jpg)
Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5, Y intercept = P/5
lower profit
higher profit
What is the highest profit for those X,Y in the feasible region?
![Page 71: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/71.jpg)
Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5, Y intercept = P/5
lower profit
higher profit
We keep moving the profit line upwards until the last feasible point is touched
![Page 72: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/72.jpg)
Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5, Y intercept = P/5
We keep moving the profit line upwards until the last feasible point is touched
It is this line!
![Page 73: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/73.jpg)
Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5, Y intercept = P/5
We keep moving the profit line upwards until the last feasible point is touched
It is this line!And this is the only point; (12, 0)
![Page 74: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/74.jpg)
Now, for X and Y in this region, which one gives the highest profit?
Y
X
12
12
6
Profit; P = 3X + 5Y Y = -(3/5)X + P/5, slope = -3/5, Y intercept = P/5
So the greatest profit is achieved when X=12 and Y=0;Profit = 36
36 = 3X + 5Y
![Page 75: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/75.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= M Y = -(1/2) X + M/2 slope = -1/2
X + Y <= N Y = -(1) X + N slope = -1
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2
![Page 76: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/76.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= M Y = -(1/2) X + M/2 slope = -1/2
X + Y <= N Y = -(1) X + N slope = -1
We want to find all possible X and Y that satisfy these two equations.
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2
![Page 77: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/77.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= MX + Y <= N
We want to find all possible X and Y that satisfy these two equations.
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2Feasible region
![Page 78: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/78.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= MX + Y <= N
Now plot the profit line: 3X + 5Y = P
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2Feasible region
![Page 79: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/79.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= MX + Y <= N
Now plot the profit line: 3X + 5Y = P (slope = -3/5)
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2
profit line
![Page 80: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/80.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= MX + Y <= N
Move the profit line up until it last touches the feasible region
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2Greatest profit line
![Page 81: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/81.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2Greatest profit line
Point (0,N)
![Page 82: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/82.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2Greatest profit line
Point (0,N)
So the greatest profit is achieved when X = 0, Y = N
![Page 83: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/83.jpg)
Now let’s look at Case 1: M >=2N N< M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2Greatest profit line
Point (0,N)
So the greatest profit is achieved when X = 0, Y = N
Make N chairs0 tables
![Page 84: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/84.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= M Y = -(1/2) X + M/2 slope = -1/2
X + Y <= N Y = -(1) X + N slope = -1
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2
![Page 85: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/85.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2
The feasible region is:
![Page 86: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/86.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2
The feasible region is
Note there is a new corner; R
R
![Page 87: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/87.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
slope = -1
slope = -1/2
Now let’s add the profit lines
R
![Page 88: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/88.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
Now let’s add the profit lines
The highest one touches R!
R
![Page 89: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/89.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
Now let’s add the profit lines
The highest one touches R!
So R is the point (X,Y) that has the greatest profit
R
![Page 90: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/90.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
What is (X,Y) at point R?
R
![Page 91: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/91.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
What is (X,Y) at point R?
It is where the two linesX+Y=N and X+2Y=Mmeet.
R
![Page 92: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/92.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
What is (X,Y) at point R?
It is where the two linesX+Y=N and X+2Y=Mmeet.
We find that X = 2N - MY = M - N
R
X=2N-M
Y=M-N
![Page 93: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/93.jpg)
Now let’s look at Case 2: M < 2N N > M/2
X + 2Y <= MX + Y <= N
Y
X
N
N
M/2
X + Y = N
X + 2Y = M
X = # tablesY = # chairsM = # small blocksN = # large blocks
R
X=2N-M
Y=M-N
So we have our answer:
If M < 2N then we makeX = 2N-M tables andY = M – N chairs
![Page 94: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/94.jpg)
tables 0 2 3 4 6 8 10 11 12
chairs 6 5 4 4 3 2 1 0 0
profit 30 31 29 32 33 34 35 33 36
M = 12N = 12M < 2N
tables 2 4 5 6 7 8 10 11 12
chairs 9 8 7 6 5 4 2 1 0
profit 51 52 50 48 46 44 40 38 36
M = 20N = 12M < 2N
tables 0 4 5 6 7 8 10 11 12
chairs 12 8 7 6 5 4 2 1 0
profit 60 52 50 48 46 44 40 38 36
M = 25N = 12M > 2N
X = 2N – M = 12Y = M – N = 0
X = 2N – M = 4Y = M – N = 8
X = 0 Y = N = 12
table chair$3 $5
X Y
![Page 95: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/95.jpg)
A problem for you . . .
![Page 96: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/96.jpg)
A problem for you . . .
1 0 4
0 2 12
3 2 18
3 2.5
hours per kg hours availableColumbian Mexican
roaster Aroaster Bgrind/package
profit/kg
![Page 97: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/97.jpg)
A problem for you . . .
1 0 4
0 2 12
3 2 18
3 2.5
hours per kg hours availableColumbian Mexican
roaster Aroaster Bgrind/package
profit/kg
Variables: X Y
![Page 98: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/98.jpg)
X <= 4 roaster A
![Page 99: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/99.jpg)
X
Y
X <= 4 roaster A
![Page 100: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/100.jpg)
X
Y
4
X <= 4 roaster A
X = 4
![Page 101: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/101.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
X = 4
Y = 6
![Page 102: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/102.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
![Page 103: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/103.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
![Page 104: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/104.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
![Page 105: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/105.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging3X + 2.5Y = P profit line, slope = -1.2
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
![Page 106: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/106.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging3X + 2.5Y = P profit line, slope = -1.2
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit line
![Page 107: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/107.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging3X + 2.5Y = P profit line, slope = -1.2
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit line
![Page 108: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/108.jpg)
X
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging3X + 2.5Y = P profit line, slope = -1.2
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit line
![Page 109: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/109.jpg)
X
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging3X + 2.5Y = P profit line, slope = -1.2
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit lineBest point is (2,6);Solution is X = 2
Y = 6
![Page 110: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/110.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging3.5X + 2.5Y = P profit line, slope = -1.4
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit line
![Page 111: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/111.jpg)
X
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging3.5X + 2.5Y = P profit line, slope = -1.4
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit line
![Page 112: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/112.jpg)
X
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging3.5X + 2.5Y = P profit line, slope = -1.4
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit lineBest solution is still
X = 2Y = 6
![Page 113: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/113.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging◊X + 2.5Y = P profit line, slope < -1.5 (steeper than -1.5)
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
![Page 114: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/114.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging◊X + 2.5Y = P profit line, slope < -1.5 (steeper than -1.5)
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit line
![Page 115: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/115.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging◊X + 2.5Y = P profit line, slope < -1.5 (steeper than -1.5)
X = 4
Y = 6
3X + 2Y = 18 slope = -1.5
(4,3)
(2,6)
profit line
![Page 116: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/116.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging◊X + 2.5Y = P profit line, slope < -1.5 (steeper than -1.5)
X = 4
Y = 6
3X + 2Y = 18
(4,3)
(2,6)
profit line
Best solution now is X = 4Y = 3
![Page 117: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/117.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging◊X + 2.5Y = P profit line, slope < -1.5 (steeper than -1.5)
X = 4
Y = 6
3X + 2Y = 18
(4,3)
(2,6)
profit line
Best solution now is X = 4Y = 3
So need
◊2.5
<- - 32
![Page 118: Using Mathematics to Solve Real World Problems](https://reader034.fdocuments.us/reader034/viewer/2022052607/5895aec31a28ab71688b7925/html5/thumbnails/118.jpg)
X
Y
6
4
X <= 4 roaster A2Y <= 12 roaster B
3X + 2Y <= 18 grinding, packaging◊X + 2.5Y = P profit line, slope < -1.5 (steeper than -1.5)
X = 4
Y = 6
3X + 2Y = 18
(4,3)
(2,6)
profit line
Best solution now is X = 4Y = 3
So need
◊2.5
<- - 32
◊ > 3 (2.5/2) = 3.75