2 matrices and applications - pbworksdustintench.pbworks.com/f/...chapter+2+.pdf · chapter 2 l...

Chapter 2 l Matrices and Applications 55

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

2

2.1 Calories and LunchMatrices and Problem Solving | p. 57

2.2 Regions and VerticesSystems of Linear Inequalities | p. 65

2.3 Constraints and SolutionsLinear Programming | p. 75

2.4 Spies a Lot!Introduction to Directed

Graphs | p. 83

2.5 Electricity and Water!Directed Graphs and

Problem Solving | p. 89

Delicatessens sell many varieties of sandwiches, salads, and other foods. To offer the freshest

products, delicatessen managers must frequently monitor customer purchases and order new

ingredients. You will use matrices to calculate consumption of various ingredients in a restaurant.

Matrices and Applications2CHAPTER

56 Chapter 2 l Matrices and Applications

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

2

Lesson 2.1 l Matrices and Problem Solving 57

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

2

2.1 Calories and LunchMatrices and Problem Solving

ObjectivesIn this lesson you will:

l Use matrices to model and solve problems.

l Interpret matrix elements in the context of a problem situation.

Problem 1 Calorie Counting 1. The following table lists nutritional information about one serving of

various foods.

Food Type

Carbohydrates (grams)

Protein (grams)

Fat (grams)

Calories

Milk 12 6 2 90

Meat 0 28 22 310

Vegetables 5 2 0 28

Fruit 15 0 0 60

Grain 15 3 1 81

a. Define matrix A to represent the nutritional information in the table.

b. What does the element a33

of matrix A represent in the problem

situation?

c. What does the element a21


situation?

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

d. What does the element a54


situation?

2. Mike’s daily diet includes 3 servings of milk, 2 servings of meat,

4 servings of vegetables, 2 servings of fruit, and 8 servings of grain.

Define row matrix B to represent the number of servings of each food

type in Mike’s daily diet.

a. What does the element in the third column of matrix B represent in the

problem situation?

b. What does the element in the first column of matrix B represent in the

problem situation?

3. Calculate the following using matrix A and matrix B.

a. The number of grams of carbohydrates Mike consumes on a typical day.

b. The number of grams of protein Mike consumes on a typical day.

c. The number of grams of fat Mike consumes on a typical day.

d. The number of calories Mike consumes on a typical day.

e. Explain how you calculated your answer using the matrices.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

4. Let x � the number of calories in a gram of carbohydrates, y � the

number of calories in a gram of protein, and z � the number of calories

in a gram of fat.

a. Determine matrix D and matrix E for the matrix equation

D � x

y

z � � E using the information from the table. Keep in mind three

variables need to be found and what size matrices D and E need to be.

b. Calculate x, y, and z by solving the matrix equation.

5. For optimal health, the national guidelines suggest 10% of the daily

calories consumed should be from fats.

a. Calculate the percentage of calories consumed by Mike that are from

fats using your answers from Question 4.

b. Compare Mike’s percentage to the national guidelines.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

Problem 2 Lunch Time! 1. You work in a local deli. One of your responsibilities is to monitor all

food products. The following table lists possible items at the deli and the

exact amounts of meat, cheese, and vegetables to make each. The last

column lists the cost to make each item.

SelectionsMeat

(ounces)Cheese (ounces)

Vegetables (ounces)

Cost (dollars)

Wraps 2 1 3 1.35

Sandwiches 4 1 1 1.75

Salads 0 2 4 1.00

a. Define matrix F to represent the information in the table.

b. What does the element f22

of matrix F represent in the problem situation?

c. What does the element f31

of matrix F represent in the problem situation?

2. On a typical day, the deli serves 16 wraps, 34 sandwiches, and 7 salads.

Define row matrix G to represent the number of each item sold daily.

3. Calculate the following using matrix F and matrix G.

a. The total amount of meat used on a typical day.

b. The total amount of cheese used on a typical day.

c. The total amount of vegetables used on a typical day.

d. The total cost of these items used on a typical day.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

4. Let x � the cost of an ounce of meat, y � the cost of an ounce of

cheese, and z � the cost of an ounce of vegetables.

a. Determine matrix D and matrix E for the matrix equation

D � x

y

z � � E using the information from the table.

b. Calculate x, y, and z by solving the matrix equation.

Problem 3 Exercise Anyone?The following table represents the average calories burned per hour for various

activities and adult weights.

ActivityAverage Calories Burned per Hour of Person Weighing

110 lbs. 130 lbs. 155 lbs. 190 lbs. 220 lbs.

Aerobics 294 354 422 518 623

Basketball 383 472 563 690 831

Bicycling 821 944 1128 1380 1523

Dancing 272 354 422 518 607

Frisbee 354 472 563 690 841

1. Define matrix W to represent the information in the table.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

Leticia, Shania, Jose, Oliver, and Taki exercise regularly. Each person tracked

their exercise for the past week. The following table lists the time, in hours,

each person spent doing various activities.

Person Aerobics Basketball Bicycling Dancing Frisbee

Leticia 2 0 1.5 2 0

Shania 0 10 2 0 3

Jose 1 4 2 0 1.5

Oliver 2 3 0 3 0.5

Taki 3 10 4 0 4

2. Define matrix E to represent the information in the table.

3. Calculate EW, the product of matrix E and matrix W.

4. What is the element EW23

of the product matrix? What does this element

represent in terms of the problem situation?

5. What is the element in the fifth row and the fourth column of the

product matrix? What does this element represent in terms of the

problem situation?

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

6. If all five people weighed 155 pounds, how many calories would each

person burn in one week?

7. Leticia weighs 110 pounds, Shania weighs 130 pounds, Jose weighs

155 pounds, Oliver weighs 190 pounds, and Taki weighs 220 pounds.

How many calories would each person burn in one week?

Be prepared to share your methods and solutions.


2

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

Lesson 2.2 l Systems of Linear Inequalities 65

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

2

2.2 Regions and VerticesSystems of Linear Inequalities


l Solve systems of linear inequalities.

l Maximize linear expressions on a region in the coordinate plane.

Problem 1 Systems of Two Linear Inequalities

1. Graph the solution set for each system of linear inequalities. Label the

point of intersection of the boundary lines.

a. � y � 2x � 5

2x � 3y � 7

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

b. � 3x � 2y � 6

x � 2y � 2

c. � x � y � 11

y � x � 5

d. � 2x � 2y � 25

y � �6x � 12

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

e. � 2x � 5y � 10

y � 2x

Systems of linear inequalities can consist of more than two linear inequalities.

To solve a system of inequalities consisting of more than two inequalities,

use the same procedure that you used to solve a system consisting of two

inequalities. Graph each inequality and determine the portion of the coordinate

plane that satisfies all of the inequalities of the system. As with systems of

two inequalities, a system of three inequalities may have no solution.

Solve each of the following systems of linear inequalities by graphing each

one separately and determining which region is contained in all the solution

sets. Determine all the points of intersection for the boundary lines that make

the vertices of the solution region.

2. Graph the solution set for each system of linear inequalities. Label all

points of intersection of the boundary lines.

a. �

x � 0

y � 0

x � y � 6

y � 2x � 5

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

b.

� x � 0

y � 0

y � �2x � 8

c.

� y � 2x � 5

x � y � 3

2x � 3y � 6

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

Systems can consist of a combination of linear inequalities and linear

equations. The solution to a system of linear inequalities is a portion of

the coordinate plane bounded by line segments. The solution to a system

consisting of inequalities and a single equation is a line segment.

3. Graph the solution set for each system. Label all points of intersection of

the boundary lines.

a.

� y � 0

x � 0

x � y � 12

y � 2x �10

x � 2y � 8

b.

�

y � 0

x � 0

5x � 6y � 30

y � �6x � 24

x � y � 2

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

Problem 2 Maximizing Linear ExpressionsPreviously, you solved systems of linear inequalities and systems containing

linear inequalities and linear equations. These systems can be used to solve

specific problems. Before solving these problems, let’s explore some families

of linear equations.

1. Graph each system of linear equations.

a.

� x � y � 4

x � y � 6

x � y � 8

x � y � 10

b.

� x � 2y � 2

x � 2y � 5

x � 2y � 10

x � 2y � 12

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

c. �

3x � 2y � 6

3x � 2y � 12

3x � 2y � 18

3x � 2y � 24

2. What do you notice about the lines defined by each system of

equations?

3. What do you notice about the equations of the lines as the distance to

the origin increases?

4. Two parallel lines are written in the form ax � by � c. If one line is

further from the origin than the second line, what can you conclude

about the coefficients and constants?

a. The values of a and b

b. The value of c

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

5. Graph the system of linear inequalities.

� y � 6

y � 0

x � 6

x � 0

a. Describe the solution region.

b. What is true about all points in the solution region?

c. Graph the following lines on the same graph from Question 5. For

each line, highlight the intersection of the line and the solution region

to the system of inequalities. Label the endpoints of each intersection.

i. x � y � 4

ii. x � y � 8

iii. x � y � 12

iv. x � y � 14

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

d. Describe the intersection of each line with the solution region to the

system of inequalities.

i. x � y � 4

ii. x � y � 8

iii. x � y � 12

iv. x � y � 14

e. What is the maximum value of the expression x � y in this region?

Explain.

6. Graph the system of linear inequalities.

� y � �2x � 12

y � 0

y � �x � 8

x � 0

a. Describe the solution region.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

b. Graph the following lines on the same graph from Question 6. For

each line, highlight the intersection of the line and the solution region

to the system of inequalities. Label the endpoints of each intersection.

i. x � 2y � 5

ii. x � 2y � 8

iii. x � 2y � 13

iv. x � 2y � 16

c. Describe the intersection of each line with the solution region to the

system of inequalities.

i. x � 2y � 5

ii. x � 2y � 8

iii. x � 2y � 13

iv. x � 2y � 16

d. What is the maximum value of the expression x � 2y in this region? Explain.

7. What conclusions can you make about the maximum value of a linear

expression of the form ax � by in a region representing the solution of a

system of linear inequalities?


Lesson 2.3 l Linear Programming 75

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

2

2.3 Constraints and SolutionsLinear Programming


l Analyze and solve problems using linear

programming.

l Maximize linear expressions on a region

in the coordinate plane to solve a problem.

Key Termsl linear programming

l constraints

Problem 1 Maximizing a Linear ExpressionPreviously, you made a conjecture that the maximum value of a linear

expression occurs at a vertex of the region defined by a system of linear

inequalities and the linear expression.

1. Graph the solution set for the system of linear inequalities. Label all


� x � 0

y � 0

y � �2x � 12

y � x � 8

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

2. Evaluate 3x � 4y at each of the vertices of the solution region. Identify

the point at which the expression has a maximum value and the point at

which the expression has a minimum value.

3. Evaluate 4x � y at each of the vertices of the solution region. Identify the

point at which the expression has a maximum value.

4. Evaluate 8x � 4y at each of the vertices of the solution region. Identify

the point at which the expression has a maximum value.

5. How many points in this region maximized:

a. 3x � 4y?

b. 4x � y?

c. 8x � 4y?

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

Linear programming is a branch of mathematics that determines the

maximum and minimum value of linear expressions on a region produced by

a system of linear inequalities. The inequalities are constraints because the

values of the expression are “constrained” to lie within the region.

Problem 2 Linear Programming 1. A company makes and sells two different television models. One model

takes 2 person-hours to make. The other takes 3 person-hours to make.

The company has 24 employees, each working 8 hours a day, which is

equivalent to 192 person-hours per day. Define a variable for the

number of televisions made of each model and then write a linear

inequality to represent the number of person-hours to produce both

models.

2. The company’s total manufacturing capacity is 72 televisions per day.

Write an inequality to represent this constraint.

3. The company cannot make a negative number of televisions. Write two

linear inequalities to represent this constraint.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

4. In Questions 1 through 3, you wrote four inequalities. Graph the solution

set for the system of linear inequalities. Label all points of intersection of

the boundary lines.

5. The company sells the first model for $250 and the second model for

$300. Assume that all televisions that are made are also sold. How

many of each model should be made to maximize income? What is the

maximum income?

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

6. The company is considering changing the price of the second model

to $375. Assume that all televisions that are made are also sold. How

many of each model should be made to maximize income? What is the

maximum income?

7. If the company sells the first model for $250 and the second model for

$275, how many of each model should be made to maximize income?

What is the maximum income?

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

Problem 3 StereosA company produces two models of stereos, one with an MP3 player input

and the other without. The assembly lines can produce a total of 180 stereos

each day. The stereo without the input requires 3 person-hours and $75 of

materials to produce. The stereo with the input requires 4 person-hours and

$60 of materials to produce. The company has 640 person-hours of labor

available daily. The company has budgeted $12,900 for materials each day.

The profit for the stereo without the input is $35. The profit for the stereo with

the input is $27.50.

How many of each model should be produced to maximize profit?

1. Write the constraints as a system of linear inequalities.

2. Write an expression for the profit.

3. Graph the solution set for the system of linear inequalities. Label all


© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

4. How many of each model should be made to maximize profit? What is the

maximum profit?

5. A competitor had reduced the price of their stereo with MP3 input. In order to

compete, the company would have to decrease their profit on that model to

$25. How would this effect the number of stereos made?


Lesson 2.4 l Introduction to Directed Graphs 83

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

2

2.4 Spies a Lot!Introduction to Directed Graphs


l Define directed graphs.

l Use directed graphs to model situations.

l Define and interpret digraph notation.

l Define and interpret incident matrices.

Key Termsl directed graphs

l digraphs

l arcs of digraphs

l directed edges of digraphs

l vertices of digraphs

l nodes of digraphs

l origin of an arc

l initial node of an arc

l destination of an arc

l terminal node of an arc

l successor

l parallel arcs

l loop

l simple digraph

l path

l length of a path

l adjacency matrix

Problem 1 Spies!During times of war, countries often use spy networks to gather information

about enemies and allies. These networks have multiple paths for gathering

and distributing information as a means of verifying the information's

authenticity. Sources, called assets, report to several different handlers, who

pass on the information. Sometimes, information is passed directly to the

handler's supervisor.

Consider a spy network consisting of one supervisor, S, two handlers, H1 and

H2, and six assets, A

1, A

2, A

3, A

4, A

5, and A

6. Assets A

1, A

2, A

3, and A

4 report

to H1. Assets A

4, A

5, and A

6 report to H

2. Asset A

3 also reports directly to the

supervisor, as did H1 and H

2. Finally H

1 and H

2 report any information either

receives to each other.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

1. Draw arrows to show how information is passed between the assets,

handlers, and supervisor.

S

A1

A5

A2A3

A6

A4

H1

H2

Directed graphs or digraphs, G, are graphs that show the set of arcs, directed edges, or paths, E, between a set of vertices, nodes, or points, V.

The arcs are also called directed edges because the direction matters.

A digraph can be written as G(V, E). The set of vertices can be written as G(V).

The set of directed edges can be written as G(E).

If e1 � (H

1, S) is a directed edge, then:

l e1 begins at H

1 and ends at S,

l H1 is the origin or initial node of e

1,

l S is the destination or terminal node of e1,

l S is the successor of H1,

l H1 is adjacent to S,

l S is adjacent from H1.

2. Define the set of vertices, G(V), for the digraph in Question 1.

G(V) � � �

3. Label each arc in the digraph in Question 1.

4. Define the set of edges, G(E), for the digraph in Question 1.

G(E) � � �

A picture of a digraph is a representation in a plane, so the complete diagram

in Question 1 is a picture of G.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

Problem 2 PathsDigraph H is shown.

A D

C

B

a2

a3

a1a5a4

a6

1. Define H(V ).

H(V) �

2. Define H(E ).

H(E) �

3. Define each edge as an ordered pair.

Edges a2 and a

3 are parallel arcs because they both begin at D and end at A.

Edge a6 is a loop because it begins and ends at B. A simple digraph is a digraph

without parallel arcs or loops.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

4. Digraph K is defined as:

l K(V) � � W, X, Y, Z �

l K(E) � � a, b, c, d, e, f, g �

l a � (X, Y ), b � (Y, X ), c � (W, Y ), d � (X, W ), e � (Z, Y ), f � (X, Z ), g � (W, Z )

Sketch digraph K. Label each node and arc.

A path is an alternating sequence of nodes and arcs. For example, path P in

digraph K can be written as P � (W, c, Y, b, X) and defines the path that starts

at W, passes through Y, and ends at X. The length of a path is the number of

arcs in the path.

5. Is path P the only path from W to X in digraph K? Explain.

6. How long is path P? Explain.

For a simple digraph with vertices v1, v

2, ..., v

m, the adjacency matrix is an

m × m matrix where each row and each column is a node; and any element aij

is 1 if there is an arc from vi and v

j, and 0 if there is not an arc.

7. Complete the adjacency matrix A for digraph K.

W X Y Z

A �

W

X

Y

Z� �

8. Calculate the square of this matrix.

A2 � � �

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

The square of an adjacency matrix represents the number of paths of length 2

between each pair of vertices. For example,

l a21

� 0 so there are no paths of length 2 between vertex X and vertex W.

l a24

� 1 so there is one path of length 2 between vertex X and vertex Z.

The path is (X, d, W, g, Z).

l a23

� 2 so there are two paths of length 2 between vertex X and vertex Y.

9. Name the two paths from X to Y.

10. Calculate A3.

A3 � � �

11. What do you think the cube of an adjacency matrix represents?

12. How many paths of length three are there from Y to Y? List them.

13. Calculate A4.

A4 � � �

14. What do you think the fourth power of an adjacency matrix represents?

Lesson 2.5 l Directed Graphs and Problem Solving 89

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.

2

2.5 Electricity and Water!Directed Graphs and Problem Solving


l Model problems using directed graphs.

l Solve problems using directed graphs.

Problem 1 ElectricityAn electric company has four different transmission centers, A, B, C, and D.

The adjacency matrix T represents the centers that can transmit electricity.

A B C D

T �

A

B

C

D

� 0

1

1

0

1

0

1

0

1

0

0

1

0

0

1

0

� 1. Draw a digraph for matrix T.

2. Calculate T 2.

3. How many 2-step routes exist from center A to center A? From center D

to center A? List them.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

4. Calculate T � T 2.

5. What is the element t23

of the sum matrix from Question 4? What does

this element represent in the problem situation?

6. What is the element t32


this element represent in the problem situation? List them.

7. Calculate T 3.

8. How many 3-step routes exist from center A to center C? List them.

9. Calculate T � T 2 � T 3.

10. How many zero elements exist in the matrix from Question 9?

What does this mean in the problem situation?

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

11. What is the element t 31


this element represent in the problem situation? List them.

Problem 2 WaterThe following map shows five lakes in northern Quebec. Archeologists have

found artifacts from early Native American settlements at each of these

lakes. Using the artifacts, archeologists have been able to hypothesize about

established trading patterns. Each arrow represents a trading relationship from

one settlement to another.

A E

QUEBEC

C

DB

1. Define an adjacency matrix N to represent the trading patterns.

2. What does the sum of each column of matrix N represent?

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

3. What does the sum of each row of matrix N represent?

4. Calculate N 2.

5. Calculate N � N 2.

6. Did artifacts from every settlement reach every other settlement in two or

fewer transactions? If not, list them.

7. Calculate N 3.

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


2

8. How many different paths of length three exist from settlement E to

settlement A? List them.

9. Calculate N � N 2 � N 3.

10. What does the sum of each column of the matrix from Question 9

represent?

11. What does the sum of each row of the matrix from Question 9

represent?

12. Did artifacts from every settlement reach every other settlement in three

or fewer transactions? If not, list them.

13. How many paths of length three or less are there between B and E? List them.


2 matrices and applications - pbworksdustintench.pbworks.com/f/...chapter+2+.pdf · chapter 2 l...

Documents