## Engage NY Eureka Math Precalculus Module 2 Lesson 1 Answer Key

### Eureka Math Precalculus Module 2 Lesson 1 Exploratory Challenge Answer Key

Exploratory Challenge 1

A network diagram depicts interrelated objects by using circles to represent the objects and directed edges drawn as segments or arcs between related objects with arrows to denote direction. The network diagram below shows the bus routes that run between four cities, forming a network. The arrows indicate the direction the buses travel.

Figure 1

a. How many ways can you travel from City 1 to City 4? Explain how you know.

Answer:

There are three ways to travel from City 1 to City 4. According to the arrows, you can travel from City 1 to City 2 to City 4, from City 1 to City 3 to City 4, or from City 1 to City 3 to City 2 to City 4.

b. What about these bus routes doesn’t make sense?

Answer:

It is not possible to leave City 4. The direction of the arrows show that there are no bus routes that lead from City 4 to any other city in this network. It is not possible to travel to City 1. The direction of the arrows show that there are no bus routes that lead to City 1.

c. It turns out there was an error in printing the first route map. An updated network diagram showing the bus routes that connect the four cities is shown below in Figure 2. Arrows on both ends of an edge indicate that buses travel in both directions.

Figure 2

How many ways can you reasonably travel from City 4 to City 1 using the route map in Figure 2? Explain how you know.

Answer:

There is only one reasonable way. You must go from City 4 to City 2 to City 3 to City 1. The arrows indicate that there is only one route to City 1, which comes from City 3. However, it is possible to travel from City 4 to City 2 to City 3 as many times as desired before traveling from City 4 to City 2 to City 1.

Exploratory Challenge 2

A rival bus company offers more routes connecting these four cities as shown in the network diagram in Figure 3.

Figure 3

a. What might the loop at City 1 represent?

Answer:

This loop could represent a tour bus that takes visitors around City 1 but does not leave the city limits.

b. How many ways can you travel from City 1 to City 4 if you want to stop in City 2 and make no other stops?

Answer:

There are three bus routes from City 1 to City 2 and two bus routes from City 2 to City 4, so there are 6 possible ways to travel from City 1 to City 4.

c. How many possible ways are there to travel from City 1 to City 4 without repeating a city?

Answer:

City 1 to City 4 with no stops: no routes

City 1 to City 4 with a stop in City 2: 6 routes

City 1 to City 4 with a stop in City 3: 1 route

City 1 to City 4 via City 3 then City 2: 2 routes

City 1 to City 4 via City 2 then City 3: 6 routes

Total ways: 0+6+1+2+6=15 possible ways to travel from City 1 to City 4 without visiting a city more than once.

Exploratory Challenge 3

Let’s consider a “direct route” to be a route from one city to another without going through any other city. Organize the number of direct routes from each city into the table shown below. The first row showing the direct routes between City 1 and the other cities is complete for you.

Answer:

### Eureka Math Precalculus Module 2 Lesson 1 Exercise Answer Key

Exercises

Exercise 1.

Use the network diagram in Figure 3 to represent the number of direct routes between the four cities in a matrix R.

Answer:

Exercise 2.

What is the value of r_{2,3}? What does it represent in this situation?

Answer:

The value is 2. It is the number of direct routes from City 2 to City 3.

Exercise 3.

What is the value of r_{2,3}∙r_{3,1}, and what does it represent in this situation?

Answer:

The value is 4. It represents the number of one-stop routes between City 2 and City 1 that pass through City 3.

Exercise 4.

Write an expression for the total number of one-stop routes from City 4 and City 1, and determine the number of routes stopping in one city.

Answer:

r_{4,2}∙r_{2,1}+r_{4,3}∙r_{3,1}=2∙2+1∙2=6

Exercise 5.

Do you notice any patterns in the expression for the total number of one-stop routes from City 4 and City 1?

Answer:

The middle indices in each expression are the same in each term and represent the cities where a stop was made.

Exercise 6.

Create a network diagram for the matrices shown below. Each matrix represents the number of transportation routes that connect four cities. The rows are the cities you travel from, and the columns are the cities you travel to.

a.

Answer:

b.

Answer:

Here is a type of network diagram called an arc diagram.

Suppose the points represent eleven students in your mathematics class, numbered 1 through 11. Suppose the arcs above and below the line of vertices 1–11 represent people who are friends on a social network.

Exercise 7.

Complete the matrix that shows which students are friends with each other on this social network. The first row has been completed for you.

Answer:

a. Number 1 is not friends with number 10. How many ways could number 1 get a message to number 10 by only going through one other friend?

Answer:

There are 2 ways. Number 1 to number 5 and then number 5 to number 10 or number 1 to number 7 and numbers 7 to number 10.

b. Who has the most friends in this network? Explain how you know.

Answer:

Number 1 is friends with 7 people, and that’s more than anyone else, so number 1 has the most friends in this network.

c. Is everyone in this network connected at least as a friend of a friend? Explain how you know.

Answer:

No. Number 2 is not connected to number 4 as a friend of a friend because number 2 is only friends with number 7, and number 7 and number 4 are not friends.

d. What is entry a_{2,3}? Explain its meaning in this context.

Answer:

The entry is 0. Number 2 is not friends with number 3.

### Eureka Math Precalculus Module 2 Lesson 1 Problem Set Answer Key

Question 1.

Consider the railroad map between Cities 1, 2, and 3, as shown.

a. Create a matrix R to represent the railroad map between Cities 1, 2, and 3.

Answer:

b. How many different ways can you travel from City 1 to City 3 without passing through the same city twice?

Answer:

r_{1,2}∙r_{2,3}+r_{1,1}∙r_{1,3}+r_{1,3}=2∙0+0∙1+1=1

c. How many different ways can you travel from City 2 to City 3 without passing through the same city twice?

Answer:

r_{2,3}+r_{2,1}∙r_{1,3}=0+1∙1=1

d. How many different ways can you travel from City 1 to City 2 with exactly one connecting stop?

Answer:

r_{1,3}∙r_{3,2}=1∙3=3

e. Why is this not a reasonable network diagram for a railroad?

Answer:

More trains arrive in City 2 than leave, and more trains leave City 3 than arrive.

Question 2.

Consider the subway map between stations 1, 2, and 3, as shown.

a. Create a matrix S to represent the subway map between stations 1, 2, and 3.

Answer:

s = \(\left[\begin{array}{lll}

0 & 2 & 1 \\

1 & 0 & 2 \\

2 & 1 & 0

\end{array}\right]\)

b. How many different ways can you travel from station 1 to station 3 without passing through the same station twice?

Answer:

s_{1,3}+s_{1,2}∙s_{2,3}=1+2∙2=5

c. How many different ways can you travel directly from station 1 to station 3 with no stops?

Answer:

s_{1,3}=1

d. How many different ways can you travel from station 1 to station 3 with exactly one stop?

Answer:

s_{1,2}∙s_{2,3}=2∙2=4

e. How many different ways can you travel from station 1 to station 3 with exactly two stops? Allow for stops at repeated stations.

Answer:

s_{1,2}∙s_{2,1}∙s_{1,3} + s_{1,3}∙s_{3,2}∙s_{2,3}+s_{1,3}∙s_{3,1}∙s_{1,3}=2∙1∙1+1∙1∙2+1∙2∙1=6

Question 3.

Suppose the matrix R represents a railroad map between cities 1, 2, 3, 4, and 5.

a. How many different ways can you travel from City 1 to City 3 with exactly one connection?

Answer:

r_{1,2}∙r_{2,3}+r_{1,4} ∙r_{4,3}+r_{1,5} ∙ r_{5,3}=1∙1+1∙0+1∙3=4

b. How many different ways can you travel from City 1 to City 5 with exactly one connection?

Answer:

r_{1,2}∙r_{2,5}+r_{1,3}∙r_{3,5}+r_{1,4} ∙ r_{4,5}=1∙0+2∙2+1∙2=6

c. How many different ways can you travel from City 2 to City 5 with exactly one connection?

Answer:

r_{2,1} ∙r_{1,5}+r_{2,3} ∙r_{3,5}+r_{2,4}∙r_{4,5}=2∙1+1∙2+1∙2=6

Question 4.

Let B=\(\left[\begin{array}{lll}

0 & 2 & 1 \\

1 & 1 & 2 \\

2 & 1 & 1

\end{array}\right]\) represent the bus routes between 3 cities.

a. Draw an example of a network diagram represented by this matrix.

Answer:

b. Calculate the matrix B^{2}=BB.

Answer:

B^{2}=\(\left[\begin{array}{lll}

4 & 3 & 5 \\

5 & 5 & 5 \\

3 & 6 & 5

\end{array}\right]\)

c. How many routes are there between City 1 and City 2 with one stop in between?

Answer:

b_{1,1} b_{1,2}+b_{1,2} b_{2,2}+b_{1,3} b_{3,2}=0∙1+2∙1+1∙1=3

d. How many routes are there between City 2 and City 2 with one stop in between?

Answer:

b_{2,1} b_{1,2}+b_{2,2} b_{2,2}+b_{2,3} b_{3,2}=1∙2+1∙1+2∙1=5

e. How many routes are there between City 3 and City 2 with one stop in between?

Answer:

b_{3,1} b_{1,2}+b_{3,2} b_{2,2}+b_{3,3} b_{3,2}=2∙2+1∙1+1∙1=6

f. What is the relationship between your answers to parts (b)–(e)? Formulate a conjecture.

Answer:

The numbers 3, 5, and 6 appear in the second column of matrix B^{2}. It seems that the entry in row i and column j of matrix B^{2} is the number of ways to get from city i to city j with one stop.

Question 5.

Consider the airline flight routes between Cities 1, 2, 3, and 4, as shown.

a. Create a matrix F to represent the flight map between Cities 1, 2, 3, and 4.

Answer:

b. How many different routes can you take from City 1 to City 4 with no stops?

Answer:

f_{1,4=2}

c. How many different routes can you take from City 1 to City 4 with exactly one stop?

Answer:

f_{1,2} f_{2,4}+f_{1,3} f_{3,4}=2∙2+1∙1=5

d. How many different routes can you take from City 3 to City 4 with exactly one stop?

Answer:

f_{3,1} f_{1,4}+f_{3,2} f_{2,4}=1∙2+1∙2=4

e. How many different routes can you take from City 1 to City 4 with exactly two stops? Allow for routes that include repeated cities.

Answer:

f_{1,2} f_{2,1} f_{1,4}+f_{1,2} f_{2,3} f_{3,4}+f_{1,3} f_{3,1} f_{1,4}+f_{1,3} f_{3,2} f_{2,4}+f_{1,4} f_{4,1} f_{1,4}+f_{1,4} f_{4,2} f_{2,4}+f_{1,4} f_{4,3} f_{3,4}=

(2∙1∙2)+(2∙1∙1)+(1∙1∙2)+(1∙1∙2)+(2∙2∙2)+(2∙2∙2)+(2∙1∙1)=28

f. How many different routes can you take from City 2 to City 4 with exactly two stops? Allow for routes that include repeated cities.

Answer:

f_{2,1} f_{1,2} f_{2,4}+f_{2,1} f_{1,3} f_{3,4}+f_{2,3} f_{3,1} f_{1,4}+f_{2,3} f_{3,2} f_{2,4}+f_{2,4} f_{4,1} f_{1,4}+f_{2,4} f_{4,2} f_{2,4}+f_{2,4} f_{4,3} f_{3,4}=

(1∙2∙2)+(1∙1∙1)+(1∙1∙2)+(1∙1∙2)+((2∙) ̇2∙2)+(2∙2∙2)+(2∙1∙1)=27

Question 6.

Consider the following directed graph representing the number of ways Trenton can get dressed in the morning (only visible options are shown):

a. What reasons could there be for there to be three choices for shirts after “traveling” to shorts but only two after traveling to pants?

Answer:

It could be that Trenton has shirts that only make sense to wear with one or the other. For instance, maybe he does not want to wear a button-up shirt with a pair of shorts.

b. What could the order of the vertices mean in this situation?

Answer:

The order of the vertices is probably the order Trenton gets dressed in.

c. Write a matrix A representing this directed graph.

Answer:

d. Delete any rows of zeros in matrix A, and write the new matrix as matrix B. Does deleting this row change the meaning of any of the entries of B? If you had deleted the first column, would the meaning of the entries change? Explain.

Answer:

Deleting the row did not change the meaning of any of the other entries. Each entry b_{i,j} still says how to get from article of clothing i to article of clothing j. If we had deleted the first column, then each entry b_{i,j} would represent how to get from article of clothing i-1 to article of clothing j.

e. Calculate b_{1,2}∙b_{2,4}∙b_{4,5}. What does this product represent?

Answer:

2∙2∙1=4

The product represents the number of outfits that Trenton can wear assuming he wears pants instead of shorts.

f. How many different outfits can Trenton wear assuming he always wears a watch?

Answer:

b_{1,2}∙b_{2,4}∙b_{4,5}+b_{1,3}∙b_{3,4}∙b_{4,5}=4+9=13

Question 7.

Recall the network representing bus routes used at the start of the lesson:

Faced with competition from rival companies, you have been tasked with considering the option of building a toll road going directly from City 1 to City 4. Once built, the road will provide income in the form of tolls and also enable the implementation of a nonstop bus route to and from City 1 and City 4.

Analysts have provided you with the following information (values are in millions of dollars):

a. Express this information in a matrix P.

Answer:

\(\left[\begin{array}{ccc}

-63 & 65 & 100 \\

-5 & 0.75 & 1.25

\end{array}\right]\)

b. What are the dimensions of the matrix?

Answer:

2×3

c. Evaluate p_{1,1}+p_{1,2}. What does this sum represent?

Answer:

-63+65=2

$2,000 is the worst-case profit of the road after one year.

d. Solve p_{1,1}+t∙p_{1,2}=0 for t. What does the solution represent?

Answer:

-63+65t=0

65t=63

t=\(\frac{63}{65}\)≈0.9692

It will take about 1 year for the road to break even if we assume the worst-case profit.

e. Solve p_{1,1}+t∙p_{1,3}=0 for t. What does the solution represent?

Answer:

-63+100t=0

100t=63

t=0.63

It will take about 7.5 months for the road to break even if we assume the best-case profit every year.

f. Summarize your results to parts (d) and (e).

Answer:

It will take between 7.5 months and 1 year for the road to break even.

g. Evaluate p_{1,1}+p_{2,1}. What does this sum represent?

Answer:

-63+-5=-68

The total cost of the new road and new bus route is $68,000,000.

h. Solve p_{1,1}+p_{2,1}+t(p_{1,2}+p_{2,2})=0 for t. What does the solution represent?

Answer:

-68+65.75t=0

65.75t=68

t=\(\frac{68}{65.75}\)≈1.034

It will take about 1 year for the road and new bus route to break even assuming the worst-case scenario for profit.

i. Make your recommendation. Should the company invest in building the toll road or not? If they build the road, should they also put in a new bus route? Explain your answer.

Answer:

Answers will vary but should include the length of time it will take for the company to be profitable after the initial investment. Other factors can include considering the positive press for the company from building and maintaining a nonstop route between the cities as well as how this would affect the other routes of their buses.

### Eureka Math Precalculus Module 2 Lesson 1 Exit Ticket Answer Key

The following directed graph shows the major roads that connect four cities.

Question 1.

Create a matrix C that shows the direct routes connecting the four cities.

Answer:

Question 2.

Use the matrix to determine how many ways are there to travel from City 1 to City 4 with one stop in City 2.

Answer:

c_{1,2}∙c_{2,4}=3∙2=6

Question 3.

What is the meaning of c_{2,3}?

Answer:

Since c_{2,3}=1, it means that there is just one road that goes to City 3 from City 2 without going through the other cities.

Question 4.

Write an expression that represents the total number of ways to travel between City 2 and City 3 without passing through the same city twice. (You can travel through another city on the way from City 2 to City 3.)

Answer:

We will count the routes by the number of stops in intermediate cities.

Direct routes between City 2 and City 3: c_{2,3}=1

One-stop routes through City 1: c_{2,1}∙c_{1,3}=1∙2=2

One-stop routes through City 4: c_{2,4}∙c_{4,3}=2∙2=4

Two-stop routes through City 1 and then City 4: c_{2,1}∙c_{1,4}∙c_{4,3}=1∙0∙2=0

Two-stop routes through City 4 and then City 1: c_{2,4}∙c_{4,1}∙c_{1,3}=2∙0∙2=0

c_{2,3}+c_{2,1}∙c_{1,3}+c_{2,4}∙c_{4,3}+c_{2,1}∙c_{1,4}∙c_{4,3}+c_{2,4}∙c_{4,1}∙c_{1,3}=1+2+4+0+0=7

So there are 7 ways to travel between City 2 and City 3 without passing through the same city twice.