## Engage NY Eureka Math 8th Grade Module 6 Lesson 1 Answer Key

### Eureka Math Grade 8 Module 6 Lesson 1 Example Answer Key

Example 2: Another Rate Plan
A second wireless access company has a similar method for computing its costs. Unlike the first company that Lenore was considering, this second company explicitly states its access fee is $0.15, and its usage rate is$0.04 per minute.
Total Session Cost = $0.15 +$0.04 (number of minutes)
→ How is this plan presented differently?
In this case, we are given the access fee and usage rate with an equation. In the first example, just data points were given.

→ Based on the work with the first set of problems, how do you think the two plans are different?
The values for the access fee and usage charge per minute are different, or the initial value and the rate of change are different.

### Eureka Math Grade 8 Module 6 Lesson 1 Exercise Answer Key

Exercises 1–6

Exercise 1.
Lenore makes a table of this information and a graph where number of minutes is represented by the horizontal axis and total session cost is represented by the vertical axis. Plot the three given points on the graph. These three points appear to lie on a line. What information about the access plan suggests that the correct model is indeed a linear relationship? The amount charged for the minutes connected is based upon a constant usage rate in dollars per minute. Exercise 2.
The rate of change describes how the total cost changes with respect to time.
a. When the number of minutes increases by 10 (e.g., from 10 minutes to 20 minutes or from 20 minutes to 30 minutes), how much does the charge increase?
When the number of minutes increases by 10 (e.g., from 10 minutes to 20 minutes or from 20 minutes to 30 minutes), the cost increases by $0.30 (30 cents). b. Another way to say this would be the usage charge per 10 minutes of use. Use that information to determine the increase in cost based on only 1 minute of additional usage. In other words, find the usage charge per minute of use. Answer: If$0.30 is the usage charge per 10 minutes of use, then $0.03 is the usage charge per 1 minute of use (i.e., the usage rate). Since the usage rate is constant, students should use what they have learned in Module 4. Exercise 3. The company’s pricing plan states that the usage rate is constant for any number of minutes connected to the Internet. In other words, the increase in cost for 10 more minutes of use (the value that you calculated in Exercise 2) is the same whether you increase from 20 to 30 minutes, 30 to 40 minutes, etc. Using this information, determine the total cost for 40 minutes, 50 minutes, and 60 minutes of use. Record those values in the table, and plot the corresponding points on the graph in Exercise 1. Answer: Consider the following table and graphs. Exercise 4. Using the table and the graph in Exercise 1, compute the hypothetical cost for 0 minutes of use. What does that value represent in the context of the values that Lenore is trying to figure out? Answer: Since there is a$0.30 decrease in cost for each decrease of 10 minutes of use, one could subtract $0.30 from the cost value for 10 minutes and arrive at the hypothetical cost value for 0 minutes. That cost would be$0.10. Students may notice that such a value follows the regular pattern in the table and would represent the fixed access fee for connecting. (This value could also be found from the graph after completing Exercise 6.)

Exercise 5.
On the graph in Exercise 1, draw a line through the points representing 0 to 60 minutes of use under this company’s plan. The slope of this line is equal to the constant rate of change, which in this case is the usage rate. Exercise 6.
Using x for the number of minutes and y for the total cost in dollars, write a function to model the linear relationship between minutes of use and total cost.
y = 0.03x + 0.10

Exercises 7–16

Exercise 7.
Let x represent the number of minutes used and y represent the total session cost in dollars. Construct a linear function that models the total session cost based on the number of minutes used.
y = 0.04x + 0.15

Exercise 8.
Using the linear function constructed in Exercise 7, determine the total session cost for sessions of 0, 10, 20, 30, 40, 50, and 60 minutes, and fill in these values in the table below.  Exercise 9.
Plot these points on the original graph in Exercise 1, and draw a line through these points. In what ways does the line that represents this second company’s access plan differ from the line that represents the first company’s access plan?
The second company’s plan line begins at a greater initial value. The same plan also increases in total cost more quickly over time; in other words, the slope of the line for the second company’s plan is steeper. Exercises 10–12

MP3 download sites are a popular forum for selling music. Different sites offer pricing that depends on whether or not you want to purchase an entire album or individual songs à la carte. One site offers MP3 downloads of individual songs with the following price structure: a $3 fixed fee for a monthly subscription plus a charge of$0.25 per song.
Exercise 10.
Using x for the number of songs downloaded and y for the total monthly cost in dollars, construct a linear function to model the relationship between the number of songs downloaded and the total monthly cost.
Since $3 is the initial cost and there is a 25 cent increase per song, the function would be y = 3 + 0.25x or y = 0.25x + 3. Exercise 11. Using the linear function you wrote in Exercise 10, construct a table to record the total monthly cost (in dollars) for MP3 downloads of 10 songs, 20 songs, and so on up to 100 songs. Answer: Exercise 12. Plot the 10 data points in the table on a coordinate plane. Let the x-axis represent the number of songs downloaded and the y-axis represent the total monthly cost (in dollars) for MP3 downloads. Answer: A band will be paid a flat fee for playing a concert. Additionally, the band will receive a fixed amount for every ticket sold. If 40 tickets are sold, the band will be paid$200. If 70 tickets are sold, the band will be paid $260. Exercise 13. Determine the rate of change. Answer: The points (40,200) and (70,260) have been given. So, the rate of change is 2 because $$\frac{260-200}{70-40}$$ = 2. Exercise 14. Let x represent the number of tickets sold and y represent the amount the band will be paid in dollars. Construct a linear function to represent the relationship between the number of tickets sold and the amount the band will be paid. Answer: Using the rate of change and (40,200): 200 = 2(40) + b 200 = 80 + b 120 = b Therefore, the function is y = 2x + 120. Exercise 15. What flat fee will the band be paid for playing the concert regardless of the number of tickets sold? Answer: The band will be paid a flat fee of$120 for playing the concert.

Exercise 16.
How much will the band receive for each ticket sold?
The band receives $2 per ticket. ### Eureka Math Grade 8 Module 6 Lesson 1 Problem Set Answer Key Question 1. Recall that Lenore was investigating two wireless access plans. Her friend in Europe says that he uses a plan in which he pays a monthly fee of 30 euro plus 0.02 euro per minute of use. a. Construct a table of values for his plan’s monthly cost based on 100 minutes of use for the month, 200 minutes of use, and so on up to 1,000 minutes of use. (The charge of 0.02 euro per minute of use is equivalent to 2 euro per 100 minutes of use.) Answer: b. Plot these 10 points on a carefully labeled graph, and draw the line that contains these points. Answer: c. Let x represent minutes of use and y represent the total monthly cost in euro. Construct a linear function that determines the monthly cost based on minutes of use. Answer: y = 30 + 0.02x d. Use the function to calculate the cost under this plan for 750 minutes of use. If this point were added to the graph, would it be above the line, below the line, or on the line? Answer: The cost for 750 minutes would be €45. The point (750,45) would be on the line. Question 2. A shipping company charges a$4.45 handling fee in addition to $0.27 per pound to ship a package. a. Using x for the weight in pounds and y for the cost of shipping in dollars, write a linear function that determines the cost of shipping based on weight. Answer: y = 4.45 + 0.27x b. Which line (solid, dotted, or dashed) on the following graph represents the shipping company’s pricing method? Explain. Answer: The solid line would be the correct line. Its initial value is 4.45, and its slope is 0.27. The dashed line shows the cost decreasing as the weight increases, so that is not correct. The dotted line starts at an initial value that is too low. Question 3. Kelly wants to add new music to her MP3 player. Another subscription site offers its downloading service using the following: Total Monthly Cost = 5.25 + 0.30(number of songs). a. Write a sentence (all words, no math symbols) that the company could use on its website to explain how it determines the price for MP3 downloads for the month. Answer: “We charge a$5.25 subscription fee plus 30 cents per song downloaded.”

b. Let x represent the number of songs downloaded and y represent the total monthly cost in dollars. Construct a function to model the relationship between the number of songs downloaded and the total monthly cost.
y = 5.25 + 0.30x

The cost of downloading 10 songs is $8.25. Question 4. Li Na is saving money. Her parents gave her an amount to start, and since then she has been putting aside a fixed amount each week. After six weeks, Li Na has a total of$82 of her own savings in addition to the amount her parents gave her. Fourteen weeks from the start of the process, Li Na has $118. a. Using x for the number of weeks and y for the amount in savings (in dollars), construct a linear function that describes the relationship between the number of weeks and the amount in savings. Answer: The points (6, 82) and (14, 118) have been given. So, the rate of change is 4.5 because $$\frac{118-82}{14-6}$$ = $$\frac{36}{8}$$ = 4.5. Using the rate of change and (6, 82): 82 = 4.5(6) + b 82 = 27 + b 55 = b The function is y = 4.5x + 55. b. How much did Li Na’s parents give her to start? Answer: Li Na’s parents gave her$55 to start.