Harbor square athletic club prices

Lesson 3.4

Introduction

Course Objectives

This lesson will address the following course outcomes:

· 20. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).

· 22. Identify when a linear model is reasonable for a given situation and, when appropriate, formulate a linear model. In the context of the situation interpret the slope and intercepts and determine the reasonable domain and range.

· 25. Use functional models to make predictions and solve problems.

Specific Objectives

Students will understand

· that linear models are appropriate when the situation has a constant rate of increase/decrease or can be approximated by a constant rate.

· that the rate of change (slope) has units in context.

· the difference between a positive slope and a negative slope.

· that the linear models for authentic situations have limitations in using them to make predictions.

Students will be able to

· make a linear model when given data or information in context.

· calculate a slope given data or information in context.

· estimate and calculate the value that makes two linear models equivalent.

Weight Loss

Problem Situation 1: Weight Loss Challenge

Four friends decided to do a joint weight loss challenge. They weighed in at the start, and again after 3 or 4 weeks. Their results are shown in the graph below.

https://s3.amazonaws.com/wamapdata/ufiles/2/M96-L3-4-g1.JPG

#1 Points possible: 5. Total attempts: 5

In this context, what would the rate of change (slope) mean? What does the vertical intercept mean? Give an answer, then compare it to ours.

Recall from the last lesson that we can compute rate of change (slope) as change of outputchange of inputchange of outputchange of input

#2 Points possible: 20. Total attempts: 5

Calculate each person's rate of change. Give your answers as fractions or integers. Do not use mixed numbers.

Marcel:

Jonas:

Jamie:

Carlie:

The units on these numbers is:

#3 Points possible: 12. Total attempts: 5

Consider how the rate of change relates to the shape of the graph.

If the rate of change is positive, then the line will be

If the rate of change is negative, then the line will be

If the rate of change is zero, then the line will be

#4 Points possible: 5. Total attempts: 5

If Jamie continues losing weight at the same rate, find a formula for a linear model for Jamie's weight, W, after t weeks.

#5 Points possible: 5. Total attempts: 5

Using your equation, predict how much Jamie will weigh after 5 weeks. Give your answer as a decimal, to the nearest pound.

pounds

#6 Points possible: 5. Total attempts: 5

Using your equation, predict how long it will take until she reaches her target weight of 173 pounds. Give your answer to one decimal place.

weeks

Slope From 2 Points

In the previous problem, you were able to calculate the rate of change by counting on the graph how much each person's weight changed over 3 or 4 weeks. This approach works well with clear graphs or simple numbers, but becomes more problematic in other cases.

For example, suppose that another friend, Raj, decides to join the challenge. Being a mathematician and a technology fan, he has a much more accurate scale and tracks things more closely. Since he joined late, he doesn't know what his weight was when the others started, but he knows after 0.5 weeks his weight was 193.4 pounds, and after 4 weeks his weight was 187.6.

https://s3.amazonaws.com/wamapdata/ufiles/2/M96-L3-4-g2.JPG

To compute his rate of change, we have to determine the change in output (weight) and change in input (weeks). This is harder to just count from the graph, so instead we can use the values themselves and find the difference:

187.6 − 193.4 = −5.8 He lost 5.8 pounds

4 − 0.5 = 3.5 He lost it over 3.5 weeks

So the slope is change of outputchange of input=187.6−193.44−0.5=−5.83.5≈−1.66change of outputchange of input=187.6-193.44-0.5=-5.83.5≈-1.66 pounds per week.

In general, if we have two points (x1, y1) and (x2, y2), then we can calculate the slope as

slope=change of outputchange of input=y2−y1x2−x1slope=change of outputchange of input=y2-y1x2-x1

It doesn't matter which point you call (x1, y1) and which you call (x2, y2), but it is important that when calculating the differences, both start with values from the same point.

#7 Points possible: 5. Total attempts: 5

Find the slope between the points (50, 600) and (65, 480).

Slope =

Comparing Consumption

Problem Situation 2: Milk and Soft Drink Consumption

Since 1950, the U.S. per-person consumption of milk and soft drinks has changed drastically. For example, in 1950, the number of gallons of milk consumed per person was 36.4 gallons; in 2000 that number had decreased to 22.6 gallons. Meanwhile, the number of gallons of soft drinks consumed per person in 1950 was 10.8 gallons. By 2000, this number had increased to 49.3 gallons per person.

#8 Points possible: 5. Total attempts: 5

Graph the lines for milk consumption and soft drink consumption. Plotting the exact points will not be possible, so just do the best you can. The horizontal axis is in years after 1950 and the vertical axis is in gallons.

Clear All Draw: Line

#9 Points possible: 5. Total attempts: 5

From the graphs, estimate the year in which the consumption (per person) of milk equaled the consumption (per person) of soft drinks. Estimate to the nearest year.

#10 Points possible: 10. Total attempts: 5

Find equations for the linear models for milk consumption, M, and soda consumption, S, both in terms of t, years since 1950. Keep at least 3 decimal places on any values calculated.

Try the problem on your own first. If you are having trouble after 2 tries, we will break it down.

Equation for Milk consumption: (this equation should involve M and t)

Equation for Soft drink consumption: (this equation should involve S and t)

#11 Points possible: 8. Total attempts: 5

Use your equations to solve for the year when the consumption (per person) of milk equaled the consumption (per person) of soft drinks.

Algebraically, the answer is t = (to one decimal place).

This means that the consumption will be equal in the year

This is a good time to mention the limitations of the models. Although the vertical intercepts have meaning (the amounts consumed in 1950 if 1950 is time = 0), the horizontal intercepts do not always have meaning in reality. When the linear model for milk equals 0 (crosses the horizontal axis), this means that the milk consumption per person is none. However, in reality, the consumption will not likely go to 0 any time soon. So, there are limits to the time for which the model is reliable and accurate. There are limitations to using mathematical models (linear or nonlinear), but the models can still be extremely useful - as long as they are used for a period of time.

HW 3.4

#1 Points possible: 5. Total attempts: 5

Which of the following was one of the main mathematical ideas of the lesson?

· A slope can be written as a fraction or a decimal.

· A slope is a ratio with units that describes how two variables change in relationship to each other.

· A slope tells you how much one variable has changed.

· Milk consumption is decreasing as soft drink consumption increases.

#2 Points possible: 9. Total attempts: 5

You have learned about absolute change, relative change, and average rate of change. Identify which type of change best describes each of the following values and explain your answer.

a. 30%

b. $30/year

c. $30

#3 Points possible: 25. Total attempts: 5

The three graphs below model the relationship between time spent in class and time spent studying out of class for three students. Use the graphs to answer the following questions.

http://s3.amazonaws.com/wamapdata/ufiles/2/4-1-2-3a.jpg http://s3.amazonaws.com/wamapdata/ufiles/2/4-1-2-3b.jpg http://s3.amazonaws.com/wamapdata/ufiles/2/4-1-2-3c.jpg

a. Which of the graphs has a slope of 2?

b. Which graph shows a relationship of 1 hour studying for each hour in class?

c. Which graph matches the following table?

Hours in class

1

4

7

9

Hours studying

2

8

14

18

d.

e. What is the equation for Graph B? Use C for hours in class and s for hours studying.

f. Which graph would contain the point (20, 10) if it were extended?

#4 Points possible: 15. Total attempts: 5

For each of the following, sketch a line that meets the conditions.

(a) Any line with a positive slope greater than 1

Clear All Draw: Line

(b) Any line with a positive slope smaller than 1

Clear All Draw: Line

(c) Any line with a slope of −1

Clear All Draw: Line

#5 Points possible: 15. Total attempts: 5

A parents' group is writing a grant to support an afterschool program for their children. They want to make the point that government funding for afterschool programs in their school has decreased over the last decade while the cost of offering programs has increased. They have the data given below.

2000

2010

Government funding for afterschool programs ($/child)

$3,000

$1,800

Cost of offering afterschool programs ($/child)

$3,800

$5,100

a. Which of the following is the best interpretation of the average rate of change in the government funding?

· Government funding has decreased by an average of $1,800 per child each year.

· Government funding has changed by an average of $120 each year.

· Government funding has decreased by an average of $1,200 per child each year.

· Government funding has decreased by an average of $120 per child each year.

b. Complete the interpretation below of the average rate of change in the cost of offering afterschool programs. The cost of offering afterschool programs has by an average of $ per child each year.

c. Sketch a graph showing linear models of the two sets of data, using years since 2000 on the horizontal axis.

$/child

Clear All Draw: Line

Years since 2000

d.

#6 Points possible: 24. Total attempts: 5

During the summer, Tamir likes to swim at his local pool a few times a week. He is debating if he should buy a “season pass” that allows him to swim as many times as he wants or just pay each time he goes swimming. The pool charges $4.50 each day if he pays each time. The "season pass" is $207, which is good for the entire summer (91 days).

a. Create a table to compare the two options by picking a set of inputs (Number of Swim Days) and calculating the corresponding cost of a season pass and cost of paying per day.

No. Swim Days

Cost of Season Pass ($)

Cost of Paying Per Day ($)

b.

c. How many times would Tamir have to swim to make the "season pass" less expensive than paying for each visit individually? He would have to swim more than times

d. Which of the graphs below models each situation? Season Pass: Paying per day: https://s3.amazonaws.com/wamapdata/ufiles/2/4-1-1-6.jpg

#7 Points possible: 15. Total attempts: 5

Delilah wants to join a gym, so she shops around to find the one with the lowest overall price (she is not sure how long she will be a member). She finds the Harbor Square Athletic Club is running a special, and only charges a $25 initiation fee plus $87 a month to be a member. The local YMCA charges $100 to join, and has a monthly fee of $72.

a. Find the equations for the linear models for the costs of the Harbor Square Athletic Club and the YMCA. Use C = cost ($) and m = months. Athletic Club: YMCA:

b. Under what conditions would it be less expensive to join the YMCA? The YMCA will be less expensive if she remains a member for months

#8 Points possible: 5. Total attempts: 5

Janey bought a used bicycle for $640. The bike was 3 years old when she bought it, and cost $850 new.

Assuming it decreases in value by the same amount each year, write a linear equation for the value of the bicycle, V, when it is t years old (so t = 0 was when it was new).

#9 Points possible: 20. Total attempts: 5

In Lesson 2.6, you used the following data about population changes in the United States. Suppose state planners in Indiana and Michigan think that the average rate of change will continue through 2020.

State

2000 Population

2010 Population

Michigan

9,938,444

9,883,640

Indiana

6,080,485

6,483,802

a. Write equations to model the populations for each state. Use M for the population of Michigan; I for the population of Indiana and t for time in years after 2000. Michigan: Indiana:

b. Use your equations to find the projected population for Michigan and Indiana in 2020. Michigan: Indiana:

#10 Points possible: 25. Total attempts: 5

Netflix made headlines in September 2011 when it split its streaming video service and its DVD by-mail service into two companies. This decision was based on projections that streaming videos will replace DVDs in the future. In this problem, you will explore these projections.

Total DVD and Blu-ray disc sales in 2009 were $8.73 billion. For the purposes of this problem, you will combine DVD and Blue-ray disk sales into one category of DVD sales. A study by In-Stat predicted that "physical disc sales will decline by $4.6 billion by 2014," and that "streaming, on the other hand, should grow from its current $2.3 billion to $6.3 billion over the same time period (2009–2014)."1 2

a. What is the slope, including units, of the model for DVD sales? Write a statement interpreting the meaning of the slope in the context of the problem.

b. Write an equation that represents the linear model for DVD sales. Let D = the sales of DVDs in billions of dollars, and t = the number of years after 2009.

c. What is the slope, including units, of the model for streaming videos? Write a statement interpreting the meaning of the slope in the context of the problem.

d. Write an equation that represents the linear model for streaming videos. Let S = the total sales of streamed videos in billions of dollars, and t = the number of years after 2009.

e. In what year do the models predict streaming video sales to exceed DVD sales?

#11 Points possible: 5. Total attempts: 5

Given a line passing through (19,1)(19,1) and (11,17)(11,17), which of the following is the correct slope of the line?

· m=(17)−(−1)(11)−(−19)m=(17)-(-1)(11)-(-19)

· m=(11)−(19)(17)−(−1)m=(11)-(19)(17)-(-1)

· m=(17)−(−1)(11)−(19)m=(17)-(-1)(11)-(19)

· m=(17)−(1)(11)−(19)m=(17)-(1)(11)-(19)

· m=(11)−(19)(17)−(1)m=(11)-(19)(17)-(1)

· m=(11)−(−19)(17)−(1)m=(11)-(-19)(17)-(1)

· m=(17)−(1)(11)−(−19)m=(17)-(1)(11)-(-19)

· m=(11)−(−19)(17)−(−1)m=(11)-(-19)(17)-(-1)

#12 Points possible: 5. Total attempts: 5

Determine the slope of the line passing through the points (9,2)(9,2) and (−3,−1)(-3,-1). m=m=

#13 Points possible: 5. Total attempts: 5

Determine the slope of the line passing through the points (7,−3)(7,-3) and (−4,6)(-4,6). m=m=

#14 Points possible: 5. Total attempts: 5

Given the points (0,−5)(0,-5) and (−9,−2)(-9,-2) and on a line, find its equation in the form y=mx+by=mx+b. y=y=