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MAKING CONNECTIONS TO THE COLLABORATION
(1) Which of the following was one of the main mathematical ideas of the collaboration?
(i) Understanding heart rate is important to prevent heart attacks.
(ii) A linear model has a vertical intercept.
(iii) If a model based on data is created correctly, it will accurately predict values beyond the data.
(iv) Models created from data are based on assumptions. It is important to ask if the assumptions are correct or will remain correct over time.
DEVELOPING SKILLS AND UNDERSTANDING
The flight times and distances from Memphis, Tennessee to various cities are given in the table and graph below. Use this information to answer Questions 3–5.
Time to Fly from Memphis to Various Cities^{3}
City | Distance (miles) | Time (minutes) |
Atlanta | 332 | 82 |
Baltimore | 787 | 162 |
Boston | 1139 | 191 |
Cincinnati | 403 | 100 |
Detroit | 611 | 121 |
(2) Use what you know about the relationship of the equation to the graph to select the model from the list below that is most representative of the data. Do not actually find a linear model to make your choice. In each option, d represents the distance in miles and t represents the time in minutes.
(i) t = –0.13d + 40
(ii) t = 0.13d + 40
(iii) t = 0.13d + 110
(iv) t = –0.13d + 110
(3) Using the model, calculate the flight time from Memphis to Philadelphia if the distance between the two cities is 874 miles. Round your answer to the nearest tenth of an hour.
(4) Based on the model you chose, select the response that best tells how long it will take a Delta plane flying at full speed to travel 1 mile.
(i) The plane goes 0.13 miles in one minute so it takes between 7 and 8 minutes to go 1 mile.
(ii) It takes 40 minutes to go 1 mile.
(iii) It takes 0.13 minutes or around 8 seconds to go 1 mile.
(iv) It takes 40.13 minutes to go 1 mile.
WORLD RECORD SPEEDS OVER TIME
The table below shows the world record times in the men’s 100-meter dash from 1912 to 2009. Use this to answer Questions 5–8.
Men’s 100-meter dash^{4}
Year | World Record Time (nearest tenth of a second) |
1912 | 10.6 |
1921 | 10.4 |
1930 | 10.3 |
1936 | 10.2 |
1956 | 10.1 |
1960 | 10 |
1968 | 9.9 |
1999 | 9.8 |
2008 | 9.7 |
2009 | 9.6 |
(5) (a) Create a scatterplot using the data relating “years since 1900” and “World Record Time” for the Men’s 100-meter dash.
(b) Write an equation for a linear model representing the data. Define the variables you use in your equation.
(6) Use the model you created to predict in what year the world record will fall below 9.0 seconds.
(7) Identify which of the following statements are correct based on your model. Note that the values given are approximations. There may be more than one correct answer.
(i) On average, the men’s world record in the 100-meter dash has decreased about one-hundredth of a minute each year.
(ii) On average, the men’s world record in the 100-meter dash has decreased about one-tenth of a minute every 10 years.
(iii) On average, the men’s world record in the 100-meter dash has decreased about one-hundredth of a second each year.
(iv) On average, the men’s world record in the 100-meter dash has decreased about one-tenth of a second every 10 years.
(8) Models often only work well for a limited range of input values. Outside that range, the model is said to break down. Which of these explains why your world record model might break down? There may be more than one correct answer.
(i) The data do not form a perfect line.
(ii) Eventually the model will predict negative times for the race.
(iii) Changes in equipment or training might make for sudden improvements in times and change the trend in the current data.
(iv) The times are not really accurate.
(9) The following chart gives sunrise times for New York City in hours after midnight as measured in Eastern Standard Time (EST) on the fifteenth day of each month. The first month is August 2010, the second month is September 2010, and so on. Would it be appropriate to use a linear model to represent these data? Explain.
MAKING CONNECTIONS ACROSS THE COURSE
In baseball, a team must score more runs than its opponent to win the game. Players have to reach base for a team to score runs. How often a player reaches base is measured by the on-base percentage. However, a player who reaches a base may or may not score. The scatterplot gives data relating the number of runs and the on-base percentage. Use the graph to answer Questions 10 and 11.
(11) Katy used the graph to create a linear model that she could use to predict the number of runs scored using the on-base percentage. Her model was r = 30b + 450, where r = number of runs and b = on-base percentage (i.e., for 32%, b = 32). What is wrong with Katy’s model?
(i) The data are not nearly linear so a linear model should not be used.
(ii) Katy put the rate and the vertical intercepts in the wrong places in the equation.
(iii) Katy used the vertical intercept in her equation where she should have used the horizontal intercept.
(iv) The vertical intercept of 450 runs is incorrect because the scale on the horizontal axis does not start at zero.
(12) Based on the slope in Katy’s model, if on-base percentage for a team goes up by 1% then by about how much does the number of runs scored go up?
(i) 480 runs
(ii) 450 runs
(iii) 30 runs
(iv) 0.3 runs
(12) The graph below shows the cost of basic cable service from 2002 to 2010.^{5}
(a) An employee of the cable company made a linear equation of the data using the 2002 cost as a starting value and the 2005 cost as a second data point. What is the company’s equation? Use C for the cost per month and t for time in years since 2002. Round the slope to two decimal places.
(b) A consumer advocate made a linear equation of the data using the 2002 cost as a starting value and the 2009 cost as a second data point. What is the advocate’s equation? Use C for the cost per month and t for time in years since 2002. Round the slope to two decimal places.
(c) What will the cost per month be in 2015 based on the company’s model found in (a)? Based on the consumer advocate’s model found in (b)? (Hint: Remember that t is the number of years since 2002.)
(i) Company's projection:
(ii) Consumer advocate’s projection:
(d) Which of the following is the most accurate estimate of the relative increase in the monthly cost of cable services from 2002 to 2010?
(i) 90–100%
(ii) 75–85%
(iii) 50–60%
(iv) 35–45%
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^{3} Flight times retrieved for flights leaving July 28, 2011 from Delta.com.
^{4} http://en.Wikipedia.org/wiki/Men%27s_100_metres_world_record_progression