The Ultimate Race: Usian Bolt holds the Olympic record for the 100 meter (m) sprint finishing in 9.69 seconds (s). He is challenging all comers to a race and has agreed to give his competitors a head start. How much of a head start would you need to win a race against Usian? Let's investigate! Go to part a to begin. a. The table below shows Usian's distance in meters at constant time intervals during a warm-up run. What is Usian's average speed over any given time interval? Time (s) Distance (m) 0 0 2 20 4 40 6 60 8 80 10 100 b. Let t represent time in seconds and d be distance in meters. Use the rate found in part a to construct a linear equation that will model Usain's distance in meters as a function of time in seconds during a 100 meter race. c. Now suppose some dude has an average sprint speed of 5 meters per second (m/s). If Usian gives his challenger a 30 meter head start will the dude win the race? Complete the following table to help answer this question.Assume Usain's average speedis constant as in his warm-up in part a. Distance (m) from start Time (s) Usian Dude 0 1 2 3 4 5 6 7 8 9 10 d. Who won the race and how far behind was the loser? Explain using complete sentences. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ e. Construct a second linear equation that will model the dude's distanced, in meters, as a function of time t, in seconds, during a 100 meter race. f. The slope-intercept form of a linear equation is y+mxtb. The equation for part e should be in the form d=mt+b where m is the slope and b is the y-intercept. Explain what the parameters m and b represent in the context of the 100 meter race. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ g. Graph the linear equations from parts b and e. Label each line with the correct name using Usian or Dude. Mark the point where the two lines intersect with an ordered pair. h. What does the point of intersection indicate about the race between Usian and the Dude. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ i. If all head starts are measured to the nearest meter, then what is the smallest head start needed for the dude to win a 100 meter race against Usian. Assume Usain's average speed is a constant 10 meters per second (m/s) and the dude is a constant 5 m/s. j. Bonus: Construct an Excel spreadsheet that allows the user to enter any speed in meters per second for the slope m and a head start in meters for the y-intercept band then automatically builds a table similar to part c that shows Usian in a 100 m race with the user. Are NECC students finishing degrees and certificates? NECC was the first college in Massachusetts to commit to the American Association of Community Colleges \"Call to Action,\" pledging to \"produce 50 percent more students with high quality degrees and certificates by 2020, while increasing access and quality.\" This means a goal of 1500 graduates by 2020. Given below is the number of graduates obtaining degrees and certificates in the years from 2007 to 2012. Observe the data and go to part a to begin. Year Graduate s 2007 748 200 8 843 200 2010 9 885 993 2011 2012 104 8 1141 a. State the two related variables in the table above and indicate which is independent and which is dependent. Explain your reasoning using complete sentences. Also define each variable using an appropriate letter or symbol. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ b. Construct a scatter plot of the above data using the grid below. Let 2007 be year 0, 2008 be year 1, 2009 be year 3 and so on. Be sure to label and scale each axis and include a title. Check your work by using technology to graph the same data. c. Observe the graph you created and explain any trend or relationship you see in the two-variable data. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ d. Use a straightedge to draw a line of best fit for the data you graphed in part b. This is a line that comes as close as possible to all the data points but does not necessarily pass through any of them. Then select two points on your best fit line and record the ordered pairs below. e. Use the selected points from part d to find the slope of your best fit line. f. Use the information from parts d and e to find the vertical intercept (y-intercept) of your best fit line. g. Use the results of parts e and f to write an equation of your best fit line. h. Use your best fit equation to predict the number of graduates obtaining degrees and certificates in 2013. i. Use technology to find the linear regression line that models the given data. State the equation below and explain how your equation of best fit compares with the linear regression equation produced by technology. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ j. Use function notation to write the linear regression line provided by technology. Then predict the number of graduates obtaining degrees and certificates in 2020 and express your result using function notation. k. Explain how NECC did in meeting their goal of 1500 graduates by 2020. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________