## Question

Project on Linear Functions

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?

0 2 4 6 8 10

0 20 40 60 80 100

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 speed is constant as in his warm-up in part a.

Distance (m) from start

Time (s) Usian Dude

1

3

5

7

9

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 distance d, 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 = mx + b. 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.

(To easily make a graph in Excel instead, see instructions bĂ©low for letter ''')

100

90

80

70

60

50

40

30

20

10

1 2 3 4 5 6 7 8 9 10

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.

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