INVESTIGATION: Running a marathon

In this investigation, we will determine how long it would take you to run a marathon. As we go, we will be exploring rational functions in real-life, including their characteristics and their graphs.

 

Materials

• Tape
• Stopwatch
• Paper
• Different coloured pens
• Measuring Tape
• Internet

 

Instructions

You can work with a partner for this investigation.

1. The average marathon is about 42 kilometres long. The following function indicates how long it would take someone to run a marathon with respect to the rate at which they run: t = \frac{42}{r} where t is the time it takes you to run the marathon (in hours), and r is the rate at which you run (in kilometres per hour).
2. Fill in the table of values for the given function.
    

   Running Rate in km/h (x)	1	2	5	10	15	20	25
   Time Taken in hours (y)

3. Time how long it takes you to run 25 metres. You may want to use your tape to mark a starting and ending location.
4. Divide the distance you ran (25 metres) by the total amount of time it took you to determine your rate.
5. Convert the rate you just found into kilometres per hour instead of metres per second.
6. Draw an arrow to indicate where you fall on the graph you have created.

 

Questions

1. In what quadrant(s) does a portion of the function that you graphed lie?
2. If you plot the entire function of t = \frac{42}{r}  on the Cartesian plane what quadrant(s) would your graph lie in?
3. Does it make sense to graph the entire function for this scenario? Why or why not?
4. What is the domain of the function that you graphed?
5. What is the range of the function that you graphed?
6. What are the vertical and horizontal asymptotes of the graph that you created? Draw them in on your graph using a dotted line drawn by a different coloured pen from the one you previously used.
7. Interpret the horizontal and vertical asymptotes in terms of the situation. Why does it make sense that there are asymptotes?
8. How long would it take someone to run the marathon approximately if they were running at a rate of 32 kilometres per hour? Is this plausible? Why or why not? You can use the internet to look up any information that may be relevant to answering this question. 
9. How realistic do you think your predicted marathon rate would be?
10. Compare with a friend! If you both could maintain your rate for the entire marathon who would run the marathon faster?