Write and interpret linear functions

Question

Tracy, a long distance runner, times herself as she runs a half marathon, which is 13.1 miles long. The distance is measured in miles and the time is measured in minutes. The table shows DISTANCE (M) and has the numbers 1,2,3,4,5 listed at the top. Underneath is TIME (MIN) and has the numbers 8.5, 16.2, 24.6, 33.1, 41.8 underneath.

1. Create a linear function to model this situation.

2. Based on your model, how will it take Tracy to run the half-marathon rounded to the nearest minute?

Answer 1

Hi, Elizabeth.

Part 1 of this problem asks you to create a linear function that models the given data. In this case, distance d (in miles) is the independent variable, and time t (in minutes) is the dependent variable. So, the function you're looking for has the form t(d) = md + b where m and b are constants. You need to figure out what the values of m and b are.

First, it makes sense to assume that the function should output a time of 0 minutes when the distance run is 0 miles. This means that t(0) = 0. Since t(0) = m(0) + b = b, you can conclude that b should be 0.

Finding the value of m is trickier because Tracy is not running at a constant speed. You can see that her speed varies by observing that the d-values in the table are equally spaced (they differ by 1), but the corresponding times are not equally spaced. Find the differences in consecutive times like so:

16.2 - 8.5 = 7.7

24.6 - 16.2 = 8.4

33.1 - 24.6 = 8.5

41.8 - 33.1 = 8.7

If these differences had been constant, you could simply let m equal that constant. Given that they're not constant, you must decide what's the best constant to use for the value of m. For instance, you might decide to take the average of the differences: (7.7 + 8.4 + 8.5 + 8.7)/4 = 8.325. But this isn't the only value of m possible. For instance, if you add the data point (d, t) = (0, 0) to the table, you can obtain another difference:

8.5 - 0 = 8.5

And if you include that difference in your average, you get (8.5 + 7.7 + 8.4 + 8.5 + 8.7)/5 = 8.36 as your value for m.

Let's go ahead and use 8.36 as the value of m. So, the model becomes t(d) = 8.36d + 0 = 8.36d. (Note that m is measured in minutes per mile because you're going to multiply 8.36 by the number of miles run in order to obtain the number of minutes it takes Tracy to run that distance.)

Part 2 of the problem asks you to predict how long it takes Tracy to run 13.1 miles. The model predicts that the time will be t(13.1) = 8.36(13.1) = 109.516 minutes, or about 110 minutes when rounded to the nearest minute. (Note that if you had used the model t(d) = 8.325, you'd get t(13.1) = 8.325(13.1) = 109.0575, or about 109 minutes when rounded to the nearest minute.)

One last thing: If you use the Desmos graphing calculator, you can enter Tracy's data into a table and perform what's called a linear regression using a model of the form t(d) = md. I did this myself and got a value of m equal to 8.29273. The regression model t(d) = 8.29273d predicts a time of 109 minutes (to the nearest minute) for Tracy to run 13.1 miles.

As you can see from this problem, modeling involves making decisions and assumptions, and you may get different results based on what decisions and assumptions you make.

Doug