Question

the table shows the relationship between time spent running and distance traveled. running distance over time

time (minutes)	distance (feet)
2	1,050
3	1,600
4	2,110
5	2,650

which type of model best describes the relationship?
o linear, because the r value for the linear model is closest to 1
o exponential, because the r value for the exponential model is closest to 0
o linear, because the rate of change between each pair of points is exactly 520
o exponential, because the rate of change between each pair of points is 1.98

Explanation:

Step1: Calculate rate of change

For a linear relationship, the rate of change (slope) between points should be constant. Calculate the differences in distance and time.
For time from 1 - 2 minutes: Distance change = $1050 - 530=520$ feet.
For time from 2 - 3 minutes: Distance change = $1600 - 1050 = 550$ feet.
For time from 3 - 4 minutes: Distance change = $2110 - 1600=510$ feet.
For time from 4 - 5 minutes: Distance change = $2650 - 2110 = 540$ feet.
The rate of change is not exactly 520.

Step2: Analyze linear - r value concept

The $r$ - value (correlation coefficient) measures the strength and direction of a linear relationship. A value close to 1 or - 1 indicates a strong linear relationship. Without calculating the $r$ - value exactly, we can see that the data points seem to follow a linear trend more than an exponential one. In an exponential relationship, the ratio of consecutive $y$ - values should be constant. Here, the ratios are not constant.

Answer:

linear, because the $r$ value for the linear model is closest to 1