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

Explanation:

Step1: Calculate rate of change

To check if it's linear, find the rate of change between consecutive points. For example, from $t = 1$ to $t=2$: Rate of change $=\frac{1050 - 530}{2 - 1}=\frac{520}{1}=520$. From $t = 2$ to $t = 3$: Rate of change $=\frac{1600 - 1050}{3 - 2}=\frac{550}{1}=550$. Since the rate of change is not exactly the same between each pair of points, the statement about a constant rate - of - change of 520 is incorrect.

Step2: Consider $r$ - value concept

The $r$ - value (correlation coefficient) measures the strength and direction of a linear relationship. A value close to 1 indicates a strong positive linear relationship, and a value close to 0 indicates a weak or non - linear relationship. We are not given the $r$ - values for the linear and exponential models in the problem statement, but we can analyze the data's behavior. Let's assume we calculate the differences in distances for consecutive time intervals: For $t = 1$ to $t = 2$: $\Delta d=1050 - 530 = 520$ For $t = 2$ to $t = 3$: $\Delta d=1600 - 1050 = 550$ For $t = 3$ to $t = 4$: $\Delta d=2110 - 1600 = 510$ For $t = 4$ to $t = 5$: $\Delta d=2650 - 2110 = 540$ The differences are approximately constant, which is a characteristic of a linear relationship. A linear model is appropriate when the change in the dependent variable (distance) with respect to the independent variable (time) is approximately constant. And a strong linear relationship would have an $r$ value close to 1.

Answer:

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