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? linear, because the r - value for the linear model is closest to 1 exponential, because the r - value for the exponential model is closest to 0 linear, because the rate of change between each pair of points is exactly 520 exponential, because the rate of change between each pair of points is 1.98

Explanation:

Step1: Recall model - type criteria

A linear model has a constant rate of change and a high correlation coefficient (r - value close to 1 or - 1). An exponential model has a non - constant multiplicative rate of change.

Step2: Analyze rate of change

For the given data:
When time changes from 1 to 2 minutes, distance changes from 530 to 1050 feet. The change in distance $\Delta d_1=1050 - 530 = 520$ feet.
When time changes from 2 to 3 minutes, distance changes from 1050 to 1600 feet. The change in distance $\Delta d_2=1600 - 1050 = 550$ feet.
Since the rate of change is not exactly constant, we rely on the r - value. A high r - value (close to 1 for a positive linear relationship) indicates a good fit for a linear model. In general, an r - value close to 0 for an exponential model means it is not a good fit, and we expect a non - constant multiplicative rate of change for an exponential model which is not the case here. So, a linear model is appropriate because the r - value for the linear model is closest to 1.

Answer:

A. linear, because the r - value for the linear model is closest to 1