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 two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For $(x_1 = 1,y_1 = 530)$ and $(x_2 = 2,y_2 = 1050)$: $m_1=\frac{1050 - 530}{2 - 1}=520$.
For $(x_1 = 2,y_1 = 1050)$ and $(x_2 = 3,y_2 = 1600)$: $m_2=\frac{1600 - 1050}{3 - 2}=550$.
For $(x_1 = 3,y_1 = 1600)$ and $(x_2 = 4,y_2 = 2110)$: $m_3=\frac{2110 - 1600}{4 - 3}=510$.
For $(x_1 = 4,y_1 = 2110)$ and $(x_2 = 5,y_2 = 2650)$: $m_4=\frac{2650 - 2110}{5 - 4}=540$.
The average rate of change is approximately constant.
We are not given $r$ - values in the problem, so we can't use the $r$ - value criterion. But since the rate of change between pairs of points is approximately constant, it is a linear relationship.

Answer:

linear, because the rate of change between each pair of points is approximately constant (close - to 520 on average)