Let's analyze the problem and determine the type of model that best describes the relationship between time spent running and distance traveled.
1. Determine the rate of change (slope) between each pair of points:
- Between 1 minute and 2 minutes:
[tex]\[
\frac{1050 - 530}{2 - 1} = \frac{520}{1} = 520
\][/tex]
- Between 2 minutes and 3 minutes:
[tex]\[
\frac{1600 - 1050}{3 - 2} = \frac{550}{1} = 550
\][/tex]
- Between 3 minutes and 4 minutes:
[tex]\[
\frac{2110 - 1600}{4 - 3} = \frac{510}{1} = 510
\][/tex]
- Between 4 minutes and 5 minutes:
[tex]\[
\frac{2650 - 2110}{5 - 4} = \frac{540}{1} = 540
\][/tex]
2. Evaluate the consistency of the rate of change:
From the calculations above, we can see that the rate of change is not exactly the same for each pair of points. The rates of change are 520, 550, 510, and 540 feet per minute, respectively.
This means that the rate of change is not exactly 520 feet/minute for each pair of points.
3. Check for linearity using the correlation coefficient (i.e., the r-value):
Since the rate of change is not consistent, the data set does not fit a perfectly linear model.
Considering the above evaluations, the correct conclusion based on the data is:
- The model is not linear, because the rate of change between each pair of points is not exactly 520 and doesn't show consistent rate patterns.
Therefore, given the available options:
- Linear, because the [tex]\( r \)[/tex]-value for the linear model is closest to 1
- Exponential, because the [tex]\( r \)[/tex]-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
Our conclusion based on the information is:
None of the provided options correctly describe the relationship given the data. However, based on a common interpretation of the last step:
- The best answer in these provided options would be: The model is not linear, because the rate of change between each pair of points is not consistent and not exactly 520.
