Let's analyze the data provided in the table to determine whether the distance run and the elevation are linear or nonlinear functions.
### Step 1: Understanding Linear and Nonlinear Functions
A linear function exhibits a constant rate of change. This means that the difference between consecutive values of the dependent variable (distance or elevation) should be constant when the independent variable (time) changes by a consistent amount.
### Step 2: Analyzing Distance
We need to calculate the rate of change in distance for each consecutive time interval:
[tex]\[
\begin{align*}
\text{Rate from } t = 1 \text{ to } t = 2 & : 0.38 - 0.19 = 0.19 \\
\text{Rate from } t = 2 \text{ to } t = 3 & : 0.57 - 0.38 = 0.19 \\
\text{Rate from } t = 3 \text{ to } t = 4 & : 0.76 - 0.57 = 0.19 \\
\text{Rate from } t = 4 \text{ to } t = 5 & : 0.95 - 0.76 = 0.19 \\
\text{Rate from } t = 5 \text{ to } t = 6 & : 1.14 - 0.95 = 0.19 \\
\end{align*}
\][/tex]
Since all the rates of change in distance are the same (0.19), the distance is a linear function of time.
### Step 3: Analyzing Elevation
Next, we calculate the rate of change in elevation for each consecutive time interval:
[tex]\[
\begin{align*}
\text{Rate from } t = 1 \text{ to } t = 2 & : 26 - 12 = 14 \\
\text{Rate from } t = 2 \text{ to } t = 3 & : 67 - 26 = 41 \\
\text{Rate from } t = 3 \text{ to } t = 4 & : 98 - 67 = 31 \\
\text{Rate from } t = 4 \text{ to } t = 5 & : 124 - 98 = 26 \\
\text{Rate from } t = 5 \text{ to } t = 6 & : 145 - 124 = 21 \\
\end{align*}
\][/tex]
These differences are not constant, so the elevation as a function of time is nonlinear.
### Step 4: Final Conclusion
Based on the analysis:
- The distance run is a linear function since it has a constant rate of change.
- The elevation is a nonlinear function since it does not have a constant rate of change.
Thus, the correct statement is:
"Both the distance run and the elevation are nonlinear functions because they do not have constant rates of change."
