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This table shows data collected by a runner.

\begin{tabular}{|c|c|c|}
\hline \begin{tabular}{c}
Time \\
(minutes)
\end{tabular} & \begin{tabular}{c}
Distance \\
(miles)
\end{tabular} & \begin{tabular}{c}
Elevation \\
(meters)
\end{tabular} \\
\hline 1 & 0.19 & 12 \\
\hline 2 & 0.38 & 26 \\
\hline 3 & 0.57 & 67 \\
\hline 4 & 0.76 & 98 \\
\hline 5 & 0.95 & 124 \\
\hline 6 & 1.14 & 145 \\
\hline
\end{tabular}

Which statement about the scenario represented in the table is true? Assume time is the independent variable.

A. The distance run is a nonlinear function because it does not have a constant rate of change.

B. The elevation is a nonlinear function because it does not have a constant rate of change.

C. Both the distance run and the elevation are nonlinear functions because they do not have constant rates of change.

D. Both the distance run and the elevation are linear functions because they have a constant rate of change.



Answer :

GinnyAnswer
To determine which statement about the scenario represented in the table is true, let’s analyze the rates of change for both distance and elevation.

### Distance Analysis
We need to calculate the rate of change of the distance over time. This is done by finding the differences between consecutive distances and dividing by the time interval.

Calculate the differences in distance:
- [tex]\((0.19 - 0)\)[/tex] miles at [tex]\((1 - 0)\)[/tex] minute = [tex]\(0.19\)[/tex] miles/minute
- [tex]\((0.38 - 0.19)\)[/tex] miles at [tex]\((2 - 1)\)[/tex] minute = [tex]\(0.19\)[/tex] miles/minute
- [tex]\((0.57 - 0.38)\)[/tex] miles at [tex]\((3 - 2)\)[/tex] minute = [tex]\(0.19\)[/tex] miles/minute
- [tex]\((0.76 - 0.57)\)[/tex] miles at [tex]\((4 - 3)\)[/tex] minute = [tex]\(0.19\)[/tex] miles/minute
- [tex]\((0.95 - 0.76)\)[/tex] miles at [tex]\((5 - 4)\)[/tex] minute = [tex]\(0.19\)[/tex] miles/minute
- [tex]\((1.14 - 0.95)\)[/tex] miles at [tex]\((6 - 5)\)[/tex] minute = [tex]\(0.19\)[/tex] miles/minute

We see that the rate of change (0.19 miles/minute) is constant for each interval, indicating that the distance-time relationship is linear.

### Elevation Analysis
We need to calculate the rate of change of the elevation over time. Again, this involves finding the differences between consecutive elevations and dividing by the time interval.

Calculate the differences in elevation:
- [tex]\((26 - 12)\)[/tex] meters at [tex]\((2 - 1)\)[/tex] minute = [tex]\(14\)[/tex] meters/minute
- [tex]\((67 - 26)\)[/tex] meters at [tex]\((3 - 2)\)[/tex] minute = [tex]\(41\)[/tex] meters/minute
- [tex]\((98 - 67)\)[/tex] meters at [tex]\((4 - 3)\)[/tex] minute = [tex]\(31\)[/tex] meters/minute
- [tex]\((124 - 98)\)[/tex] meters at [tex]\((5 - 4)\)[/tex] minute = [tex]\(26\)[/tex] meters/minute
- [tex]\((145 - 124)\)[/tex] meters at [tex]\((6 - 5)\)[/tex] minute = [tex]\(21\)[/tex] meters/minute

Here, the rate of change varies for each interval, indicating that the elevation-time relationship is nonlinear.

### Conclusion
Based on the analysis:
- The distance-time relationship is linear because the rate of change is constant (0.19 miles/minute).
- The elevation-time relationship is nonlinear because the rate of change is not constant.

Therefore, the correct statement is:
The elevation is a nonlinear function because it does not have a constant rate of change.

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