Question

the table shows the relationship between time spent running and distance traveled.

running distance over time
\

$$\begin{tabular}{|c|c|} \\hline time (minutes) & distance (feet) \\\\ \\hline 1 & 530 \\\\ \\hline 2 & 1,050 \\\\ \\hline 3 & 1,600 \\\\ \\hline 4 & 2,110 \\\\ \\hline 5 & 2,650 \\\\ \\hline \\end{tabular}$$

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:

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<reasoning>

Calculate rates of change

Using the Linear Rate of Change knowledge point
\[

$$\begin{aligned} m_1 &= \frac{1050 - 530}{2 - 1} = 520 \\ m_2 &= \frac{1600 - 1050}{3 - 2} = 550 \\ m_3 &= \frac{2110 - 1600}{4 - 3} = 510 \\ m_4 &= \frac{2650 - 2110}{5 - 4} = 540 \end{aligned}$$

\]

Evaluate model options

Using the Linear Modeling knowledge point
\[

$$\begin{aligned} &\text{The rates of change are approximately constant (around 520-550).}\\ &\text{Since the rate of change is not exactly 520, option 3 is incorrect.}\\ &\text{A linear model is appropriate, and its correlation coefficient } r \text{ will be closest to 1.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) linear, because the \(r\) value for the linear model is closest to 1</mcq-correct>
<mcq-option>(B) exponential, because the \(r\) value for the exponential model is closest to 0</mcq-option>
<mcq-option>(C) linear, because the rate of change between each pair of points is exactly 520</mcq-option>
<mcq-option>(D) exponential, because the rate of change between each pair of points is 1.98</mcq-option>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Calculate rates of change

Using the Linear Rate of Change knowledge point
\[

$$\begin{aligned} m_1 &= \frac{1050 - 530}{2 - 1} = 520 \\ m_2 &= \frac{1600 - 1050}{3 - 2} = 550 \\ m_3 &= \frac{2110 - 1600}{4 - 3} = 510 \\ m_4 &= \frac{2650 - 2110}{5 - 4} = 540 \end{aligned}$$

\]

Evaluate model options

Using the Linear Modeling knowledge point
\[

$$\begin{aligned} &\text{The rates of change are approximately constant (around 520-550).}\\ &\text{Since the rate of change is not exactly 520, option 3 is incorrect.}\\ &\text{A linear model is appropriate, and its correlation coefficient } r \text{ will be closest to 1.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) linear, because the \(r\) value for the linear model is closest to 1</mcq-correct>
<mcq-option>(B) exponential, because the \(r\) value for the exponential model is closest to 0</mcq-option>
<mcq-option>(C) linear, because the rate of change between each pair of points is exactly 520</mcq-option>
<mcq-option>(D) exponential, because the rate of change between each pair of points is 1.98</mcq-option>
</answer>

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