Answer :
To determine which type of model best describes the relationship between time spent running and distance traveled, we need to examine the rate of change between each pair of points. We will calculate the rate of change (or slope) between consecutive time points and check for consistency to see if the relationship is linear.
Consider the given data points:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time (minutes)} & \text{Distance (feet)} \\ \hline 1 & 530 \\ \hline 2 & 1050 \\ \hline 3 & 1600 \\ \hline 4 & 2110 \\ \hline 5 & 2650 \\ \hline \end{array} \][/tex]
1. Calculate the rate of change between each pair of points:
- From \( t = 1 \) to \( t = 2 \):
[tex]\[ \text{Rate of change} = \frac{1050 - 530}{2 - 1} = 520 \, \text{feet per minute} \][/tex]
- From \( t = 2 \) to \( t = 3 \):
[tex]\[ \text{Rate of change} = \frac{1600 - 1050}{3 - 2} = 550 \, \text{feet per minute} \][/tex]
- From \( t = 3 \) to \( t = 4 \):
[tex]\[ \text{Rate of change} = \frac{2110 - 1600}{4 - 3} = 510 \, \text{feet per minute} \][/tex]
- From \( t = 4 \) to \( t = 5 \):
[tex]\[ \text{Rate of change} = \frac{2650 - 2110}{5 - 4} = 540 \, \text{feet per minute} \][/tex]
2. Compare the rate of change values:
- The rates of change are \( 520 \), \( 550 \), \( 510 \), and \( 540 \) feet per minute.
3. Assess the consistency of the rate of change:
- The rates of change between the points are not exactly consistent—they vary from each other.
Given the calculated rates of change, the relationship is not linear because the rate of change is not constant. A linear model would require the rate of change to be exactly the same between each pair of points. In our case, the rates vary (520, 550, 510, 540).
Therefore, based on your choices:
- The correct conclusion is that the relationship is not linear because the rate of change between each pair of points is not exactly 520.
Given these calculations and analysis, the best answer to the question is:
- None of the provided statements are accurate for describing the relationship with the given data, as it does not perfectly fit the criteria for linearity due to the varying rate of change.
Consider the given data points:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time (minutes)} & \text{Distance (feet)} \\ \hline 1 & 530 \\ \hline 2 & 1050 \\ \hline 3 & 1600 \\ \hline 4 & 2110 \\ \hline 5 & 2650 \\ \hline \end{array} \][/tex]
1. Calculate the rate of change between each pair of points:
- From \( t = 1 \) to \( t = 2 \):
[tex]\[ \text{Rate of change} = \frac{1050 - 530}{2 - 1} = 520 \, \text{feet per minute} \][/tex]
- From \( t = 2 \) to \( t = 3 \):
[tex]\[ \text{Rate of change} = \frac{1600 - 1050}{3 - 2} = 550 \, \text{feet per minute} \][/tex]
- From \( t = 3 \) to \( t = 4 \):
[tex]\[ \text{Rate of change} = \frac{2110 - 1600}{4 - 3} = 510 \, \text{feet per minute} \][/tex]
- From \( t = 4 \) to \( t = 5 \):
[tex]\[ \text{Rate of change} = \frac{2650 - 2110}{5 - 4} = 540 \, \text{feet per minute} \][/tex]
2. Compare the rate of change values:
- The rates of change are \( 520 \), \( 550 \), \( 510 \), and \( 540 \) feet per minute.
3. Assess the consistency of the rate of change:
- The rates of change between the points are not exactly consistent—they vary from each other.
Given the calculated rates of change, the relationship is not linear because the rate of change is not constant. A linear model would require the rate of change to be exactly the same between each pair of points. In our case, the rates vary (520, 550, 510, 540).
Therefore, based on your choices:
- The correct conclusion is that the relationship is not linear because the rate of change between each pair of points is not exactly 520.
Given these calculations and analysis, the best answer to the question is:
- None of the provided statements are accurate for describing the relationship with the given data, as it does not perfectly fit the criteria for linearity due to the varying rate of change.
