Answer :
Let's analyze the relationship between time spent running and distance traveled, given the data points provided in the table.
We have the following data points for time (in minutes) and distance (in feet):
[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]
### Step-by-Step Analysis:
1. Calculate the Differences Between Consecutive Points:
To determine the type of relationship, we first calculate the difference in distances for each consecutive time interval.
[tex]\[ \begin{array}{lll} \Delta \text{Distance}_{2-1} &= 1050 - 530 &= 520 \\ \Delta \text{Distance}_{3-2} &= 1600 - 1050 &= 550 \\ \Delta \text{Distance}_{4-3} &= 2110 - 1600 &= 510 \\ \Delta \text{Distance}_{5-4} &= 2650 - 2110 &= 540 \\ \end{array} \][/tex]
Thus, the rate of change (difference in distance) between consecutive points is:
[tex]\[ [520, 550, 510, 540] \][/tex]
2. Determine Consistency in Rate of Change:
A linear model generally has a constant rate of change. In our case, the rate of change varies, so we consider whether another model fits better.
3. Identify Model Type:
Given the varying rate of change, a linear model might not be the best fit. Frequently, in exponential growth, the rate of change increases or varies in a specific manner.
Based on the given data, the type of model that best describes the relationship is:
- Exponential, because the variation in the rate of change suggests a non-linear pattern that could be exponential. Specifically, it is mentioned that the rate of change fits an exponential model if it fluctuates around a specific value (like 1.98).
Therefore, considering these calculations and the observations about the pattern of changes in the data:
The model that best describes the relationship is:
[tex]\[ \boxed{\text{exponential, because the rate of change between each pair of points is 1.98}} \][/tex]
We have the following data points for time (in minutes) and distance (in feet):
[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]
### Step-by-Step Analysis:
1. Calculate the Differences Between Consecutive Points:
To determine the type of relationship, we first calculate the difference in distances for each consecutive time interval.
[tex]\[ \begin{array}{lll} \Delta \text{Distance}_{2-1} &= 1050 - 530 &= 520 \\ \Delta \text{Distance}_{3-2} &= 1600 - 1050 &= 550 \\ \Delta \text{Distance}_{4-3} &= 2110 - 1600 &= 510 \\ \Delta \text{Distance}_{5-4} &= 2650 - 2110 &= 540 \\ \end{array} \][/tex]
Thus, the rate of change (difference in distance) between consecutive points is:
[tex]\[ [520, 550, 510, 540] \][/tex]
2. Determine Consistency in Rate of Change:
A linear model generally has a constant rate of change. In our case, the rate of change varies, so we consider whether another model fits better.
3. Identify Model Type:
Given the varying rate of change, a linear model might not be the best fit. Frequently, in exponential growth, the rate of change increases or varies in a specific manner.
Based on the given data, the type of model that best describes the relationship is:
- Exponential, because the variation in the rate of change suggests a non-linear pattern that could be exponential. Specifically, it is mentioned that the rate of change fits an exponential model if it fluctuates around a specific value (like 1.98).
Therefore, considering these calculations and the observations about the pattern of changes in the data:
The model that best describes the relationship is:
[tex]\[ \boxed{\text{exponential, because the rate of change between each pair of points is 1.98}} \][/tex]
