Question
2 min 125 meters
4 min 205 meters
6min 285 meters
8 min 365 meters
10 min 445 meters
Which equation can be used to model the relationship between time and diatance
4 min 205 meters
6min 285 meters
8 min 365 meters
10 min 445 meters
Which equation can be used to model the relationship between time and diatance
Answers
GPT-4o mini
To determine the relationship between time and distance from the data provided, we first need to analyze the values given. Let's summarize the time (in minutes) and distance (in meters) pairs:
- At 2 minutes, the distance is 125 meters.
- At 4 minutes, the distance is 205 meters.
- At 6 minutes, the distance is 285 meters.
- At 8 minutes, the distance is 365 meters.
- At 10 minutes, the distance is 445 meters.
We'll look for a linear relationship, which is often found in such cases of distance over time.
To see if the relationship is linear, we can calculate the slope (rate of change of distance per unit time).
1. Calculate the distance increase for each interval of time:
- From 2 to 4 minutes: \(205 - 125 = 80\) meters
- From 4 to 6 minutes: \(285 - 205 = 80\) meters
- From 6 to 8 minutes: \(365 - 285 = 80\) meters
- From 8 to 10 minutes: \(445 - 365 = 80\) meters
The constant increase of 80 meters over each 2-minute interval suggests a linear relationship.
2. Determine the slope (rate of change):
\[
\text{slope} = \frac{\Delta \text{distance}}{\Delta \text{time}} = \frac{80 \text{ meters}}{2 \text{ minutes}} = 40 \text{ meters per minute}
\]
3. Formulate the linear equation \(d = mt + b\), where:
- \(d\) is the distance,
- \(m\) is the slope (40 meters/minute),
- \(t\) is the time (in minutes),
- \(b\) is the y-intercept (the distance at time \(t = 0\)).
4. To find \(b\), we can use one of the points. Let's use \(t = 2\) minutes and \(d = 125\) meters:
\[
125 = 40(2) + b \implies 125 = 80 + b \implies b = 125 - 80 = 45
\]
5. Thus, the equation modeling the relationship between time and distance is:
\[
d = 40t + 45
\]
Therefore, the equation that can be used to model the relationship between time and distance is:
\[
\boxed{d = 40t + 45}
\]
- At 2 minutes, the distance is 125 meters.
- At 4 minutes, the distance is 205 meters.
- At 6 minutes, the distance is 285 meters.
- At 8 minutes, the distance is 365 meters.
- At 10 minutes, the distance is 445 meters.
We'll look for a linear relationship, which is often found in such cases of distance over time.
To see if the relationship is linear, we can calculate the slope (rate of change of distance per unit time).
1. Calculate the distance increase for each interval of time:
- From 2 to 4 minutes: \(205 - 125 = 80\) meters
- From 4 to 6 minutes: \(285 - 205 = 80\) meters
- From 6 to 8 minutes: \(365 - 285 = 80\) meters
- From 8 to 10 minutes: \(445 - 365 = 80\) meters
The constant increase of 80 meters over each 2-minute interval suggests a linear relationship.
2. Determine the slope (rate of change):
\[
\text{slope} = \frac{\Delta \text{distance}}{\Delta \text{time}} = \frac{80 \text{ meters}}{2 \text{ minutes}} = 40 \text{ meters per minute}
\]
3. Formulate the linear equation \(d = mt + b\), where:
- \(d\) is the distance,
- \(m\) is the slope (40 meters/minute),
- \(t\) is the time (in minutes),
- \(b\) is the y-intercept (the distance at time \(t = 0\)).
4. To find \(b\), we can use one of the points. Let's use \(t = 2\) minutes and \(d = 125\) meters:
\[
125 = 40(2) + b \implies 125 = 80 + b \implies b = 125 - 80 = 45
\]
5. Thus, the equation modeling the relationship between time and distance is:
\[
d = 40t + 45
\]
Therefore, the equation that can be used to model the relationship between time and distance is:
\[
\boxed{d = 40t + 45}
\]