Question
4. Given the table for x, time in minutes, and y, distance traveled:x: time 8 5 7 2 9y: distance 664 415 581 166 747 (4 points)a. Every minute meters are traveled.b. Write an equation to show the relationship between x and y: .c. This relationship is because the y-intercept is .
Answers
GPT-4o mini
To analyze the relationship between time (x in minutes) and distance traveled (y in meters), we can first summarize the given data:
| Time (x) | Distance (y) |
|----------|--------------|
| 8 | 664 |
| 5 | 415 |
| 7 | 581 |
| 2 | 166 |
| 9 | 747 |
### a. Find the rate of meters traveled per minute.
To determine the rate at which distance is traveled per minute, we can calculate the distance traveled per minute for each data point:
1. For \( x = 8: \, \frac{664}{8} = 83 \, \text{meters per minute}\)
2. For \( x = 5: \, \frac{415}{5} = 83 \, \text{meters per minute}\)
3. For \( x = 7: \, \frac{581}{7} = 83 \, \text{meters per minute}\)
4. For \( x = 2: \, \frac{166}{2} = 83 \, \text{meters per minute}\)
5. For \( x = 9: \, \frac{747}{9} = 83 \, \text{meters per minute}\)
As we can see, the rate of distance traveled per minute is consistently \(83\) meters per minute.
### b. Write an equation to show the relationship between x and y.
The relationship can be described using the linear equation \( y = mx + b \), where:
- \( m \) is the slope (the distance per minute),
- \( b \) is the y-intercept (the initial distance traveled when time is zero).
Given that \( m = 83 \) and assuming the y-intercept \( b \) is \(0\) (if we extrapolate and assume no distance is traveled when no time has passed), the equation becomes:
\[ y = 83x + 0 \]
or simply:
\[ y = 83x \]
### c. This relationship is **linear** because the y-intercept is **0**.
The equation \( y = 83x \) indicates a linear relationship where the distance increases directly with time, starting from zero when no time has passed, hence the y-intercept is \(0\).
| Time (x) | Distance (y) |
|----------|--------------|
| 8 | 664 |
| 5 | 415 |
| 7 | 581 |
| 2 | 166 |
| 9 | 747 |
### a. Find the rate of meters traveled per minute.
To determine the rate at which distance is traveled per minute, we can calculate the distance traveled per minute for each data point:
1. For \( x = 8: \, \frac{664}{8} = 83 \, \text{meters per minute}\)
2. For \( x = 5: \, \frac{415}{5} = 83 \, \text{meters per minute}\)
3. For \( x = 7: \, \frac{581}{7} = 83 \, \text{meters per minute}\)
4. For \( x = 2: \, \frac{166}{2} = 83 \, \text{meters per minute}\)
5. For \( x = 9: \, \frac{747}{9} = 83 \, \text{meters per minute}\)
As we can see, the rate of distance traveled per minute is consistently \(83\) meters per minute.
### b. Write an equation to show the relationship between x and y.
The relationship can be described using the linear equation \( y = mx + b \), where:
- \( m \) is the slope (the distance per minute),
- \( b \) is the y-intercept (the initial distance traveled when time is zero).
Given that \( m = 83 \) and assuming the y-intercept \( b \) is \(0\) (if we extrapolate and assume no distance is traveled when no time has passed), the equation becomes:
\[ y = 83x + 0 \]
or simply:
\[ y = 83x \]
### c. This relationship is **linear** because the y-intercept is **0**.
The equation \( y = 83x \) indicates a linear relationship where the distance increases directly with time, starting from zero when no time has passed, hence the y-intercept is \(0\).