Question

question: Given the table for x, time in minutes, and y, distance traveled: x: time 8 5 7 2 9 y: 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 .

Answer 1

This relationship is (BLANK, Porportional or non) because the y-intercept is . (0,0), (0,83).

Answer 2

To analyze the given data and establish a relationship between time (x) and distance traveled (y), we can follow these steps:

### Given Data:
- x (time in minutes): 8, 5, 7, 2, 9
- y (distance in meters): 664, 415, 581, 166, 747

### a. Every minute meters are traveled.
To determine how many meters are traveled per minute, we can calculate the average distance per minute across the given data.

1. Calculate the total distance:
- Total distance (y) = 664 + 415 + 581 + 166 + 747 = 2573 meters

2. Calculate the total time:
- Total time (x) = 8 + 5 + 7 + 2 + 9 = 31 minutes

3. Calculate the average speed (meters per minute):
- Average speed = Total distance / Total time = 2573 m / 31 min ≈ 83.32 meters/minute.

### b. Write an equation to show the relationship between x and y:
Based on the average speed, the relationship between distance (y) and time (x) can generally be expressed by the equation of motion:
\[ y = mx + b \]
Where:
- \( m \) is the slope (average speed)
- \( b \) is the y-intercept

Using the average speed we calculated:
\[ y \approx 83.32x + b \]

To find the y-intercept \( b \), we can plug in a known point from the data. Let's use the point (2, 166):
\[ 166 = 83.32(2) + b \]
\[ 166 = 166.64 + b \]
\[ b = 166 - 166.64 = -0.64 \]

Thus, the linear equation becomes:
\[ y \approx 83.32x - 0.64 \]

### c. This relationship is because the y-intercept is .
The y-intercept is approximately \(-0.64\). This signifies that if no time has passed (x=0), theoretically, the distance traveled would almost be zero; however, the slight negative value indicates that our model does not perfectly fit the data at the origin, likely due to real-world conditions not accounted for in a linear model.

In summary, your answers are:

a. Every minute, approximately **83.32 meters** are traveled.
b. The equation is \( y \approx 83.32x - 0.64 \).
c. This relationship is because the y-intercept is approximately **-0.64**.

Answer 3

In the context of a linear relationship, whether the relationship is proportional or non-proportional can be identified by examining the y-intercept.

1. **Proportional Relationship**: A relationship is proportional if it can be represented by the equation \(y = mx\), where \(b = 0\), meaning that if you plug in \(x = 0\), then \(y\) must also be \(0\). The graph of a proportional relationship passes through the origin (0, 0).

2. **Non-Proportional Relationship**: A relationship is non-proportional if the y-intercept \(b\) is not equal to \(0\). In this case, the line does not pass through the origin.

Given your previous question, it seems you found a linear relationship with a y-intercept (approximately) \(-0.64\).

### Fill in the blanks:

This relationship is **non-proportional** because the y-intercept is **approximately -0.64**.

If you are considering specific points such as (0, 0) or (0, 83), if the y-intercept were 0 (which would indicate that as time approaches 0, distance also approaches 0), it could be called proportional.

### Conclusion:
- If the equation's y-intercept is (0,0): **Proportional**
- If the equation's y-intercept is (0, 83) or a negative value like (-0.64): **Non-Proportional**.

In your specific case, since we derived a non-zero y-intercept, the answer remains as stated above.