Question

4. 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

To analyze the data and answer the questions, we can follow these steps:

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

### a. Every minute, ___ meters are traveled.

To determine the average speed of travel in meters per minute, we can calculate the average distance traveled per minute based on the provided data.

1. **Calculate Total Distance and Total Time**:
- Total distance \( = 664 + 415 + 581 + 166 + 747 = 2573 \, \text{meters} \)
- Total time \( = 8 + 5 + 7 + 2 + 9 = 31 \, \text{minutes} \)

2. **Average Speed (meters per minute)**:
\[
\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{2573 \, \text{meters}}{31 \, \text{minutes}} \approx 83.3 \, \text{meters/minute}
\]

**Answer**: Every minute **approximately 83.3** meters are traveled.

### b. Write an equation to show the relationship between x and y:

To find a linear equation that models the relationship between distance (y) and time (x), we can use the form of the linear equation:

\[
y = mx + b
\]

Where:
- \( m \) is the slope (average speed)
- \( b \) is the y-intercept (distance at time 0)

Using the average speed we just calculated:
- \( m \approx 83.3 \)

To calculate \( b \), we can inspect the data and see if we can derive the y-intercept. If we were to extrapolate back to \( x = 0 \) using the average speed, we can treat \( b \) as 0 for simplification in the absence of more specific rules-data.

However, to derive the exact relationship and y-intercept, we commonly perform linear regression or calculate based on existing data points. Given the nature, if we take, for simplicity, \( b = 0 \):

Thus, an approximate equation might be:
\[
y = 83.3x + 0
\]

**Answer**: The equation is \( y = 83.3x \) (for a simple approximation). More analysis may adjust the slope and the intercept based on real data fitting.

### c. This relationship is ____ because the y-intercept is ____.

- The relationship is **linear** (assuming a straight-line correlation).
- The y-intercept is **0** (if we assume distance starts at 0 when time is 0 in a simplest context; else calculation of \( b \) may provide a more accurate non-zero intercept).

**Answer**: This relationship is **linear** because the y-intercept is **0** (or as derived from regression fitting, needs calculation).

For more accurate results, graphical analysis or statistical regression would provide precise figures.