To compare the functions that represent the paths of the two rockets launched by different lab groups, we need to go through a step-by-step analysis comparing their heights at different times, the maximum heights achieved, and the times the rockets were in the air.
Let's start by understanding the given data:
### Lab Group A Rocket Path
| Time after Launch (sec) | 0 | 2 | 5 | 8 | 10 |
|--------------------------|---|---|---|---|----|
| Height (ft) | 0 | 48 | 75 | 48 | 0 |
### Lab Group B Rocket Path
| Time after Launch (sec) | 0 | 1 | 5 | 9 | 10 |
|--------------------------|---|---|---|---|----|
| Height (ft) | 0 | 36 | 100 | 36 | 0 |
#### Analysis:
1. Initial and Final Heights (x-intercepts):
- Both rockets start at height 0 ft and end at height 0 ft.
- Time of launch and landing for both rockets is from second 0 to second 10.
2. Maximum Heights:
- Lab Group A: Maximum height is 75 ft at time 5 seconds.
- Lab Group B: Maximum height is 100 ft at time 5 seconds.
3. Times of Maximum Heights:
- Both rockets reach their maximum heights at time 5 seconds.
4. Total Time in the Air:
- Both rockets are in the air for a total of 10 seconds (from launch to landing at t = 0 to t = 10).
Based on this analysis, we can conclude:
- Since the maximum heights occurred at the same time (x-value), it means the maximum heights were attained at the same time.
- The x-intercepts are the same for both functions, indicating that both rockets were launched and landed at the same times.
Thus, the conclusion based on the given analysis is:
Since the [tex]$x$[/tex]-intercepts are the same for both functions, the rockets followed the same path.
Therefore, the correct answer is:
```plaintext
Since the [tex]$x$[/tex]-intercepts are the same for both functions, the rockets followed the same path.
```
This matches option 2 in the provided choices.
