Certainly, let's analyze the given information step-by-step according to the scenarios:
### Given Data
- Time (in minutes): 0, 30, 60, 90, 120
- Melissa's elevation (in feet): 8,342, 9,550, 11,239, 12,921, 12,921
- Corey's elevation (in feet): 10,004, 11,432, 12,921, 11,075, 10,004
### Analyzing the Scenarios
1. Melissa and Corey started on different trails.
- At 0 minutes, Melissa's elevation is 8,342 feet.
- At 0 minutes, Corey's elevation is 10,004 feet.
- Since their initial elevations are different, it indicates that they started on different trails.
- Conclusion: This scenario is true.
2. Melissa hiked up the trail for a longer period of time than Corey.
- To verify this, we need to see if Melissa's elevation increased continuously for a longer duration compared to Corey.
- Melissa's elevations at 0, 30, and 60 minutes: 8,342 -> 9,550 -> 11,239 (continuous increase)
- Corey's elevations at 0, 30, and 60 minutes: 10,004 -> 11,432 -> 12,921 (continuous increase)
- After 60 minutes, Corey's elevation starts to decrease (12,921 -> 11,075 at 90 minutes), while Melissa's elevation still increases until 90 minutes (11,239 -> 12,921).
- Conclusion: This scenario is false.
3. Melissa and Corey crossed paths between 60 and 90 minutes.
- At 60 minutes, Melissa's elevation is 11,239 feet and Corey's elevation is 12,921 feet.
- At 90 minutes, Melissa's elevation is 12,921 feet and Corey's elevation is 11,075 feet.
- Since Melissa's elevation increased from 11,239 to 12,921, and Corey's elevation decreased from 12,921 to 11,075, their elevation crossed each other between 60 to 90 minutes.
- Conclusion: This scenario is true.
4. Corey began his descent before Melissa.
- Corey's elevation starts to decrease after 60 minutes (12,921 at 60 minutes to 11,075 at 90 minutes).
- Melissa's elevation remains the same after 90 minutes (12,921 at both 90 and 120 minutes), but does not decrease until after 90 minutes.
- Conclusion: This scenario is false.
5. As Melissa's time increased from 0 to 60 minutes, her elevation decreased.
- Melissa's elevation at 0, 30, and 60 minutes: 8,342 -> 9,550 -> 11,239 (continuous increase)
- Since her elevation continuously increased as time increased from 0 to 60 minutes, it did not decrease.
- Conclusion: This scenario is false.
### Final Result
Based on the analysis, the scenarios that must be true are:
1. Melissa and Corey started on different trails.
3. Melissa and Corey crossed paths between 60 and 90 minutes.
Thus, the scenarios that are true are:
- [1]
- [3]
