Let's analyze each scenario step by step using the elevation data from the table.
### 1. Melissa and Corey started on different trails.
- Melissa's elevation at 0 minutes = 8,342 feet
- Corey's elevation at 0 minutes = 10,004 feet
Since their starting elevations are different, Melissa and Corey started on different trails.
Conclusion: This scenario is true.
### 2. Melissa hiked up the trail for a longer period of time than Corey.
- Increase in Melissa's elevation (comparing sequentially):
- From 0 to 30 min: 8,342 to 9,550 (increase)
- From 30 to 60 min: 9,550 to 11,239 (increase)
- From 60 to 90 min: 11,239 to 12,921 (increase)
- From 90 to 120 min: 12,921 to 12,921 (no change)
Melissa hiked uphill for 90 minutes.
- Increase in Corey's elevation (comparing sequentially):
- From 0 to 30 min: 10,004 to 11,432 (increase)
- From 30 to 60 min: 11,432 to 12,921 (increase)
- From 60 to 90 min: 12,921 to 11,075 (decrease)
- From 90 to 120 min: 11,075 to 10,004 (decrease)
Corey hiked uphill for 60 minutes.
Since Melissa hiked uphill for 90 minutes and Corey for 60 minutes, Melissa hiked uphill longer.
Conclusion: This scenario is true.
### 3. Melissa and Corey crossed paths between 60 and 90 minutes.
We need to check if there exists any time between 60 and 90 minutes where they have the same elevation or if they cross each other.
- At 60 minutes:
- Melissa's elevation = 11,239 feet
- Corey's elevation = 12,921 feet
- At 90 minutes:
- Melissa's elevation = 12,921 feet
- Corey's elevation = 11,075 feet
Between 60 and 90 minutes, Melissa increased in elevation from 11,239 feet to 12,921 feet, whereas Corey decreased from 12,921 feet to 11,075 feet. As they moved in opposite directions, and considering their elevations at given times, they necessarily crossed each other's paths.
Conclusion: This scenario is true.
### 4. Corey began his descent before Melissa.
- When did Corey start descending?
- He began descending after 60 minutes (since elevation at 60 min is 12,921 feet and at 90 min is 11,075 feet).
- When did Melissa start descending?
- She maintained her maximum elevation of 12,921 feet from 90 to 120 minutes. There is no descent observed from the given data.
Corey started descending after 60 minutes, and Melissa never descended during the observed period till 120 minutes.
Conclusion: This scenario is true.
### 5. As Melissa's time increased from 0 to 60 minutes, her elevation decreased.
- At 0 minutes: 8,342 feet
- At 60 minutes: 11,239 feet
Since Melissa's elevation increased from 8,342 feet to 11,239 feet from 0 to 60 minutes, her elevation did not decrease during this time frame.
Conclusion: This scenario is false.
### Summary
Based on the above analysis, the true scenarios are:
- Melissa and Corey started on different trails.
- Melissa hiked up the trail for a longer period of time than Corey.
- Melissa and Corey crossed paths between 60 and 90 minutes.
- Corey began his descent before Melissa.
The false scenario is:
- As Melissa's time increased from 0 to 60 minutes, her elevation decreased.
