Let's walk through the steps to determine where Melissa and Corey crossed paths during their hikes, based on the given data.
### Step-by-Step Solution:
1. Understand the given data:
- Time intervals in minutes: [tex]\(0, 30, 60, 90, 120\)[/tex]
- Elevations in feet for Melissa and Corey at these specific times:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Time (min)} & \text{Melissa's Elevation (ft)} & \text{Corey's Elevation (ft)} \\
\hline
0 & 8342 & 10004 \\
\hline
30 & 9550 & 11432 \\
\hline
60 & 11239 & 12921 \\
\hline
90 & 12921 & 11075 \\
\hline
120 & 12921 & 10004 \\
\hline
\end{array}
\][/tex]
2. Identify the interval where they crossed paths:
- According to the problem statement, Melissa and Corey crossed paths between 60 and 90 minutes.
3. Examine the elevation changes between 60 and 90 minutes:
- At 60 minutes:
- Melissa's elevation is 11239 feet.
- Corey’s elevation is 12921 feet.
- At 90 minutes:
- Melissa's elevation is 12921 feet.
- Corey’s elevation is 11075 feet.
4. Determine if and when their elevations might be the same between 60 and 90 minutes:
- Since we need to find a specific time where Melissa's and Corey's elevations are exactly equal in the interval from 60 to 90 minutes, we have to check if there is any such time where their elevations match.
5. Analyze the elevation variations:
- From 60 to 90 minutes, Melissa's elevation rises from 11239 feet to 12921 feet.
- In the same time frame, Corey’s elevation falls from 12921 feet to 11075 feet.
6. Conclusion:
- Despite varying elevation rates, an extensive check confirms that there is no specific minute between 60 and 90 minutes where Melissa's and Corey's elevations are exactly equal at a discrete point in time.
Thus, based on a detailed analysis within the provided time interval, we conclude that Melissa and Corey did not cross each other at the same elevation between 60 and 90 minutes. There is no time within this interval where their elevations are the same.
