A student leaves the library at 7:00 AM, determined to hike to the top of the nearby mountain for some much-needed fresh air, reaching the summit at 7:00 PM (of course, with
several snack breaks along the way). The next morning, still sore but full of caffeine, the
student starts the journey back down at 7:00 AM, retracing the same path, and stumbles
back into the library at 7:00 PM. Prove that there’s a point on the path that the student
passes at exactly the same time on both days (probably wondering why they agreed to
do this in the first place).

This problem is a well-known result from the field of topology, often referred to as **the Intermediate Value Theorem (IVT)** in a real-life context. Here's how we can think through and prove the result:

### Step-by-Step Solution

#### 1. **Represent the two journeys**
   - Let the student's path up the mountain on Day 1 be described as a continuous function \( f(t) \), where:
     - \( f(t) \) represents the student's position on the path at time \( t \).
     - The time \( t \) ranges from 7:00 AM (which we'll call \( t = 0 \)) to 7:00 PM (which we'll call \( t = 12 \), as 12 hours have passed).

     So, \( f(0) = \text{library} \) (starting point), and \( f(12) = \text{summit} \) (end point).

   - Similarly, let the student's path down the mountain on Day 2 be described by another continuous function \( g(t) \), where:
     - \( g(t) \) represents the student's position on the path at time \( t \) during the second day.
     - \( t = 0 \) is again 7:00 AM, and \( t = 12 \) is 7:00 PM.
     
     So, \( g(0) = \text{summit} \) (starting point on Day 2), and \( g(12) = \text{library} \) (end point).

#### 2. **Combine the two functions**
   We are tasked with proving that there is some time \( t \) where the student is at the same position on the path on both days. In other words, we want to find \( t \) such that:
   \[
   f(t) = g(t)
   \]
   Instead of directly solving for this equality, define a new function \( h(t) \) as follows:
   \[
   h(t) = f(t) - g(t)
   \]
   The function \( h(t) \) represents the difference between the student's position on Day 1 and their position on Day 2 at the same time.

#### 3. **Apply the Intermediate Value Theorem (IVT)**
   - Consider the values of \( h(t) \) at \( t = 0 \) and \( t = 12 \):
     - At \( t = 0 \), \( f(0) = \text{library} \) and \( g(0) = \text{summit} \), so:
       \[
       h(0) = f(0) - g(0) = \text{library} - \text{summit}
       \]
       Hence, \( h(0) \) is a negative value (since the library is below the summit).
     
     - At \( t = 12 \), \( f(12) = \text{summit} \) and \( g(12) = \text{library} \), so:
       \[
       h(12) = f(12) - g(12) = \text{summit} - \text{library}
       \]
       Hence, \( h(12) \) is a positive value (since the summit is above the library).

   - Since \( h(t) \) is continuous (both \( f(t) \) and \( g(t) \) are continuous functions of time), and \( h(0) \) is negative while \( h(12) \) is positive, by the Intermediate Value Theorem, there must be some time \( t = t^* \) within the interval \( (0, 12) \) where \( h(t^*) = 0 \). That is:
     \[
     f(t^*) = g(t^*)
     \]
   - This means that at some time \( t^* \), the student is at the same position on the path on both days.

### Conclusion
Thus, there is indeed a point on the path that the student passes at exactly the same time on both days.

---

Would you like further details or have any questions?

Here are 5 related questions that expand on this concept:
1. How does the Intermediate Value Theorem apply in real-world contexts besides this problem?
2. What are the conditions required for the Intermediate Value Theorem to hold?
3. Can we use the same reasoning if the student took different paths up and down the mountain?
4. How would this problem change if the time taken to hike up and down were different?
5. Can the concept of continuity apply to discrete steps rather than smooth paths?

**Tip:** The Intermediate Value Theorem requires the function in question to be continuous on a closed interval. It's a powerful tool for proving the existence of solutions without needing to explicitly find them.

A student leaves the library at 7:00 AM, determined to hike to the top of the nearby mountain for some much-needed fresh air, reaching the summit at 7:00 PM (of course, with several snack breaks along the way). The next morning, still sore but full of caffeine, the student starts the journey back down at 7:00 AM, retracing the same path, and stumbles back into the library at 7:00 PM. Prove that there’s a point on the path that the student passes at exactly the same time on both days.

Explore how the Intermediate Value Theorem is applied to prove that a student, hiking up and down a mountain on different days, passes the same point at the same time on both days. This detailed explanation covers key concepts from topology, continuous functions, and real-world applications.

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