Problem 69 - Calculus Early Transcendentals: Pearson New International Edition

Question

Starting at 4 A.M., a hiker slowly climbed to the top of a mountain, arriving at noon. The next day, he returned along the same path, starting at \(5 \mathrm{~A} . \mathrm{M}\). and getting to the bottom at \(11 \mathrm{~A} . \mathrm{M}\). Show that at some point along the path his watch showed the same time on both days.

Step-by-Step Solution

Answer

The hiker's watch showed the same time when both ascent and descent paths intersect.

Step 1: Understanding the Problem

The hiker climbed to the top from 4 A.M. to noon, which took 8 hours, and returned from 5 A.M. to 11 A.M., a 6-hour journey. We need to show that there exists a point on the path where the time of day was the same on both days.

Step 2: Visualizing the Paths

Imagine the path as a line on a graph where the x-axis represents time and the y-axis represents the distance along the path. The ascent is a line from point (4, 0) to (12, d), and the descent is a line from point (5, d) to (11, 0), where d is the total distance of the path.

Step 3: Model Overlapping Timelines

To find a common point in time, consider placing the ascent timeline and descent timeline on top of each other. Because both paths occupy the same total distance over varying time intervals, there must be an intersection according to the Intermediate Value Theorem.

Step 4: Apply the Intermediate Value Theorem

Since the ascent and descent form continuous paths over the mountain, the Intermediate Value Theorem assures that there is a continuous function describing both paths. Where these two functions intersect is where he would be at the same location at the same time.

Step 5: Identify the Same Time and Location

Visualize walking up the line while simultaneously tracing the down line. Since both lines cover the path at different time intervals and meet at the start and end, there must be at least one point on this path where the times read the same, corresponding to the intersection of the two paths.

Key Concepts

Continuous Functions

In mathematics, a continuous function is essentially a function that, intuitively, you can draw without lifting your pencil off the paper. It is a smooth curve where every point on the path is connected to its neighboring points without any gaps or jumps.

In the context of our hiking problem, the ascent and descent paths of the hiker can both be represented as continuous functions over time. When the hiker climbs and subsequently descends the mountain, these paths are continuous. There are no sudden teleports from one point in time to another; instead, each movement is a step along a defined path.

The continuity here is crucial for applying the Intermediate Value Theorem. It tells us that within the domain of time the hiker is moving (from start to end of each day), every point of distance reached on the ascent is smoothly mapped back on the descent path. Thus, this continuity guarantees that there exists a specific moment in time when both ascent and descent overlap at the same distance along the path.

Graph Interpretation

To visualize this problem, imagine plotting these paths on a graph. The x-axis represents time of day, while the y-axis shows the distance along the path from the mountain's base. The hiker's ascent from 4 A.M. to noon can be drawn as a line moving from the point (4, 0) up to (12, d), where d is the mountain's distance. Likewise, his descent from 5 A.M. to 11 A.M. can be depicted as another line that starts at (5, d) and moves down to (11, 0).

This graph helps in visualizing the intersection point of ascent and descent lines. It can show us that even though the rates of travel might change due to the different durations (8 hours versus 6 hours), the hiker must coincide at a certain point in time and space as per the Intermediate Value Theorem. Simple interpretation thus becomes a powerful tool in solving what might initially seem a complex problem.

Ascent and Descent Paths

The ascent and descent paths of a hiker can be thought of as metaphorical paths along the mountain. When we consider these as a function of time and apply the Intermediate Value Theorem, it shows that there is a natural intersection point somewhere along these paths.

During ascent, the hiker gradually moves up, increasing his distance from the start point over time. This path upwards is distinct from the descent, where the hiker reverses his journey, reducing his distance and eventually reaching the base.

Both ascent and descent, therefore, form inherent parts of the mountain's topographical path. As one rises and one falls back down over timed intervals, a natural crossover at some point in time cannot be avoided. The essence here is that the continuous nature of these movements and their shared geographical path means that a crossover point must exist, regardless of the difference in speed or time taken to complete each journey.