Problem 71 - Calculus Early Transcendentals

Question

Use Exercise 69 to prove that if two trails start at the same place and finish at the same place, then regardless of the ups and downs of the trails, they have the same net change in elevation.

Step-by-Step Solution

Answer

Question: Prove that if two trails start and finish at the same place, the net change in their elevation is the same, regardless of the ups and downs throughout the trails. Answer: Since both trails start and end at the same elevation, the sum of elevation changes for both trails must be equal to zero. This means that the net elevation change for both trails is the same, regardless of the ups and downs of each trail.

Step 1: Define the elevation changes for each trail

Let's denote the elevation changes for the two trails as \(x_1, x_2, x_3, ..., x_n\) for Trail 1 and \(y_1, y_2, y_3, ..., y_k\) for Trail 2. Here, positive values represent ups and negative values represent downs in elevation along the trails.

Step 2: Determine the total elevation change for each trail

Since we want to compare the net elevation change for both trails, we need to compute the sum of elevation changes in each trail. For Trail 1, the total elevation change is \(\sum_{i=1}^n x_i\), and for Trail 2, the total elevation change is \(\sum_{j=1}^k y_j\).

Step 3: Analyze the starting and ending points

Both trails start and end at the same place; this means that after completing each trail, we end up at the same elevation where we started. Mathematically, this indicates that the total elevation changes along each trail should be equal to zero. That is, \(\sum_{i=1}^n x_i = 0\) and \(\sum_{j=1}^k y_j = 0\).

Step 4: Compare the net elevation change

Since both trails return to the same elevation after completing, we know that the sum of their elevation changes must be equal. So, $$ \sum_{i=1}^n x_i = \sum_{j=1}^k y_j $$

Step 5: Conclusion

By proving that the sum of elevation changes for the two trails is equal, we've shown that regardless of the ups and downs of the Trails 1 and 2, they both have the same net change in elevation. Therefore, the statement is proved as true.

Key Concepts

Elevation Changes

Understanding elevation changes is essential for students learning about geography, earth sciences, and activities like hiking and climbing. When referring to elevation changes, we're talking about variations in height from a reference point, typically sea level. Imagine walking up a hill: every step you take uphill increases your elevation, while each step back downhill decreases it.

In our exercise, elevations changes are represented by variables like \(x_1, x_2, x_3, ..., x_n\)) and \(y_1, y_2, y_3, ..., y_k\)) for two different trails. These variables can have positive (representing an increase in elevation, or 'up') or negative values (representing a decrease in elevation, or 'down'). Students need to keep in mind that even within a single trail, these changes can vary greatly - steep climbs followed by gradual descents or vice versa.

Visualizing Elevation Changes on Graphs

A practical way to understand elevation changes is by plotting them on a graph, with upward changes showing a rise on the graph and downward changes indicating a fall. Doing so can help students visualize the concept of elevation change more concretely.

Total Elevation Change

Total elevation change is a key term referred to in various exercises, including our textbook problem. It is essentially the cumulative effect of all individual elevation changes on a trail. In simpler terms, it's the sum of all ups and downs you'd experience if you were walking the trail yourself. This is important for understanding the overall intensity and difficulty of a trail.

To calculate total elevation change, one would add up each elevation gain (ups) and loss (downs) along the path. In mathematical terms, for Trail 1, the total elevation change is the sum of \(x_i\)), denoted as \(\sum_{i=1}^n x_i\)), and for Trail 2, it's the sum of \(y_j\)), written as \(\sum_{j=1}^k y_j\)). However, since both trails start and end at the same elevation, their total elevation change must return to the starting point's elevation, therefore theoretically equating to zero.

Accounting for Different Trails

Regardless of how many elevations changes there might be along the way, the fact that both trails end at the same elevation they began means that these numerous changes will ultimately balance out. This concept is important when comparing the difficulty of two different trails, as it provides a more holistic measure than just looking at the highest point reached or the steepest climb.

Sum of Elevation Changes

The sum of elevation changes is the mathematical approach to calculating the total elevation change. It involves adding or subtracting each segment's elevation gain or loss to get a final value. This value represents the trail's net elevation change from start to finish, which should be zero if the trail loops back to its starting point.

It's crucial for students to be comfortable with the summation operation – denoted by the Greek letter sigma (\(\Sigma\)) – as it's a fundamental mathematical concept that will reappear in various contexts, not just in calculating elevation changes. Applying the principle to our textbook case, we've found that the sum of elevations changes for both trails is equal because both trails return to their starting elevation.

Implications of Elevation Summation

In the real world, this principle can reassure us that, for any route we take up a mountain and back down to our starting point, the uphill and downhill segments will ultimately cancel out. This concept can also be applied across different disciplines, including physics, engineering, and finance, where the net change over time or over a process is a critical parameter to understand.