Problem 15 - Applied Calculus

Question

Hiking on a level trail going due east, you decide to leave the trail and climb toward the mountain on your left. The farther you go along the trail before turning off, the gentler the climb. Sketch a possible topographical map showing the elevation contours.

Step-by-Step Solution

Answer

Sketch contour lines increasing in elevation leftwards with the climbing path curving upwards.

Step 1: Understand Contour Lines

Contour lines on a topographical map represent lines of equal elevation. Close lines indicate steep slopes, while widely spaced lines indicate gentle slopes. The exercise asks us to sketch topographical lines based on your directional change.

Step 2: Analyze the Scenario

The problem describes a situation where climbing starts from a level trail going east towards a mountain on your left. This indicates that the elevation increases as you go left from your eastward path.

Step 3: Sketch the Trail Path

Draw a straight horizontal line from the left to the right of your paper to represent the eastward trail. This will be our reference as we sketch the contour lines.

Step 4: Sketch Contour Lines

Begin drawing contour lines that gradually increase in elevation values as you move upwards (towards the mountain). Closer contour lines should appear near the mountain, indicating the steeper slope, while wider lines appear further away on the slope to show a gentler climb.

Step 5: Marking the Climb Path

Draw a path that starts along the trail and curves upwards (to the left), indicating your climbing path towards the mountain. Ensure the path intersects with increasingly high contour lines.

Key Concepts

Contour Lines

Contour lines are an essential feature of any topographical map. They are drawn as lines that connect points of equal elevation. This means that if you were to follow a contour line with your finger, you would not ascend or descend.
This property makes contour lines incredibly useful for understanding the terrain.

• Close contour lines represent areas of rapid elevation change, like steep hills or cliffs.
• Lines spaced further apart suggest gentle slopes and flat areas.

When sketching from the original exercise, imagine that each line you draw encapsulates a particular elevation level as you move towards the mountain. The closer these lines, the steeper the climb.
It's also important to note that contour lines never cross each other. If they did, it would mean the same point has two different elevations, which isn’t possible.

Elevation

Elevation is a measurement of vertical distance, typically above sea level. Understanding elevation is critical when navigating and interpreting topographical maps. On a map, higher elevations are shown by contour lines with higher numerical values.
As you go towards a mountain, the numbers on these contour lines increase. In the context of hiking towards the mountain from the exercise's level trail, think of elevation as your path climbs higher into the landscape's relief. This ascending data let hikers anticipate which areas they may need more energy to climb and identify potential stopping points.
Knowing these elevations helps you assess the difficulty of your climb based on how quickly elevation changes along the contour lines.

Slope Analysis

Slope analysis involves understanding and interpreting the changes in elevation over distance on a topographical map. This is key for hikers, planners, and geographers when determining the steepness of terrain.

• A steep slope will have contour lines very close together.
• A gentle slope will show wider spacing between lines.

In the exercise scenario, as you decide to climb towards the mountain, you perform a mental slope analysis. The further you walk along the eastward path before turning towards the mountain means encountering less steep slopes, as the contour lines are spaced wider apart in that direction.
Being able to visualize and analyze these variations lets you plan your hike path by selecting routes that align with your stamina and adventure goals.