The Curious Case of Walking Directions and Geographical Displacement

Have you ever encountered a problem that combines the seemingly straightforward act of walking in specific directions with complex geographical calculations? This article explores such a scenario, breaking down a series of walking instructions to find the actual displacement from the starting point. We'll delve into each step, applying geometric principles to solve the puzzle.

Problem Statement

A person walks 3km South, then 4km West, then 5km North, and finally 2km Southeast. How far is he from the starting point?

Breaking Down the Problem

Let's assume the starting point is at the origin (0,0) in a Cartesian coordinate system. We'll follow each step and calculate the new coordinates.

Starting Point

Origin (0,0)

First Move

The person walks 3km South.

New position: (0, -3)

Second Move

The person walks 4km West.

New position: (-4, -3)

Third Move

The person walks 5km North.

New position: (-4, 2) (since -3 5 2)

Fourth Move

The person walks 2km Southeast. Southeast indicates a 45-degree angle from North-East, meaning the movement is split into equal components in the x and negative y directions. Using trigonometric functions:

Change in the x-direction: 2 * cos(45°) ≈ 2 * 0.7071 ≈ 1.414

Change in the y-direction: 2 * sin(45°) ≈ 2 * 0.7071 ≈ 1.414

New position: (-4 - 1.414, 2 - 1.414) ≈ (-5.414, 0.586)

Calculating Final Distance from Origin

To find the distance from the origin (0,0) to the final position (-5.414, 0.586), we can use the distance formula:

[ d sqrt{x_2^2 y_2^2} ]

Plugging in the coordinates:

[ d sqrt{(-5.414)^2 (0.586)^2} sqrt{29.307396 0.343396} sqrt{29.650792} approx 5.445 text{ km} ]

Thus, the person is approximately 5.445 km from the starting point in this scenario.

Alternative Scenarios and Considerations

The solution can vary depending on the starting location. For instance:

Starting on the Equator: If starting on the equator at the prime meridian, the steps would follow the same geometric principles, yielding a displacement of approximately 5.445 km. Starting Near the South Pole: If the starting point is 3 km north of the South Pole, the calculations become more complex due to the curvature of the Earth. In this case, the movements would need to be recalculated considering the spherical geometry of the Earth.

This problem illustrates how geographical angles and distances can behave differently depending on the starting point and the specific location on Earth. It's an excellent example of how real-world problems can be simplified using geometric and trigonometric principles.