Navigating through Geometric Coordinates: A Walk West, South, Then East

Imagine a person embarking on a journey, moving 40 km west, then turning left and walking 30 km, and finally turning left again and walking 30 km. Where is this person relative to their starting point? This problem can be solved using basic principles of coordinate geometry and Euclidean distance calculations. Let's break down the movements and solve for the final distance from the origin, the starting point of the journey.

Step-by-Step Analysis

Starting Point: The journey begins at Point O, the person's house. First Movement: The person walks 40 km west. Now they are at Point A, 40 km west of O. Second Movement: Turning left and moving 30 km, they are now facing south and end up at Point B. Point B is 40 km west and 30 km south of O. Third Movement: Again turning left, they face east and walk 30 km, ending up at Point C. Point C is 10 km west of O since they walked 30 km east from Point B, which was 40 km west.

Calculating the Distance

Now, we need to determine the distance from Point C to Point O. While the person is 10 km west of their starting point, the total distance from the home can be calculated using the Pythagorean theorem. Point C is 30 km south and 10 km west of O.

Presentation of the Solution

Perpendicular Distances: The horizontal distance is 10 km (west), and the vertical distance is 30 km (south). Using the Pythagorean Theorem: The distance D from the final position to the starting point can be calculated as follows:

D √(302 102) √(900 100) √1000 31.62 km

Generalization and Practical Application

The same problem can be extended to understand the effect of latitude on the journey's path. If the journey were near the equator (latitude around 0 degrees), the Earth's curvature would be minimal, and the problem can be treated as a flat-Earth problem with high accuracy.

However, if the person started closer to the poles (e.g., 70 degrees latitude), the latitudinal and longitudinal distances would be affected due to the Earth's spherical shape. In such cases, one would need to consider the curvature of the Earth to ensure accurate distances and positions.

Conclusion

In summary, a person who walks 40 km west, then 30 km south, and finally 30 km east will be approximately 31.62 km away from their starting point. This solution is applicable for most practical scenarios, but for higher latitude journeys, the Earth's curvature must be considered for precise results.