Calculating the Total Walk of a Girl Using Geometry: Practical Applications in Real Life

In the real world, mathematical concepts like geometry are indispensable for solving practical challenges. The problem of a girl walking in various directions can be a perfect example to showcase the application of geometry in everyday scenarios. This article will guide you through the steps to calculate the total walk of the girl and the final direction she faced using the principles of basic geometry and trigonometry.

Understanding the Problem

The problem at hand involves a girl who walks in multiple directions, which can be described as follows:

Walks 6560 feet (approximately 1.96 km) to the East. Walks 6 km North. Walks 6260 feet (approximately 1.91 km) to the East. Walks 22960 feet (approximately 6.96 km) to the East. Walks 2 km to the West.

Our goal is to calculate the total distance she traveled and the overall direction of her walk from the starting point.

Converting Feet to Kilometers

For the sake of simplicity and uniformity, we will convert all distances from feet to kilometers (km).

1 foot 0.0003048 km

Converting the distances:

6560 feet 1.96 km 6260 feet 1.91 km 22960 feet 6.96 km

Therefore, the girl's journey can be described as follows:

1.96 km to the East 6 km to the North 1.91 km to the East 6.96 km to the East 2 km to the West

Total Distance Calculation

To find the total distance walked, we simply add all the eastward and northward distances:

Eastward: 1.96 km 1.91 km 6.96 km 10.83 km

Northward: 6 km (no westward movement in the north direction)

Total distance 6 km North 10.83 km East

Finding the Hypotenuse

The total distance is the hypotenuse of a right triangle with legs of 6 km (north) and 10.83 km (east).

Solving the Right Triangle:

Aim:

To calculate the hypotenuse (c) of the right triangle where:

a 6 km (north) b 10.83 km (east)

Using the Pythagorean theorem:

c^2 a^2 b^2

c^2 6^2 10.83^2

c^2 36 117.2889

c^2 153.2889

c sqrt(153.2889)

c ≈ 12.38 km

The girl walked approximately 12.38 kilometers from her starting point.

Determining the Direction

Determining the direction involves finding the angle θ (theta) between the northward leg and the hypotenuse.

θ arctan(a/b)

θ arctan(6/10.83)

θ ≈ 30.33°

The angle is 30.33°, which means the direction is 'north of east'.

Practical Applications of Geometry in Real Life

Understanding the principles of geometry, such as right triangles and trigonometry, can be instrumental in various real-life applications. These include:

Navigation and GPS systems Surveying and mapping Structural engineering Weather forecasting

By applying these concepts, we can solve complex problems and make informed decisions in our daily lives.

Conclusion

In conclusion, the girl's walk involved a total distance of approximately 12.38 kilometers, leading her in a direction that is just north of east. This problem serves as a practical example of how geometry and trigonometry concepts can be applied to real-life situations, making these mathematical tools not just theoretical but extremely useful in our day-to-day lives.