Boy's Journey North And East Calculate Bearing And Distance
This article explores a classic problem in geometry and trigonometry, focusing on calculating the bearing and distance of a person's final position after walking in two different directions. We will analyze the scenario where a boy walks 5 kilometers due North and then 4 kilometers due East. Our goal is to determine (a) the bearing of his current position from the starting point and (b) the total distance the boy is from his starting point. This problem elegantly demonstrates the application of fundamental concepts such as the Pythagorean theorem and trigonometric ratios in real-world scenarios. Let's delve into the step-by-step solution.
(a) Finding the Bearing of His Current Position
To determine the bearing of the boy's current position from his starting point, we need to calculate the angle formed by his path. The bearing is an angle measured clockwise from the North direction. First, visualize the boy's journey as two sides of a right-angled triangle. The 5 km walk North forms one side, and the 4 km walk East forms the other side. The line connecting the starting point to the final position forms the hypotenuse of this triangle. To find the bearing, we must first calculate the angle (θ) between the North direction and the line connecting the starting point to the final position. We can use the trigonometric tangent function to find this angle. The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the eastward walk (4 km), and the adjacent side is the northward walk (5 km). Therefore, tan(θ) = opposite / adjacent = 4 km / 5 km = 0.8. To find the angle θ, we take the inverse tangent (arctan) of 0.8. Using a calculator, arctan(0.8) ≈ 38.66 degrees. This angle represents the degrees East of North. Therefore, the bearing of the boy's current position from the starting point is approximately 038.66°. This detailed calculation involves understanding the geometry of the problem, identifying the relevant trigonometric ratio, and applying the inverse tangent function. This result gives us the precise direction the boy is located relative to his starting point.
Furthermore, it's crucial to understand that bearing is a fundamental concept in navigation and surveying. It provides a precise way to describe direction relative to a fixed point, typically North. The bearing is expressed in degrees, and it always ranges from 0° to 360°, measured clockwise from North. In this scenario, understanding the relationship between the sides of the right-angled triangle and the angle θ allows us to accurately determine the bearing. The use of the arctangent function is a key step in converting the ratio of the sides into the angle. The result, approximately 38.66 degrees, is then expressed in the standard bearing format, 038.66°, to clearly indicate the direction. This calculation demonstrates the practical application of trigonometric principles in determining directional information, a skill valuable in various fields such as geography, engineering, and even everyday navigation. The process of visualizing the problem as a geometric shape, identifying the relevant trigonometric relationships, and performing the calculations is a crucial aspect of problem-solving in mathematics and related disciplines.
Moreover, let's consider the implications of this bearing in a broader context. If the boy were to return to his starting point, he would need to travel on a reciprocal bearing. The reciprocal bearing is the opposite direction, calculated by adding 180 degrees to the original bearing if it's less than 180 degrees, or subtracting 180 degrees if it's greater than 180 degrees. In this case, to find the reciprocal bearing, we would add 180 degrees to 38.66 degrees, resulting in 218.66 degrees. Therefore, the boy would need to travel on a bearing of approximately 218.66 degrees to return to his starting point. This concept of reciprocal bearing is essential in navigation and helps ensure accurate return journeys. The understanding of bearing also extends to more complex scenarios involving multiple legs of travel, where each leg needs to be analyzed and combined to determine the overall displacement and bearing. The accurate calculation of bearing is not only a mathematical exercise but also a practical skill with real-world applications in navigation, surveying, and various other fields where directional information is crucial.
(b) Calculating the Distance from the Starting Point
Now, let's determine the distance the boy is from the starting point. As previously established, the boy's journey forms a right-angled triangle. The northward walk of 5 km and the eastward walk of 4 km are the two legs of the triangle, and the distance from the starting point to the final position is the hypotenuse. To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). The formula is expressed as: c² = a² + b². In this case, a = 5 km and b = 4 km. Plugging these values into the formula, we get: c² = (5 km)² + (4 km)² = 25 km² + 16 km² = 41 km². To find the distance (c), we take the square root of 41 km². Using a calculator, √41 ≈ 6.40 km. Therefore, the boy is approximately 6.40 kilometers away from his starting point. This calculation demonstrates a direct application of the Pythagorean theorem, a fundamental concept in geometry, to solve a practical distance problem.
Understanding the Pythagorean theorem is crucial for solving various problems involving right-angled triangles, not just in mathematics but also in fields like physics, engineering, and architecture. The theorem provides a powerful tool for calculating distances and lengths in two-dimensional space. In this particular scenario, the application of the theorem allows us to precisely determine the straight-line distance between the boy's starting and ending points, regardless of the path he took. This is a fundamental concept in understanding displacement versus distance traveled. The distance traveled is the total length of the path taken, which in this case is 5 km + 4 km = 9 km. However, the displacement, which is the straight-line distance from the starting point to the ending point, is 6.40 km. This distinction is important in many scientific and engineering applications, where understanding both the total distance and the net displacement is necessary. The Pythagorean theorem allows us to easily calculate the displacement, providing a valuable piece of information in the analysis of motion and spatial relationships.
Furthermore, the Pythagorean theorem can be applied in a variety of contexts beyond simple distance calculations. For instance, it can be used to determine the dimensions of structures, calculate the slope of a hill, or even in navigation to determine the shortest distance between two points on a map. The ability to apply this theorem effectively requires a strong understanding of geometric principles and the relationships between the sides of a right-angled triangle. In this problem, the theorem provides a straightforward method for finding the distance, but it also highlights the importance of visualizing the problem geometrically. By recognizing that the boy's journey forms a right-angled triangle, we can readily apply the Pythagorean theorem to find the desired distance. This problem-solving approach, which involves translating a real-world scenario into a mathematical model, is a fundamental skill in mathematics and its applications. The accurate calculation of the distance not only answers the specific question but also reinforces the understanding and application of a core mathematical principle.
Conclusion
In summary, we have successfully determined both the bearing and the distance of the boy's final position from his starting point. The bearing was calculated to be approximately 038.66°, and the distance was found to be approximately 6.40 kilometers. This exercise highlights the practical application of trigonometry and geometry in solving real-world problems. By understanding and applying concepts such as the tangent function and the Pythagorean theorem, we can accurately analyze and describe spatial relationships and movements. This type of problem-solving is essential in various fields, including navigation, surveying, engineering, and physics. The combination of these mathematical tools allows us to describe not only the direction but also the magnitude of the displacement, providing a comprehensive understanding of the boy's journey. This detailed analysis underscores the importance of these fundamental mathematical concepts in practical applications.