Calculating Distances Using Bearings The Observation Point Problem

In the realm of mathematics, particularly trigonometry and geometry, understanding bearings and distances is crucial for solving real-world problems. This article delves into a scenario involving two houses situated 750 meters apart, with one directly north of the other. An observation point lies due west of the first house, and its bearing from the second house is 228°. Our goal is to determine the distance of this observation point from each of the houses. Let's embark on this mathematical journey, exploring the concepts and calculations involved in finding the solution.

Understanding Bearings and the Problem

Before we dive into the calculations, it's essential to grasp the concept of bearings. In navigation, a bearing is the angle, measured clockwise, from north to a specific point. A bearing of 0° (or 360°) represents north, 90° represents east, 180° represents south, and 270° represents west. In our scenario, the bearing of 228° tells us that the observation point is located in the southwest quadrant relative to the second house. Specifically, it is 228° - 180° = 48° west of south. This understanding of bearings helps us visualize the problem and set up the geometric relationships needed for solving it.

In this specific problem, we are presented with a classic scenario that combines geometry and trigonometry. The essence of the problem lies in using the given information – the distance between the houses and the bearing of the observation point from the second house – to calculate the distances from the observation point to each house. This involves applying trigonometric principles, particularly the tangent function, to relate the angles and sides of the right-angled triangles formed in the scenario. By carefully analyzing the problem and breaking it down into smaller, manageable parts, we can develop a step-by-step approach to find the solution. This process not only helps us understand the specific problem but also enhances our problem-solving skills in mathematics and related fields.

To solve this problem effectively, it's crucial to visualize the situation accurately. Imagine two houses, one directly north of the other, separated by a distance of 750 meters. Now, picture an observation point situated due west of the first house. The key piece of information is the bearing of the observation point from the second house, which is given as 228°. This bearing essentially defines the angle formed at the second house between the north direction and the line connecting the second house to the observation point. By visualizing these elements, we can start to see the right-angled triangles that are formed and how we can use trigonometric ratios to find the distances we need. This initial visualization is a crucial step in setting up the problem for a successful solution.

Setting Up the Geometric Framework

To solve this problem effectively, we need to establish a geometric framework that represents the given information. Let's denote the first house as point A, the second house as point B, and the observation point as point C. We know that AB = 750 meters, and B is due north of A. This means that the line segment AB is a vertical line. Point C is due west of A, so the line segment AC is a horizontal line. This arrangement creates a right-angled triangle ABC, with the right angle at A. The bearing of C from B is 228°, which, as we established earlier, is 48° west of south. This angle (48°) is the angle ∠CBA within the right-angled triangle.

Now that we have a clear geometric representation, we can start to see how trigonometry comes into play. The right-angled triangle ABC is our key to unlocking the solution. We have the length of one side (AB) and one angle (∠CBA). Our goal is to find the lengths of the other two sides, AC and BC, which represent the distances of the observation point from the first and second houses, respectively. By applying the appropriate trigonometric ratios, we can relate the known information to the unknowns and solve for the distances. This step-by-step approach, starting with a clear geometric framework, is essential for tackling problems involving bearings and distances.

Understanding the relationships within the right-angled triangle is paramount to solving this problem. We have the vertical side AB, the horizontal side AC, and the hypotenuse BC. The angle ∠CBA is crucial because it connects the known side (AB) to the sides we need to find (AC and BC). Trigonometric ratios, specifically sine, cosine, and tangent, provide the tools to express these relationships mathematically. By carefully selecting the appropriate ratios based on the given information and the unknowns, we can set up equations that allow us to calculate the distances. This process highlights the power of trigonometry in translating geometric relationships into algebraic expressions, enabling us to solve for unknown quantities.

Applying Trigonometry to Find Distances

With our geometric framework in place, we can now leverage trigonometry to determine the distances. We have a right-angled triangle ABC, where AB = 750 meters, and ∠CBA = 48°. We want to find AC (the distance from the observation point to the first house) and BC (the distance from the observation point to the second house).

Finding AC (Distance from Observation Point to First House)

To find AC, we can use the tangent function. In a right-angled triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In our case, for angle ∠CBA, the opposite side is AC and the adjacent side is AB. Therefore, we have:

tan(48°) = AC / AB

We know AB = 750 meters, so we can rewrite the equation as:

AC = AB * tan(48°)

AC = 750 * tan(48°)

Using a calculator, we find that tan(48°) ≈ 1.1106. Substituting this value, we get:

AC ≈ 750 * 1.1106 ≈ 832.95 meters

Therefore, the distance from the observation point to the first house is approximately 832.95 meters.

Finding BC (Distance from Observation Point to Second House)

To find BC, we can use the cosine function. In a right-angled triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. For angle ∠CBA, the adjacent side is AB and the hypotenuse is BC. Therefore, we have:

cos(48°) = AB / BC

We know AB = 750 meters, so we can rewrite the equation as:

BC = AB / cos(48°)

Using a calculator, we find that cos(48°) ≈ 0.6691. Substituting this value, we get:

BC ≈ 750 / 0.6691 ≈ 1120.92 meters

Therefore, the distance from the observation point to the second house is approximately 1120.92 meters.

This step-by-step application of trigonometric principles demonstrates how we can use angles and side lengths in right-angled triangles to solve for unknown distances. The tangent function helped us find the distance to the first house, while the cosine function allowed us to calculate the distance to the second house. This methodical approach is crucial for solving similar problems involving bearings and distances.

Summarizing the Solution

In conclusion, by applying trigonometric principles and geometric reasoning, we have successfully determined the distances of the observation point from both houses. We found that the distance from the observation point to the first house is approximately 832.95 meters, and the distance from the observation point to the second house is approximately 1120.92 meters. These results provide a clear and precise answer to the problem posed.

The key to solving this problem was the careful application of trigonometric ratios within the right-angled triangle formed by the houses and the observation point. We used the tangent function to relate the known distance between the houses and the angle of bearing to the distance from the observation point to the first house. Similarly, we used the cosine function to find the distance from the observation point to the second house. This methodical approach, combined with a clear understanding of bearings and geometric relationships, enabled us to arrive at the accurate solution.

This problem serves as a valuable example of how mathematics, particularly trigonometry and geometry, can be applied to solve real-world problems involving distances and directions. The principles used here are fundamental in fields such as navigation, surveying, and engineering, where precise measurements and calculations are essential. By understanding and mastering these concepts, we can effectively tackle a wide range of challenges in various practical applications. The ability to visualize the problem, set up the geometric framework, and apply the appropriate trigonometric functions is a testament to the power and versatility of mathematics in our everyday lives.

#Keywords Distance, Bearing, Trigonometry, Observation Point, Houses.