Calculating Distance Between Piers Using Trigonometry

JU07/12/2025 08, 2025 by THE IDEN 54 views

Calculating the Distance Between Piers A and B: A Trigonometric Approach

In this article, we will delve into a classic trigonometry problem involving the calculation of the distance between two piers, A and B, based on the angles of sight from a boat anchored offshore. The scenario presents a boat positioned 200 feet from the shoreline, with lines of sight to piers A and B forming angles of 35° and 5° respectively, relative to the shoreline. Our objective is to determine the distance between these two piers, utilizing trigonometric principles and rounding the final answer to the nearest foot. This problem not only showcases the practical application of trigonometry in real-world scenarios but also highlights the importance of understanding angles, distances, and their relationships in geometric contexts. By breaking down the problem into manageable steps and employing trigonometric functions, we can accurately calculate the desired distance and gain a deeper appreciation for the power of mathematical tools in solving spatial challenges.

Problem Statement

A boat is situated 200 feet away from the shoreline. The angle formed by its line of sight to pier A, relative to the shoreline, is 35°. Similarly, the angle formed by the boat's line of sight to pier B, relative to the shoreline, is 5°. The question at hand is: What is the distance between pier A and pier B, rounded to the nearest foot? This problem is a classic application of trigonometry in a real-world scenario. To solve it, we'll use trigonometric functions to relate the angles of sight to the distances involved. The key here is to visualize the scenario as two right triangles sharing a common side (the distance from the boat to the shore). By calculating the distances from a point on the shore (directly opposite the boat) to each pier and then finding the difference, we can determine the distance between piers A and B. The careful application of trigonometric principles, such as the tangent function, will allow us to accurately calculate these distances and provide a solution rounded to the nearest foot.

Solution

To solve this problem, we will employ trigonometric principles, specifically the tangent function, to relate the angles of sight to the distances involved. Let's break down the solution step by step:

Visualizing the Scenario: Imagine the shoreline as a straight line, and the boat positioned 200 feet away from it. Draw a perpendicular line from the boat to the shoreline; this line represents the 200-foot distance. Piers A and B are located on the shoreline, and the lines of sight from the boat to each pier form angles of 35° and 5° with the shoreline, respectively. This setup forms two right triangles that share a common side (the 200-foot distance).
Defining Variables: Let's denote the point on the shoreline directly opposite the boat as point C. Let the distance from point C to pier A be ${ x }$ , and the distance from point C to pier B be ${ y }$ . Our goal is to find the distance between pier A and pier B, which is the absolute difference between ${ x }$ and ${ y }$ , i.e., ${ |x - y| }$ .
Applying Trigonometric Functions: In the right triangle formed by the boat, point C, and pier A, we have:

${\tan(35°) = \frac{x}{200}}$

Similarly, in the right triangle formed by the boat, point C, and pier B, we have:

${\tan(5°) = \frac{y}{200}}$
Calculating the Distances: Now, we can solve for ${ x }$ and ${ y }$ using the tangent values:

${x = 200 \times \tan(35°)}$

${y = 200 \times \tan(5°)}$

Using a calculator, we find:

${x \approx 200 \times 0.7002 \approx 140.04 \text{ feet}}$

${y \approx 200 \times 0.0875 \approx 17.50 \text{ feet}}$
Finding the Distance Between Piers: The distance between pier A and pier B is the difference between ${ x }$ and ${ y }$ :

${|x - y| = |140.04 - 17.50| = 122.54 \text{ feet}}$
Rounding to the Nearest Foot: Rounding 122.54 to the nearest foot gives us 123 feet.

Therefore, the distance between pier A and pier B is approximately 123 feet.

Detailed Explanation and Elaboration

Understanding the Trigonometric Relationships

At the heart of this problem lies the application of trigonometric functions, which provide a bridge between angles and side lengths in right triangles. Specifically, the tangent function ( ${\tan}$ ) is crucial in this scenario. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In our problem, we have two right triangles, each sharing a common side (the distance from the boat to the shore), and we are given the angles of sight from the boat to each pier. By using the tangent function, we can relate these angles to the distances from a point on the shore (directly opposite the boat) to each pier.

Step-by-Step Breakdown of the Solution

Visualizing the Scenario: The ability to visualize the problem geometrically is paramount. Imagine the shoreline as a straight line and the boat positioned 200 feet away from it. Picture a perpendicular line extending from the boat to the shoreline, representing the 200-foot distance. The piers A and B are situated on the shoreline, and the lines of sight from the boat to each pier form angles of 35° and 5° with the shoreline, respectively. This visualization helps in recognizing the two right triangles and their shared side.
Defining Variables: Assigning variables to the unknown distances simplifies the problem. We denote the point on the shoreline directly opposite the boat as point C. Let ${ x }$ represent the distance from point C to pier A, and let ${ y }$ represent the distance from point C to pier B. The objective is to determine the distance between pier A and pier B, which can be expressed as the absolute difference between ${ x }$ and ${ y }$ , denoted as ${ |x - y| }$ .
Applying Trigonometric Functions: The core of the solution lies in applying the tangent function. In the right triangle formed by the boat, point C, and pier A, the tangent of the 35° angle is the ratio of the opposite side ( ${ x }$ ) to the adjacent side (200 feet). Thus, we have ${\tan(35°) = \frac{x}{200}}$ . Similarly, in the right triangle formed by the boat, point C, and pier B, the tangent of the 5° angle is the ratio of the opposite side ( ${ y }$ ) to the adjacent side (200 feet), giving us ${\tan(5°) = \frac{y}{200}}$ .
Calculating the Distances: To find the distances ${ x }$ and ${ y }$ , we rearrange the equations derived from the tangent function. For ${ x }$ , we have ${x = 200 \times \tan(35°)}$ , and for ${ y }$ , we have ${y = 200 \times \tan(5°)}$ . Using a calculator to evaluate the tangent values, we find that ${\tan(35°) \approx 0.7002}$ and ${\tan(5°) \approx 0.0875}$ . Substituting these values, we get ${x \approx 200 \times 0.7002 \approx 140.04}$ feet and ${y \approx 200 \times 0.0875 \approx 17.50}$ feet.
Finding the Distance Between Piers: The distance between pier A and pier B is simply the difference between ${ x }$ and ${ y }$ . Calculating this difference, we have ${|x - y| = |140.04 - 17.50| = 122.54}$ feet.
Rounding to the Nearest Foot: The final step is to round the calculated distance to the nearest foot, as specified in the problem. Rounding 122.54 feet gives us 123 feet.

Conclusion of Solution

Therefore, by employing trigonometric principles and performing the necessary calculations, we have determined that the distance between pier A and pier B is approximately 123 feet. This solution demonstrates the practical application of trigonometry in solving real-world problems involving distances and angles. The use of the tangent function, along with a clear understanding of geometric relationships, allowed us to accurately calculate the desired distance and provide a solution rounded to the nearest foot.

Alternative Approaches

While the tangent function provides a straightforward solution to this problem, other trigonometric functions and geometric principles can also be employed to arrive at the same answer. Let's explore some alternative approaches:

Using the Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem that relates the sides and angles of any triangle, not just right triangles. To apply the Law of Cosines, we would need to consider the triangle formed by the boat and the two piers. However, we would first need to determine the angle at the boat. This can be found by subtracting the two given angles (35° and 5°) from 180° (the sum of angles in a triangle). The angle at the boat would be 180° - 35° - 5° = 140°. We would also need to calculate the distances from the boat to each pier using sine or cosine functions. This method, while valid, involves more steps and calculations than the direct application of the tangent function.

Using the Law of Sines

The Law of Sines is another trigonometric principle that relates the sides of a triangle to the sines of its angles. Similar to the Law of Cosines, applying the Law of Sines would require additional steps to calculate the distances from the boat to each pier and the angle at the boat. While it's a viable method, it's not as efficient as using the tangent function directly.

Geometric Construction and Measurement

Another approach, albeit less precise, involves constructing a scale diagram of the problem. By drawing the shoreline, the boat's position, and the lines of sight to the piers at the given angles, one could measure the distance between the piers on the diagram and then scale it back to the real-world dimensions. This method is useful for visualizing the problem but is prone to inaccuracies due to the limitations of drawing and measuring tools.

Comparison of Approaches

While alternative methods exist, the direct application of the tangent function offers the most efficient and accurate solution in this particular scenario. It minimizes the number of calculations and directly relates the given angles to the distances we need to find. The Law of Cosines and Law of Sines, while powerful tools in trigonometry, are more suited for problems where we have different sets of information or are dealing with non-right triangles. Geometric construction can be helpful for visualization but lacks the precision of trigonometric calculations.

Real-World Applications and Implications

The trigonometric principles applied in this problem have far-reaching applications in various real-world scenarios. Understanding angles, distances, and their relationships is crucial in fields such as navigation, surveying, engineering, and even astronomy. Let's explore some specific examples:

Navigation

In navigation, determining the position and course of a vessel or aircraft relies heavily on trigonometric calculations. Techniques such as triangulation, which involves measuring angles to known landmarks, allow navigators to accurately pinpoint their location. Similarly, the angles of elevation to celestial bodies are used in celestial navigation to determine a vessel's latitude and longitude. The problem we solved, involving a boat and piers, is a simplified version of the challenges faced by navigators in determining distances and positions at sea.

Surveying

Surveyors use trigonometry extensively to measure land and create maps. They employ instruments like theodolites and total stations to measure angles and distances, and then use trigonometric functions to calculate the coordinates of various points on the land. This information is essential for land division, construction planning, and infrastructure development. The principles used in our problem are fundamental to the work of surveyors in accurately mapping the Earth's surface.

Engineering

Engineers apply trigonometric principles in the design and construction of various structures, such as bridges, buildings, and roads. They need to calculate angles and distances to ensure the stability and safety of these structures. For example, determining the angles and lengths of truss members in a bridge requires a thorough understanding of trigonometric relationships. Similarly, in road design, engineers use trigonometry to calculate the grades and curves of the road to ensure smooth and safe travel.

Astronomy

Astronomers use trigonometry to measure the distances to stars and other celestial objects. Techniques such as parallax, which involves measuring the apparent shift in the position of a star as the Earth orbits the Sun, rely on trigonometric calculations. By measuring the parallax angle, astronomers can determine the distance to the star. Trigonometry is also used in calculating the orbits of planets and other celestial bodies.

Other Applications

Beyond these specific fields, trigonometric principles are applied in various other areas, including:

Computer Graphics: Trigonometry is essential for creating 3D models and animations in computer graphics.
Physics: Many physics concepts, such as projectile motion and wave mechanics, involve trigonometric relationships.
Architecture: Architects use trigonometry to design buildings and ensure structural integrity.

Implications for Problem-Solving

The problem we solved highlights the importance of breaking down complex problems into simpler steps. By visualizing the scenario, defining variables, applying appropriate trigonometric functions, and performing calculations systematically, we were able to arrive at the solution. This approach to problem-solving is applicable in various fields and situations. It emphasizes the need for a clear understanding of the underlying principles and the ability to apply them methodically.

Conclusion

In conclusion, the problem of calculating the distance between piers A and B, given the angles of sight from a boat, serves as a compelling example of the practical application of trigonometry. By employing the tangent function and following a step-by-step approach, we were able to determine the distance to be approximately 123 feet. This problem not only reinforces our understanding of trigonometric principles but also highlights their relevance in real-world scenarios. Furthermore, the exploration of alternative solution methods and the discussion of broader applications underscore the versatility and importance of trigonometry in various fields, ranging from navigation and surveying to engineering and astronomy. The ability to visualize the problem, apply appropriate mathematical tools, and interpret the results is crucial for effective problem-solving in mathematics and beyond.