Determining The Height Of A Telephone Pole Using Trigonometry

Introduction

In this article, we will explore a classic trigonometry problem: determining the height of a telephone pole given the distance from the observer and the angle of elevation. This problem exemplifies how trigonometric ratios can be applied in real-world scenarios to calculate heights and distances indirectly. We will walk through the problem step-by-step, explaining the trigonometric principles involved and arriving at the solution. This exercise not only reinforces understanding of trigonometry but also highlights its practical applications in fields such as surveying, engineering, and navigation. The scenario presented is a person standing 36 feet away from a telephone pole, with a 30-degree angle of elevation from the ground to the top of the pole. Our goal is to find the height of the pole using the given information and trigonometric functions. This problem is a perfect example of how trigonometry can be used to solve for unknown lengths in right triangles, making it a valuable tool for various real-world applications. Understanding the relationship between angles and sides in a right triangle is crucial for solving this type of problem, and we will delve into the specifics of how to use the tangent function to find the height of the pole.

Problem Statement

A person is standing exactly 36 feet from a telephone pole. There is a $30^{\circ}$ angle of elevation from the ground to the top of the pole. What is the height of the pole?

Options:

A. 12 ft B. $12 \sqrt{3}$ ft C. 18 ft D. $18 \sqrt{2}$ ft

Understanding the Concepts

Before we dive into the solution, it's crucial to understand the underlying trigonometric concepts. The scenario forms a right triangle, where the telephone pole is the opposite side, the distance from the person to the pole is the adjacent side, and the line of sight from the person to the top of the pole is the hypotenuse. The angle of elevation is the angle between the horizontal line of sight and the line of sight to the top of the pole. In this case, we are given the adjacent side (36 feet) and the angle of elevation ($30^{\circ}$), and we need to find the opposite side (height of the pole). The trigonometric function that relates the opposite and adjacent sides of a right triangle is the tangent function. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Therefore, we can use the formula $\tan(\theta) = \frac{opposite}{adjacent}$ to solve for the height of the pole. This fundamental trigonometric relationship is the key to solving this problem and many others involving right triangles. By understanding the definitions of the trigonometric functions and how they relate to the sides of a right triangle, we can effectively tackle a wide range of problems in geometry and trigonometry. The tangent function is particularly useful when dealing with angles of elevation and depression, as it allows us to find unknown heights or distances based on known values. This concept is widely applied in fields such as surveying, navigation, and engineering, where accurate measurements are essential.

Solution

To solve this problem, we will use the tangent function. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. In this scenario:

• The angle of elevation, $\theta$, is $30^{\circ}$.
• The distance from the person to the pole (adjacent side) is 36 ft.
• The height of the pole (opposite side) is what we want to find, let's call it h.

The formula we will use is:

$\tan(\theta) = \frac{opposite}{adjacent}$

Plugging in the values, we get:

$\tan(30^{\circ}) = \frac{h}{36}$

We know that $\tan(30^{\circ}) = \frac{1}{\sqrt{3}}$. So, the equation becomes:

$\frac{1}{\sqrt{3}} = \frac{h}{36}$

To solve for h, we multiply both sides by 36:

$h = 36 \cdot \frac{1}{\sqrt{3}}$

$h = \frac{36}{\sqrt{3}}$

To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{3}$:

$h = \frac{36}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$h = \frac{36\sqrt{3}}{3}$

$h = 12\sqrt{3}$

Therefore, the height of the pole is $12\sqrt{3}$ feet. This step-by-step solution demonstrates how the tangent function is applied in a practical context to calculate unknown heights. By understanding the relationship between angles and sides in a right triangle, we can effectively use trigonometry to solve real-world problems. The process of rationalizing the denominator is also an important algebraic technique to ensure that the final answer is in its simplest form. This example highlights the power of trigonometry as a tool for solving problems involving angles and distances, and it reinforces the importance of mastering trigonometric functions and their applications. The ability to set up and solve trigonometric equations is a valuable skill in many fields, including engineering, physics, and architecture. By breaking down the problem into smaller steps and carefully applying the relevant trigonometric principles, we can arrive at the correct solution and gain a deeper understanding of the underlying concepts.

Correct Answer

The correct answer is B. $12 \sqrt{3}$ ft.

Conclusion

In conclusion, we have successfully determined the height of the telephone pole using trigonometric principles. By understanding the relationship between the angle of elevation, the distance from the observer, and the height of the pole, we were able to apply the tangent function to solve for the unknown height. This problem illustrates the practical application of trigonometry in real-world scenarios, particularly in fields such as surveying, engineering, and architecture. The step-by-step solution provided a clear and concise method for solving similar problems involving right triangles and angles of elevation. It is essential to grasp the fundamental trigonometric ratios and their applications to effectively solve these types of problems. The ability to identify the relevant sides of the triangle and the appropriate trigonometric function is crucial for success. Furthermore, the process of rationalizing the denominator is an important algebraic technique that ensures the final answer is in its simplest form. This exercise not only reinforces the understanding of trigonometry but also highlights its versatility as a problem-solving tool. By mastering these concepts, students and professionals alike can tackle a wide range of problems involving angles, distances, and heights. The application of trigonometric principles extends beyond academic exercises and finds practical use in various fields, making it a valuable skill to acquire. The problem-solving approach used in this article can be applied to other similar scenarios, further solidifying the understanding of trigonometry and its applications. Therefore, a strong foundation in trigonometry is essential for success in many fields that require spatial reasoning and problem-solving skills.