Calculating Distance Using Angle Of Depression How Far Is The Cat From The Tower

Introduction

This article delves into a classic trigonometry problem: determining the distance of an object from the base of a tower given the angle of depression. In this specific scenario, we have a tower 5 cm high and a cat on the ground. The angle of depression from the top of the tower to the cat is 30 degrees. Our goal is to calculate the horizontal distance between the cat and the base of the tower. This problem provides a practical application of trigonometric principles, particularly the tangent function, and helps to illustrate how these concepts can be used to solve real-world distance and height problems. This exploration will not only reinforce your understanding of trigonometry but also enhance your problem-solving skills in geometry and related fields. We will break down the problem step by step, ensuring a clear and comprehensive understanding of the solution.

Understanding the Problem: Angle of Depression and Trigonometry

To effectively solve this problem, it's crucial to grasp the concept of the angle of depression and how it relates to trigonometric ratios. The angle of depression is the angle formed between the horizontal line of sight and the line of sight directed downwards to an object. Imagine standing at the top of the tower and looking down at the cat; the angle your eyes make with the horizontal plane is the angle of depression. This angle is vital because it allows us to create a right-angled triangle, a cornerstone of trigonometric problem-solving. In our case, the tower's height forms the opposite side of the triangle, the distance to the cat is the adjacent side, and the line of sight forms the hypotenuse. Trigonometric ratios, such as sine, cosine, and tangent, provide the mathematical tools to relate these sides and angles. Specifically, the tangent function, defined as the ratio of the opposite side to the adjacent side, will be instrumental in solving this problem. By understanding these fundamental principles, we can translate the given information into a solvable mathematical equation, leading us to the solution of the distance between the cat and the tower's base. Understanding these trigonometric concepts ensures that you are well-equipped to tackle similar problems and appreciate the practical applications of trigonometry in various fields.

Setting Up the Trigonometric Equation

With a clear understanding of the problem and the relevant trigonometric concepts, the next step is to translate the given information into a mathematical equation. We know the height of the tower (5 cm), the angle of depression (30 degrees), and we need to find the horizontal distance between the cat and the base of the tower. As established earlier, the tangent function is the key to unlocking this problem. The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In our scenario, the height of the tower represents the opposite side, the distance we're trying to find represents the adjacent side, and the angle of depression is the angle we're working with. Therefore, we can express the relationship as follows: tan(30°) = (height of the tower) / (distance to the cat). This equation sets the stage for solving the problem. Substituting the known values, we get tan(30°) = 5 / distance. The next step involves solving this equation for the distance, which will provide us with the answer to our problem. This methodical setup ensures that we have a clear and accurate representation of the problem in mathematical terms.

Solving for the Distance

Now that we have our trigonometric equation, tan(30°) = 5 / distance, we can proceed to solve for the unknown, which is the distance between the cat and the base of the tower. The first step in solving this equation is to isolate the 'distance' variable. We can do this by multiplying both sides of the equation by 'distance' and then dividing both sides by tan(30°). This manipulation gives us: distance = 5 / tan(30°). Next, we need to determine the value of tan(30°). From trigonometric tables or a calculator, we know that tan(30°) is approximately equal to 1/√3 or √3/3, which is about 0.577. Substituting this value into our equation, we get: distance = 5 / 0.577. Performing this division, we find that the distance is approximately 8.66 cm. Therefore, the cat is approximately 8.66 cm away from the base of the tower. This step-by-step solution demonstrates how we can use trigonometric principles and algebraic manipulation to find unknown distances in real-world scenarios. The final answer not only solves the problem but also reinforces the practical application of trigonometry in distance calculations.

Final Answer: The Cat's Distance from the Tower

Having meticulously set up the trigonometric equation and solved for the unknown, we arrive at the final answer: the cat is approximately 8.66 cm away from the base of the tower. This solution is the culmination of our step-by-step process, which began with understanding the problem, defining the angle of depression, and applying the tangent function. We successfully translated the word problem into a mathematical equation, utilizing the given information about the tower's height and the angle of depression. By solving for the distance, we've not only answered the specific question but also demonstrated the power of trigonometric principles in real-world scenarios. The accuracy of our result is contingent upon the precise application of trigonometric ratios and algebraic manipulation. This example underscores the importance of attention to detail in mathematical problem-solving. Furthermore, it reinforces the practical relevance of trigonometry in fields such as surveying, navigation, and engineering, where calculating distances and heights is a common task. Understanding and applying these concepts can provide valuable skills for various applications.

Conclusion: The Power of Trigonometry in Distance Calculation

In conclusion, solving this problem highlights the power and practicality of trigonometry in calculating distances. By understanding the concepts of angle of depression and trigonometric ratios, specifically the tangent function, we were able to determine that the cat is approximately 8.66 cm away from the base of the 5 cm high tower. This exercise not only reinforces the theoretical knowledge of trigonometry but also demonstrates its real-world applications. Trigonometry is a fundamental tool in various fields, including surveying, navigation, architecture, and engineering, where the calculation of distances and heights is crucial. The ability to translate a real-world scenario into a mathematical model and solve it using trigonometric principles is a valuable skill. Moreover, this problem-solving process encourages analytical thinking and attention to detail, essential qualities in mathematics and related disciplines. By mastering trigonometric concepts, one can gain a deeper understanding of spatial relationships and develop the ability to solve a wide range of practical problems. This example serves as a testament to the enduring relevance and utility of trigonometry in our everyday lives and professional endeavors.