Calculating Tower Height Using Angles Of Elevation
Introduction
In this article, we will explore a classic problem in trigonometry: determining the height of a tower using angles of elevation. This problem involves using trigonometric ratios, specifically the tangent function, to relate the angles of elevation to the distances and heights involved. We will break down the problem step-by-step, providing a clear and comprehensive solution. This approach is not only applicable to this specific scenario but also to a wide range of similar problems in surveying, navigation, and engineering. Mastering the concepts presented here will equip you with the tools necessary to tackle real-world challenges involving angles and distances.
Understanding Angles of Elevation
The angle of elevation is a fundamental concept in trigonometry and is used extensively in various fields. It is defined as the angle formed between the horizontal line of sight and the line of sight directed upwards to a particular point. Imagine standing on the ground and looking up at the top of a tall building; the angle your eyes make with the horizontal is the angle of elevation. This angle is crucial in determining heights and distances, especially when direct measurement is not feasible. Understanding the concept of the angle of elevation is essential for solving problems related to heights and distances in real-world scenarios. In surveying, for example, surveyors use the angle of elevation to calculate the height of mountains or buildings. Similarly, in navigation, the angle of elevation of celestial bodies is used to determine the observer's position. The angle of elevation, in conjunction with the distance to the object, allows us to form a right triangle, enabling the application of trigonometric ratios like tangent, sine, and cosine. Therefore, a solid grasp of this concept is crucial for anyone working with angles and distances.
Problem Statement
Let's consider the problem at hand: From a point C on the horizontal ground, the angle of elevation to the top of a tower is measured to be 30 degrees. We then advance 12 meters towards the tower and measure the angle of elevation again, which is now 45 degrees. Our goal is to determine the height of the tower. This is a classic problem that utilizes trigonometric principles, particularly the tangent function, to relate the angles of elevation to the distances and height of the tower. The problem's setup provides us with two right triangles, each with a different angle of elevation and base distance. By setting up equations using the tangent function for both triangles, we can create a system of equations that will allow us to solve for the unknown height of the tower and the initial distance from point C to the base of the tower. This problem serves as a practical example of how trigonometry can be used to solve real-world measurement problems, demonstrating the power and versatility of trigonometric principles.
Setting up the Equations
To solve this problem, let's denote the height of the tower as h and the initial distance from point C to the base of the tower as x. When the angle of elevation is 30 degrees, we can use the tangent function to relate the height h to the distance x. The tangent of an angle in a right triangle is defined as the ratio of the opposite side (the height of the tower) to the adjacent side (the distance from the point of observation to the base of the tower). Therefore, we have: tan(30°) = h/x. Similarly, after advancing 12 meters towards the tower, the distance from the new point of observation to the base of the tower is x - 12. The angle of elevation at this point is 45 degrees. Again, using the tangent function, we get: tan(45°) = h / (x - 12). Now we have two equations with two unknowns (h and x), which we can solve simultaneously to find the height of the tower. This setup is a crucial step in solving the problem, as it translates the geometric scenario into a set of algebraic equations that can be manipulated to find the desired solution. The use of the tangent function here is key, as it directly relates the angles of elevation to the height and distances involved, allowing us to create a solvable system of equations.
Equation 1: Initial Angle of Elevation
As established earlier, the tangent of the initial angle of elevation (30 degrees) is equal to the ratio of the tower's height (h) to the initial distance from point C (x). This can be expressed as:
tan(30°) = h / x
This equation forms the foundation of our solution. We know that tan(30°) is a standard trigonometric value, which is equal to 1/√3. Substituting this value into the equation, we get:
1/√3 = h / x
This equation provides a direct relationship between the tower's height and the initial distance. It tells us that the ratio of the height to the distance is constant and equal to 1/√3. This equation will be crucial when we solve the system of equations later on. Understanding this relationship is key to grasping the underlying principles of trigonometry and its application in real-world scenarios. By using the tangent function, we have successfully translated the geometric information into an algebraic equation, allowing us to proceed with the solution process. This step highlights the power of trigonometry in bridging the gap between geometry and algebra, enabling us to solve complex problems involving angles and distances.
Equation 2: Angle of Elevation After Advancing
After advancing 12 meters towards the tower, the angle of elevation changes to 45 degrees. This new angle provides us with another crucial piece of information that we can use to determine the height of the tower. The distance from the new point of observation to the base of the tower is now x - 12. Using the tangent function again, we can relate the height h to this new distance. The tangent of 45 degrees is the ratio of the tower's height to the new distance, which can be expressed as:
tan(45°) = h / (x - 12)
We know that tan(45°) is equal to 1, a fundamental trigonometric value. Substituting this value into the equation, we get:
1 = h / (x - 12)
This simplifies to:
h = x - 12
This equation provides a direct relationship between the tower's height and the distance x. It tells us that the height of the tower is equal to the distance x minus 12 meters. This equation, along with the first equation we derived, forms a system of two equations with two unknowns, which we can solve simultaneously to find the values of h and x. The fact that tan(45°) equals 1 simplifies the equation, making it easier to solve. This step demonstrates how understanding basic trigonometric values can significantly simplify complex problems. By setting up this second equation, we have completed the necessary groundwork to solve for the height of the tower.
Solving the System of Equations
Now that we have two equations:
- 1/√3 = h / x
- h = x - 12
We can solve this system of equations to find the values of h and x. A common method for solving such systems is substitution. From the second equation, we have h expressed in terms of x. We can substitute this expression for h into the first equation:
1/√3 = (x - 12) / x
Now we have a single equation with one unknown, x. To solve for x, we can cross-multiply:
x = √3(x - 12)
Expanding the right side, we get:
x = √3x - 12√3
Rearranging the terms to isolate x, we have:
√3x - x = 12√3
Factoring out x on the left side:
x(√3 - 1) = 12√3
Finally, solving for x:
x = (12√3) / (√3 - 1)
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator (√3 + 1):
x = (12√3(√3 + 1)) / ((√3 - 1)(√3 + 1))
Simplifying, we get:
x = (12(3 + √3)) / (3 - 1)
x = (12(3 + √3)) / 2
x = 6(3 + √3)
Now that we have found the value of x, we can substitute it back into the second equation (h = x - 12) to find the value of h:
h = 6(3 + √3) - 12
h = 18 + 6√3 - 12
h = 6 + 6√3
Therefore, the height of the tower is 6 + 6√3 meters. This step-by-step solution demonstrates the process of solving a system of equations derived from trigonometric relationships. The use of substitution and algebraic manipulation allows us to find the values of the unknowns, ultimately leading us to the solution of the problem.
Calculating the Height
We have determined that the height of the tower, h, is given by the expression:
h = 6 + 6√3
To find the numerical value of h, we need to approximate the value of √3. The square root of 3 is approximately 1.732. Substituting this value into the expression for h, we get:
h = 6 + 6(1.732)
h = 6 + 10.392
h = 16.392
Therefore, the height of the tower is approximately 16.392 meters. This calculation provides a concrete answer to the problem, giving us a clear understanding of the tower's height. The process of substituting the approximate value of √3 and performing the arithmetic operations demonstrates how we can translate a mathematical expression into a practical numerical result. This final step is crucial in problem-solving, as it provides a tangible solution that can be applied in real-world scenarios. The calculated height of the tower is a valuable piece of information that can be used for various purposes, such as construction planning, surveying, and mapping.
Conclusion
In conclusion, we have successfully determined the height of the tower using angles of elevation and trigonometric principles. By setting up equations based on the tangent function and solving the resulting system of equations, we found that the height of the tower is approximately 16.392 meters. This problem highlights the power of trigonometry in solving real-world problems involving heights and distances. The key steps in the solution process included understanding the concept of the angle of elevation, setting up trigonometric equations, solving a system of equations, and performing the necessary calculations to arrive at the final answer. The techniques and principles demonstrated in this article can be applied to a wide range of similar problems in various fields, including surveying, navigation, and engineering. Mastering these concepts is essential for anyone working with angles, distances, and heights, as it provides a powerful toolset for solving practical measurement problems. The combination of trigonometric knowledge and algebraic skills is crucial for success in these areas, and the ability to apply these principles effectively is a valuable asset in many professional and academic pursuits.