Tower Height Calculation On A Hill Trigonometry Problem Solved
Introduction
Hey guys! Let's dive into a fascinating problem today that combines trigonometry and real-world scenarios. We're going to tackle a classic problem involving a tower, a hill, and some angles. Imagine this: a tower stands proudly at the foot of a hill, and this hill slopes upwards at a 15-degree angle. A person is standing on the hill, 12 meters away from the base of the tower. When they look up at the very top of the tower, they see it at a 45-degree angle. Our mission? To figure out just how tall that tower is. This isn't just a textbook problem; it's the kind of situation you might encounter in surveying, architecture, or even just appreciating the world around you. To solve this, we'll be using some cool trigonometric principles, specifically the laws of sines and cosines, which are essential tools for handling triangles that aren't right-angled. These laws allow us to relate angles and side lengths in any triangle, which is super handy for problems like this where we're dealing with slopes and angles that aren't perfectly vertical or horizontal. So, grab your thinking caps, and let's get started on unraveling this mathematical puzzle! We'll break down the problem step-by-step, making sure to understand each part before moving on. By the end, you'll not only know the height of the tower but also have a solid understanding of how to apply trigonometry to solve real-world problems. Remember, math isn't just about numbers; it's about understanding the relationships between things, and this problem is a perfect example of that.
Setting Up the Problem
First, let's visualize the scenario. Picture a hill sloping upwards, and at the bottom of this hill sits a tower. A person is standing on the hill, a certain distance from the tower's base. To really nail this problem, we need to translate this word picture into a geometric diagram. This means sketching out a triangle (or maybe even a couple of them!) that represents the situation. Think of the tower as one side of a triangle, the hill's slope as another, and the line of sight from the person to the tower's top as the third. Key to this setup is identifying the angles we know: the 15-degree incline of the hill and the 45-degree angle of elevation to the tower's top. These angles are our starting points, our anchors in the problem. We also know the distance from the person to the tower's base – that 12-meter measurement. This is another crucial piece of information that will help us solve for the unknown height of the tower. Now, let's get down to labeling our diagram. Label the base of the tower as point A, the top of the tower as point B, and the person's position on the hill as point C. This gives us triangle ABC. The height of the tower is the length of side AB, which is what we're trying to find. The distance from the person to the tower's base (12 meters) is the length of side AC. And the angles? We know the angle at A (where the tower meets the hill) isn't a right angle because of the hill's slope. We know the angle of elevation from C to B is 45 degrees, and we know the hill's inclination is 15 degrees. With this labeled diagram in place, we've transformed a word problem into a visual and geometric challenge. We've laid the groundwork for applying our trigonometric tools, and we're one big step closer to finding the tower's height. Remember, a clear diagram is often half the battle when it comes to solving these kinds of problems.
Applying the Law of Sines
Okay, now that we have our diagram all set up, it's time to bring in the big guns of trigonometry: the Law of Sines. This law is a fantastic tool for solving triangles that aren't right-angled, which is exactly what we're dealing with here. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. In simpler terms, if we have a triangle ABC, then a / sin(A) = b / sin(B) = c / sin(C), where a, b, and c are the side lengths, and A, B, and C are the angles opposite those sides. So, how does this help us with our tower problem? Well, let's think about what we know and what we need to find. We know the length AC (12 meters), and we have information about the angles. To use the Law of Sines effectively, we need to find at least one angle in our triangle ABC. We already know the angle of elevation from the person to the tower's top (45 degrees) and the hill's inclination (15 degrees). The angle at point C, which is the angle between the person's line of sight to the tower's base and their line of sight to the tower's top, is a crucial piece of the puzzle. To find this angle, we can use the fact that the angles in a triangle add up to 180 degrees. But first, we need to figure out the angle at the base of the tower (angle BAC). Since the hill is inclined at 15 degrees, angle BAC is 90 degrees (the tower) plus 15 degrees (the hill's slope), which gives us 105 degrees. Now we can find the angle at C. It's 180 degrees minus the angle at B (45 degrees) minus the angle at A (105 degrees). That leaves us with 30 degrees at angle C! With angle C in hand, we're in a great position to use the Law of Sines. We can set up a proportion using the side we know (AC) and its opposite angle (angle B), and the side we want to find (AB, the tower's height) and its opposite angle (angle C). This will give us an equation that we can solve for the height of the tower. Let's get to those calculations!
Calculating the Tower's Height
Alright, guys, let's get down to the nitty-gritty and actually calculate the height of the tower. We've set up our problem, drawn our diagram, and armed ourselves with the Law of Sines. Now it's time to put those tools to work. Remember our Law of Sines proportion? We're going to use the relationship between side AC (12 meters), its opposite angle B (45 degrees), side AB (the tower's height, which we'll call 'h'), and its opposite angle C (30 degrees). So, our equation looks like this: 12 / sin(45°) = h / sin(30°). Now we need to solve for 'h'. The first step is to isolate 'h' on one side of the equation. We can do this by multiplying both sides of the equation by sin(30°). This gives us: h = 12 * sin(30°) / sin(45°). Next, we need to plug in the values for sin(30°) and sin(45°). If you remember your trigonometric values, sin(30°) is 0.5 and sin(45°) is approximately 0.707 (or √2 / 2). If you don't have these memorized, no worries! You can always use a calculator or a trigonometric table. Substituting these values into our equation, we get: h = 12 * 0.5 / 0.707. Now it's just a matter of doing the arithmetic. 12 times 0.5 is 6, so we have: h = 6 / 0.707. Dividing 6 by 0.707 gives us approximately 8.48. So, the height of the tower, 'h', is about 8.48 meters. But wait, we're not quite done yet! It's always a good idea to think about our answer and see if it makes sense in the context of the problem. We have a tower that's about 8.48 meters tall, standing on a hill with a 15-degree slope, and a person standing 12 meters away sees the top of the tower at a 45-degree angle. Does this sound reasonable? Given the angles and distances involved, it seems like a plausible height for the tower. And there you have it! We've successfully calculated the height of the tower using the Law of Sines. This is a great example of how trigonometry can be used to solve real-world problems.
Conclusion
Wow, we've really taken a journey through trigonometry today, haven't we? We started with a description of a tower on a hill, visualized the scenario, drew a diagram, and then applied the Law of Sines to calculate the tower's height. This problem perfectly illustrates the power of trigonometry in solving real-world problems. From architecture and surveying to navigation and engineering, trigonometric principles are used every day to measure distances, heights, and angles. What's so cool about this particular problem is how it combines several key concepts. We had to understand angles of elevation and inclination, visualize a geometric scenario, and then apply a specific trigonometric law. It's not just about memorizing formulas; it's about understanding how those formulas connect to the world around us. The Law of Sines, in particular, is a versatile tool for solving triangles that aren't right-angled. It allows us to relate angles and side lengths in a way that's incredibly useful for a wide range of problems. And remember, the key to tackling these kinds of problems is to break them down into smaller, manageable steps. Draw a clear diagram, identify the knowns and unknowns, choose the appropriate trigonometric tool, and then carefully work through the calculations. But beyond the specific steps, there's a bigger takeaway here. Math isn't just an abstract subject confined to textbooks and classrooms. It's a way of thinking, a way of analyzing the world, and a powerful tool for solving problems. By tackling challenges like this tower problem, we're not just learning trigonometry; we're developing our problem-solving skills, our spatial reasoning, and our ability to think critically. So, the next time you see a tower on a hill, you'll not only appreciate its beauty but also have the mathematical tools to figure out its height! Keep exploring, keep questioning, and keep applying those math skills to the world around you. You never know what fascinating problems you'll be able to solve.
So, the final answer is approximately 8.48 meters. Pretty neat, huh?
Calculate the height of a tower standing at the foot of a 15-degree inclined hill, given that a person 12 meters away from the tower's base observes the top of the tower at a 45-degree angle.
Tower Height Calculation on a Hill Trigonometry Problem Solved
