Scooter Distance From Building: Angle Of Depression Problem
Hey guys! Let's dive into an interesting problem involving angles of depression and distances. We've got a scenario where someone's looking down from the top of a building at a scooter parked some distance away. The angle of depression is 30 degrees, and the building's height is 40 meters. Our mission? To figure out how far the scooter is from the base of the building. Sounds like fun, right? Let's get started!
Understanding the Problem
First things first, let's break down what we know and what we need to find. The angle of depression is the angle formed between the horizontal line of sight and the line of sight looking downwards. In our case, it's the angle at which the observer at the top of the building is looking down at the scooter. We know this angle is 30 degrees. We also know the height of the building, which is 40 meters. What we don't know, and what we're trying to find, is the horizontal distance between the base of the building and where the scooter is parked.
To visualize this, imagine a right-angled triangle. The building forms the vertical side, the ground forms the horizontal side, and the line of sight from the observer to the scooter forms the hypotenuse. The angle of depression is outside the triangle, but it's equal to the angle of elevation from the scooter to the top of the building (alternate interior angles, remember your geometry?). This little trick helps us bring the angle inside our triangle, making things much easier to work with. The key here is to really picture what's happening. Draw a diagram if it helps! A visual representation can make these problems way less intimidating.
When tackling word problems like these, always start by identifying the key information. What are you given? What are you trying to find? Once you've got a clear picture of the situation, you can start thinking about which mathematical tools will help you solve it. In this case, trigonometry is going to be our best friend. Remember SOH CAH TOA? It's about to come in handy!
Setting Up the Trigonometry
Now that we've got our triangle all pictured and we know we're dealing with trigonometry, let's figure out which trig function to use. Remember SOH CAH TOA? It's a handy acronym that helps us remember the relationships between the sides and angles in a right-angled triangle:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
In our problem, we're interested in the angle of elevation (which is equal to the angle of depression), the height of the building (which is the side opposite the angle), and the distance from the building to the scooter (which is the side adjacent to the angle). So, which trig function relates the opposite and adjacent sides? You guessed it – Tangent (TOA)!
The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In mathematical terms:
tan(angle) = Opposite / Adjacent
In our case, the angle is 30 degrees, the opposite side is the height of the building (40 meters), and the adjacent side is the distance we're trying to find (let's call it 'd'). So, we can write our equation as:
tan(30°) = 40 / d
See how we've translated the word problem into a neat little equation? That's the power of trigonometry! Now, all that's left to do is solve for 'd'. Don't worry; it's not as scary as it looks. We'll take it step by step. Remember, the key to mastering these types of problems is practice. The more you set them up, the more comfortable you'll become with identifying the right trig function and translating the word problem into a solvable equation.
Solving for the Distance
Alright, we've got our equation: tan(30°) = 40 / d. Now, let's roll up our sleeves and solve for 'd', the distance of the scooter from the building. The first thing we need to do is isolate 'd'. Right now, it's in the denominator, which isn't ideal. So, let's multiply both sides of the equation by 'd' to get it out of the denominator:
d * tan(30°) = 40
Great! Now 'd' is on its own on the left side, but it's still being multiplied by tan(30°). To get 'd' completely by itself, we need to divide both sides of the equation by tan(30°):
d = 40 / tan(30°)
Now we're talking! We've got 'd' all alone, and the right side of the equation is something we can actually calculate. But wait, what's tan(30°)? Unless you've got some crazy trig memorization skills, you'll probably need to use a calculator or a trigonometric table to find the value of tan(30°). If you plug it into your calculator, you'll find that:
tan(30°) ≈ 0.577
(It's actually 1/√3, but 0.577 is a good decimal approximation for our purposes.)
Now we can substitute this value back into our equation:
d = 40 / 0.577
And finally, we can do the division to find the value of 'd':
d ≈ 69.3 meters
So, the scooter is approximately 69.3 meters away from the base of the building. Woohoo! We solved it! Isn't it satisfying when the numbers finally click into place? The key takeaway here is the importance of algebraic manipulation. Getting 'd' by itself required a couple of simple steps: multiplying and dividing. These are fundamental skills in algebra, and they're crucial for solving all sorts of math problems. So, if you're feeling a little rusty on your algebra, it's worth brushing up. It'll make your life so much easier when tackling these kinds of problems!
The Final Answer and Key Concepts
So, after all that math, we've arrived at our final answer: the scooter is approximately 69.3 meters away from the base of the building. That's a pretty good distance! It's always a good idea to think about whether your answer makes sense in the context of the problem. A distance of 69.3 meters seems reasonable for a scooter parked away from a 40-meter-tall building.
Let's recap the key concepts we used to solve this problem. First, we understood the concept of the angle of depression and how it relates to the angle of elevation. We used the fact that these angles are equal (alternate interior angles) to bring the angle into our right-angled triangle. This is a crucial step in setting up the problem correctly. Secondly, we used trigonometry, specifically the tangent function (TOA), to relate the angle, the opposite side (building height), and the adjacent side (scooter distance). Choosing the right trig function is essential for solving these problems.
Finally, we used algebraic manipulation to solve for the unknown distance. We multiplied and divided both sides of the equation to isolate the variable 'd'. This is a fundamental skill in mathematics, and it's something you'll use again and again. By mastering these concepts – angles of depression, trigonometry, and algebraic manipulation – you'll be well-equipped to tackle similar problems in the future. And remember, practice makes perfect! The more problems you solve, the more confident you'll become in your math abilities. So, keep practicing, keep asking questions, and keep having fun with math!