Ladder Height On Wall: Trigonometry Problem Solved!

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Hey guys! Let's dive into a super practical trigonometry problem that you might encounter in your math class or even in real-life situations. We're talking about a ladder leaning against a wall – a classic scenario that perfectly illustrates how trigonometry works. Specifically, we're going to figure out how high up the wall a 14-meter ladder reaches when it's placed at a 60-degree angle to the ground. This isn't just about crunching numbers; it's about understanding the relationships between angles and sides in a right-angled triangle. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, so here's the setup: we've got a 14-meter ladder, and it's leaning against a wall. The angle between the ladder and the ground is 60 degrees. Our mission, should we choose to accept it (and we do!), is to find out how high the ladder reaches on the wall. In other words, we want to calculate the vertical distance from the ground to the top of the ladder. This is a classic right triangle problem, and trigonometry is our trusty tool for solving it. Think of the ladder as the hypotenuse (the longest side) of the triangle, the wall as the opposite side (the side opposite the 60-degree angle), and the ground as the adjacent side (the side next to the 60-degree angle). We know the length of the hypotenuse (14 meters) and the angle (60 degrees), and we need to find the length of the opposite side. This is where trigonometric functions like sine, cosine, and tangent come into play. Remember SOH CAH TOA? It's going to be our best friend here. So, let's break down which function is most useful for this problem and why.

Choosing the Right Trigonometric Function

To solve this problem effectively, the key is picking the right trigonometric function. Remember the acronym SOH CAH TOA? It’s a lifesaver! Let’s break it down:

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

In our ladder scenario, we know the length of the ladder (the hypotenuse) and the angle between the ladder and the ground (60 degrees). We want to find the height the ladder reaches on the wall, which is the opposite side to the 60-degree angle. Looking at SOH CAH TOA, we see that SOH (Sine = Opposite / Hypotenuse) is the perfect fit because it relates the opposite side and the hypotenuse, both of which are involved in our problem. Cosine, on the other hand, involves the adjacent side, which we don’t need right now. Tangent also uses the adjacent side, so it’s not the best choice either. By selecting the sine function, we can set up a simple equation that allows us to calculate the height of the ladder on the wall. This strategic approach simplifies the problem and makes it much easier to solve. So, let’s move on to setting up the equation and crunching those numbers!

Setting up the Equation

Alright, now that we've figured out that the sine function is our best bet, let's set up the equation. We know that the sine of an angle is equal to the Opposite side divided by the Hypotenuse. In our case, the angle is 60 degrees, the hypotenuse (the ladder) is 14 meters, and the opposite side is the height we're trying to find – let's call it h. So, we can write the equation like this:

sin(60°) = h / 14

This equation is the key to unlocking the solution. It directly relates the angle we know (60 degrees), the length of the ladder (14 meters), and the height we're trying to find (h). To solve for h, we need to isolate it on one side of the equation. This involves a bit of algebraic manipulation, but it's nothing we can't handle. We'll multiply both sides of the equation by 14 to get h by itself. Before we do that, though, let's talk about the value of sin(60°). This is a common trigonometric value that you might have memorized, or you can easily find it using a calculator or a trigonometric table. Knowing this value is crucial for solving the equation and getting the correct answer. So, let's dive into the value of sin(60°) and then get back to solving for h.

The Value of sin(60°)

Before we can solve our equation, we need to know the value of sin(60°). This is one of those trigonometric values that’s super handy to have memorized, but if you don’t, no worries! You can always look it up in a trigonometric table or use a calculator. sin(60°) is equal to √3 / 2 (that's the square root of 3 divided by 2). This value comes from the special 30-60-90 triangle, which is a fundamental concept in trigonometry. In a 30-60-90 triangle, the sides are in a specific ratio, and this ratio helps us determine the sine, cosine, and tangent of these angles. Knowing that sin(60°) = √3 / 2 allows us to plug this value into our equation and get one step closer to finding the height the ladder reaches on the wall. It’s like having a key piece of the puzzle – without it, we can’t complete the picture. Now that we have this value, we can substitute it into our equation and solve for h. So, let’s get back to our equation and see how we can use this information to find our answer!

Solving for the Height

Okay, we're armed with the value of sin(60°), which is √3 / 2. Let's plug that into our equation:

(√3 / 2) = h / 14

Now, to isolate h (the height), we need to get rid of that 14 in the denominator. We can do this by multiplying both sides of the equation by 14. This is a basic algebraic principle – whatever you do to one side of the equation, you have to do to the other to keep it balanced. When we multiply both sides by 14, the equation becomes:

14 * (√3 / 2) = h

Now we can simplify this. 14 divided by 2 is 7, so we have:

7√3 = h

And there you have it! We've solved for h, which represents the height the ladder reaches on the wall. The height is 7√3 meters. This is an exact answer, which is often preferred in math problems. If you need a decimal approximation, you can plug √3 into a calculator (it's approximately 1.732) and multiply it by 7. But for now, let's stick with the exact answer, as it's more precise. So, let's recap what we've done and see how this answer fits into the context of our problem.

Checking the Answer

We've arrived at the answer of 7√3 meters for the height the ladder reaches on the wall. It's always a good idea to take a moment and check if our answer makes sense in the context of the problem. First, let's think about the value of √3. It's approximately 1.732, so 7√3 is roughly 7 * 1.732, which is about 12.12 meters. This height is less than the length of the ladder (14 meters), which is a good sign – the ladder can't reach higher on the wall than its own length! Another way to check our answer is to think about the 60-degree angle. A 60-degree angle is quite steep, so we'd expect the height on the wall to be a significant portion of the ladder's length. Our answer of approximately 12.12 meters seems reasonable in this context. Furthermore, we can consider the properties of a 30-60-90 triangle. In such a triangle, the side opposite the 60-degree angle is √3 / 2 times the hypotenuse. This is exactly what we've calculated! This reinforces our confidence in the correctness of our solution. By taking these steps to check our answer, we can be sure that we've not only solved the problem but also understood the underlying concepts.

Conclusion

So, there you have it, guys! We've successfully navigated through this trigonometry problem and found that the height the 14-meter ladder reaches on the wall is 7√3 meters. We started by understanding the problem and identifying the relevant information: the length of the ladder and the angle it makes with the ground. Then, we strategically chose the sine function (SOH) from our trusty SOH CAH TOA to relate the opposite side (the height) to the hypotenuse (the ladder). We set up the equation, plugged in the value of sin(60°), and solved for the height. Finally, we took a moment to check our answer and make sure it made sense in the context of the problem. This whole process highlights the power of trigonometry in solving real-world problems. Whether it's ladders against walls, the height of a tree, or the distance to a ship at sea, trigonometry provides us with the tools to measure the world around us. And remember, practice makes perfect! The more you work through these types of problems, the more confident you'll become in your trigonometry skills. Keep practicing, and you'll be a math whiz in no time! Now go forth and conquer those triangles!