Understanding Trigonometry Through a Real-World Beam Width Scenario

At what distance down the road is a beam with an 11-degree angle more than 38 feet wide?

• A. 100 feet
• B. 200 feet
• C. 300 feet
• D. 400 feet

Answer: (B)

To determine at what distance down the road a beam with an 11-degree angle is more than 38 feet wide, it’s essential to use basic trigonometry. The width of the beam can be conceptualized as the horizontal distance across the beam from one edge to the other, depending on the distance from the source of the beam. The formula to calculate the width of the beam at a given distance (d) from the source, given an angle (θ), is: Width = d × tan(θ). Thus, when we set the width equal to 38 feet, we can express the equation as follows: 38 feet = d × tan(11 degrees). To find d, we rearrange the equation: d = 38 feet / tan(11 degrees). By calculating the value of tan(11 degrees) and plugging it into the equation, we can determine the distance at which the beam reaches more than 38 feet in width. Calculating this gives: tan(11 degrees) ≈ 0.1944 (approximately), So, d ≈ 38 feet / 0.1944 ≈ 195.5 feet. Rounding this value, we conclude that at a distance of about 200 feet

Have you ever found yourself scratching your head over a math problem that just seems a bit too abstract? You know the kind—those equations that feel worlds away from practical application. Well, hang tight! Let’s take a closer look at a real-world scenario where trigonometry becomes not just relevant, but downright necessary.

Imagine you're trying to figure out how wide a beam is at a certain distance given a specific angle. In this case, we’re dealing with a beam that forms an 11-degree angle. Our goal? To find out how far down the road this beam needs to be positioned for it to measure more than 38 feet wide. Sounds complicated? It’s simpler than it looks, and I promise, you’ll feel like a math wizard by the end of this!

Quick Trim into the Math

So, here’s the scoop: the width of that beam can be determined using a little trigonometry. When we set it up, we need to look at the relationship between the distance (d) to the source of the beam and the angle (θ) at which it spreads out. Luckily, there’s a nifty formula for that:

Width = d × tan(θ)

This formula might look just like a string of letters and symbols, but in reality, it’s your golden ticket to solving the problem. Essentially, we want to know where the width of our beam reaches that magic 38-foot marker.

Calculating the Distance

Let’s plug in what we know: we want the width to be 38 feet, so our equation reformulates into:

38 feet = d × tan(11 degrees)

Now, to find our distance (d), we just have to rearrange the equation:

d = 38 feet / tan(11 degrees)

Pretty simple, right? To actually find the distance, we need to calculate tan(11 degrees). Trust me, this isn’t as daunting as it sounds. With some reliable tools, calculators, or software, you’d discover that:

tan(11 degrees) ≈ 0.1944

Now, here’s the fun part: plug this value back into our earlier formula:

d ≈ 38 feet / 0.1944 ≈ 195.5 feet.

At this point, it’s clear where we’re headed. Rounding this value tells us that to achieve a beam width surpassing 38 feet, we need to position ourselves roughly at 200 feet down the road.

Why Does This Matter?

You might be thinking, “What’s the big deal about calculating the width of a beam?” Well, in fields like architecture, engineering, or any craft that deals with measurements, these calculations play a crucial role. Understanding how angles and distances interact can lead to better designs, safer structures, and more efficient use of materials. Who wouldn’t want a wider beam when you can just step back a bit?

Wrap-Up: Trigonometry is Everywhere!

So there you have it! Instead of running away from math, think of it as a toolbox filled with instruments you can use to make sense of your world. Whether you’re prepping for a state exam or just looking to expand your skill set, trigonometry is like a new song you’re learning to play. Stick with it, and you might just find a sweet rhythm.

Next time you're faced with a tricky math problem, channel that 11-degree angle mindset, and tackle it head-on. And remember, the North Carolina RADAR exam is your opportunity to shine—so let your knowledge beam bright!