Solving Trigonometry Problem Building Height And Distance

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Hey guys! Let's dive into a classic trigonometry problem that involves angles of elevation and distances. This is a common type of problem you might encounter in math class, physics, or even real-world scenarios like surveying or construction. We're going to break it down step-by-step, so it's super easy to understand.

Problem Statement

Imagine this: A person is standing some distance away from a building and looks up at the top edge of the building's cornice. They observe the top of the cornice at an angle of elevation of 30 degrees. Then, this person walks 25 meters in a straight line towards the building's entrance. After walking closer, they look up again at the cornice and now observe it at an angle of elevation of 45 degrees. Our goal is to figure out the height of the building (specifically, the height of the cornice) and the initial distance the person was standing from the building.

Visualizing the Problem

Before we jump into the math, it's always helpful to visualize what's going on. Think of it like this:

  1. The Building: We have a vertical line representing the building, and the cornice is the top point we're interested in.
  2. The Ground: The ground is a horizontal line extending from the base of the building.
  3. The Person's Initial Position: The person starts at a certain distance from the building, creating a right triangle with the building's height and the ground.
  4. Angle of Elevation (30 degrees): This is the angle between the ground and the line of sight from the person's initial position to the top of the cornice.
  5. The Person's New Position: The person moves 25 meters closer to the building, creating a new, smaller right triangle.
  6. Angle of Elevation (45 degrees): The angle between the ground and the line of sight from the person's new position to the top of the cornice is now 45 degrees.

Drawing a diagram is super helpful here! You'll have two right triangles sharing a common vertical side (the building's height). This shared side is key to solving the problem.

Setting Up the Equations

Okay, now for the fun part – the math! We'll use trigonometry, specifically the tangent function, because it relates the opposite side (the building's height) to the adjacent side (the distance from the person to the building).

Let's define our variables:

  • h = the height of the building (what we want to find)
  • x = the initial distance the person was standing from the building

Now, we can set up two equations based on the two right triangles:

  • Triangle 1 (initial position): tan(30°) = h / x
  • Triangle 2 (new position): tan(45°) = h / (x - 25)

Why x - 25 in the second equation? Because the person moved 25 meters closer to the building, so their new distance is the initial distance x minus 25 meters.

Solving the Equations

We have two equations and two unknowns (h and x), which means we can solve for them! There are a couple of ways to do this. One common method is to use substitution.

  1. Solve the first equation for h: h = x * tan(30°)

    Remember that tan(30°) is a known value (it's approximately 1/√3 or 0.577). We'll keep it as tan(30°) for now to keep things neat.

  2. Substitute this expression for h into the second equation: tan(45°) = (x * tan(30°)) / (x - 25)

    Now we have one equation with one unknown (x).

  3. Solve for x: First, remember that tan(45°) = 1. So our equation becomes: 1 = (x * tan(30°)) / (x - 25)

    Multiply both sides by (x - 25): x - 25 = x * tan(30°)

    Rearrange the equation to get all the x terms on one side: x - x * tan(30°) = 25

    Factor out x: x * (1 - tan(30°)) = 25

    Finally, divide both sides to solve for x: x = 25 / (1 - tan(30°))

    Now we can plug in the value of tan(30°) (approximately 0.577): x = 25 / (1 - 0.577) x ≈ 25 / 0.423 x ≈ 59.1 meters

    So, the person's initial distance from the building was approximately 59.1 meters.

  4. Solve for h: Now that we know x, we can plug it back into our equation h = x * tan(30°): h = 59.1 * tan(30°) h ≈ 59.1 * 0.577 h ≈ 34.1 meters

    Therefore, the height of the building (the cornice) is approximately 34.1 meters.

The Answers

Alright, we've solved it! Here are our answers:

  • Height of the building (cornice): Approximately 34.1 meters
  • Initial distance from the building: Approximately 59.1 meters

Key Concepts and Takeaways

This problem highlights some important concepts in trigonometry:

  • Angles of Elevation: Understanding what an angle of elevation is and how it relates to the sides of a right triangle.
  • Trigonometric Ratios (Tangent): Knowing the definitions of sine, cosine, and tangent, and when to use each one. In this case, the tangent was perfect because we had the opposite and adjacent sides.
  • Solving Systems of Equations: Being able to set up and solve multiple equations with multiple unknowns is a crucial skill in math and science.
  • Visualizing the Problem: Drawing a diagram makes a HUGE difference! It helps you see the relationships between the different parts of the problem.

Real-World Applications

Trigonometry isn't just something you learn in a classroom; it has tons of real-world applications. Surveyors use it to measure distances and heights, architects use it to design buildings, and even video game developers use it to create realistic 3D environments.

Imagine you're planning to build a ramp for skateboarding. You need to know the angle of the ramp and the length of the ramp to make it safe and fun. Trigonometry to the rescue!

Practice Makes Perfect

The best way to get comfortable with trigonometry is to practice! Try solving similar problems with different angles and distances. You can also look for real-world examples where you can apply these concepts.

If you get stuck, don't be afraid to ask for help! Your teacher, classmates, or even online resources like Khan Academy can be great resources.

So there you have it! We've tackled a trigonometry problem involving angles of elevation, distances, and a building. Hopefully, this explanation has made the concepts clear and you feel confident tackling similar problems in the future. Keep practicing, and you'll be a trig whiz in no time!