Mountain Height: Calculating From Two Points

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Hey guys! Ever wondered how surveyors figure out the height of a mountain without actually climbing it? It's all about using some clever trigonometry! Let's dive into how we can determine the height of a mountain from two different observation points on flat ground, using angles of elevation. This is a classic problem that combines geometry and trigonometry, and it's super useful in real-world applications like surveying and mapping.

Setting Up the Problem

Okay, so here’s the scenario: Imagine we have two observation posts on flat ground, and they are 200 meters apart. From the first post, the angle of elevation to the mountain's peak is 32 degrees. From the second post (which is closer to the mountain), the angle of elevation is 50 degrees. Our mission, should we choose to accept it, is to find the height of the mountain. Sounds like fun, right?

First things first, let's visualize the situation. Draw a horizontal line representing the ground, and mark two points (our observation posts) 200 meters apart. Then, draw a vertical line from the mountain's peak down to the ground, forming a right angle. This vertical line represents the height of the mountain, which is what we want to find. Now, draw lines from each observation post to the peak of the mountain, creating two triangles. We have two right triangles that share a common side (the height of the mountain), and we know the distance between the observation posts and the angles of elevation.

Let's label everything to make it easier to work with. Let's call the height of the mountain h. Let's call the distance from the second observation post (the one closer to the mountain) to the base of the mountain x. Therefore, the distance from the first observation post to the base of the mountain is x + 200. We now have all the information we need to set up some trigonometric equations. Remember, the angle of elevation is the angle between the horizontal line of sight and the line of sight to the top of the mountain. We have two angles of elevation: 32 degrees from the first post and 50 degrees from the second post.

Trigonometric Equations

Now, let's use the tangent function (tan) to relate the angles of elevation to the sides of the triangles. Remember, tan(angle) = opposite / adjacent. In our case, the opposite side is the height of the mountain (h), and the adjacent side is the distance from the observation post to the base of the mountain.

For the first observation post, we have: tan(32°) = h / (x + 200)

For the second observation post, we have: tan(50°) = h / x

We now have two equations with two unknowns (h and x). We can solve this system of equations to find the height of the mountain. There are several ways to solve this system, but one common method is to solve one equation for one variable and then substitute that expression into the other equation.

Solving the Equations

Let's solve the second equation for h: h = x * tan(50°)

Now, substitute this expression for h into the first equation:

tan(32°) = (x * tan(50°)) / (x + 200)

Next, we solve for x. Multiply both sides by (x + 200):

tan(32°) * (x + 200) = x * tan(50°)

Distribute tan(32°) on the left side:

x * tan(32°) + 200 * tan(32°) = x * tan(50°)

Now, isolate x terms on one side:

200 * tan(32°) = x * tan(50°) - x * tan(32°)

Factor out x:

200 * tan(32°) = x * (tan(50°) - tan(32°))

Finally, solve for x:

x = (200 * tan(32°)) / (tan(50°) - tan(32°))

Using a calculator, we find that:

tan(32°) ≈ 0.6249

tan(50°) ≈ 1.1918

So,

x ≈ (200 * 0.6249) / (1.1918 - 0.6249)

x ≈ 124.98 / 0.5669

x ≈ 220.46 meters

Now that we have the value of x, we can find the height h using the equation h = x * tan(50°):

h ≈ 220.46 * 1.1918

h ≈ 262.74 meters

Therefore, the height of the mountain is approximately 262.74 meters. Cool, right? We did it without any climbing!

Alternative Approach: Using the Law of Sines

There's also another cool way to solve this problem using the Law of Sines. This method involves setting up a different set of triangles and using the properties of sine to find the unknown lengths and angles.

First, let's consider the triangle formed by the two observation points and the peak of the mountain. The distance between the observation points is 200 meters. The angles of elevation from these points to the peak are 32° and 50°. We need to find the angle at the peak of this triangle.

Let's call the angle at the peak θ. We know that the sum of the angles in a triangle is 180°. So, we have:

θ = 180° - (180° - 50°) - 32°

θ = 180° - 130° - 32°

θ = 18°

Now we know all three angles in the triangle formed by the observation points and the peak. We can use the Law of Sines to find the distance from the second observation point to the peak (let's call this distance d):

d / sin(32°) = 200 / sin(18°)

d = (200 * sin(32°)) / sin(18°)

Using a calculator, we find that:

sin(32°) ≈ 0.5299

sin(18°) ≈ 0.3090

So,

d ≈ (200 * 0.5299) / 0.3090

d ≈ 105.98 / 0.3090

d ≈ 342.98 meters

Now that we have the distance d from the second observation point to the peak, we can use the sine function in the right triangle formed by the mountain's height, the distance d, and the vertical line from the peak to the ground:

sin(50°) = h / d

h = d * sin(50°)

Using a calculator, we find that:

sin(50°) ≈ 0.7660

So,

h ≈ 342.98 * 0.7660

h ≈ 262.72 meters

Again, we find that the height of the mountain is approximately 262.72 meters, which is very close to our previous result. The small difference is due to rounding errors in the calculations.

Key Takeaways

  • Trigonometry is powerful: By using angles of elevation and basic trigonometric functions (tan, sin), we can determine the height of objects from a distance.
  • Visualizing the problem is crucial: Drawing diagrams and labeling the components helps in setting up the equations correctly.
  • Multiple methods exist: The same problem can often be solved using different approaches, such as using tangent functions or the Law of Sines. Choosing the right method depends on the information available and the desired level of complexity.
  • Real-world applications: This technique is widely used in surveying, navigation, and other fields where measuring heights and distances is essential.

Conclusion

So there you have it! We've successfully calculated the height of a mountain from two observation points using both trigonometric functions and the Law of Sines. This example illustrates the power and versatility of trigonometry in solving real-world problems. Whether you're a surveyor, a mathematician, or just a curious mind, these techniques can come in handy. Keep exploring and happy calculating!