Roller coaster triangulation problem

[drobtj2]

Homework Statement

Derive a formula to calculate the height of the top of a hill on a roller coaster using triangulation. Final answer should be in terms of d, angle 1, and angle 2. There are two points on the ground. Angle 1 is the angle formed between the ground and a line that goes from the point closer to the hill to the top of the hill. Angle 2 is the angle formed between the ground and a line that goes from the point further away to the top. d is the distance between the two points.

Homework Equations

(trigonomotry)

The Attempt at a Solution

 

[robertm]

Just because you post the same question in two different forums doesn't mean anyone will help without at lest some effort on your part.

You are being asked to derive Height h using angles 1 and 2, and Dis. d. So how do you proceed?

What are the relationships between a right triangles side lengths and its angles? Try and discover the fundamental relationships of triangles and then derive a formula. Or simply read a little of your textbook/internet and find some applicable formula then test them with some arbitrary values.

Try anything really, and once you have then EXPLAIN fully your difficulties and what you do and do not understand. Heres a picture to help with the visualization:

 

[Moetasim]

To derive a formula for calculating the height of the top of a hill on a roller coaster using triangulation, we can use basic trigonometry principles. First, we need to define the variables given in the problem. Angle 1 and Angle 2 are both angles formed between the ground and a line going to the top of the hill from two different points on the ground. We can label these points as A and B, with A being the closer point and B being the further point. d represents the distance between points A and B.

Using trigonometric ratios, we can set up the following equations:

tan(angle 1) = height/d

tan(angle 2) = height/(d + x)

Where x is the distance from point B to the top of the hill.

We can then rearrange these equations to solve for the height:

height = d * tan(angle 1)

height = (d + x) * tan(angle 2)

Since both equations equal the height, we can set them equal to each other and solve for x:

d * tan(angle 1) = (d + x) * tan(angle 2)

d * tan(angle 1) = d * tan(angle 2) + x * tan(angle 2)

x * tan(angle 2) = d * (tan(angle 1) - tan(angle 2))

x = d * (tan(angle 1) - tan(angle 2)) / tan(angle 2)

Now that we have solved for x, we can substitute it back into one of the original equations to solve for the height:

height = (d + x) * tan(angle 2)

height = (d + d * (tan(angle 1) - tan(angle 2)) / tan(angle 2)) * tan(angle 2)

height = d * (1 + (tan(angle 1) - tan(angle 2)))

Simplifying further, we get the final formula for calculating the height of the top of a hill on a roller coaster using triangulation:

height = d * (1 + tan(angle 1) - tan(angle 2))

This formula can be used to calculate the height of the top of a hill on a roller coaster given the distance between two points on the ground and the angles formed between those points and the top of the hill.