Roller Coaster Hill Height Formula | Triangulation Problem Solution

[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

 

[Dick]

That's an angle-side-angle problem in, you guessed it, trigonometry. Get started. No one will help you unless you try to start.

 

[Architeuthis Dux]

To calculate the height of the top of the hill on a roller coaster using triangulation, we can use the trigonometric functions of sine and tangent. We can start by drawing a right triangle, with the top of the hill being the vertex of the right angle. The adjacent side of the right angle will be the distance between the two points, d. The opposite side will be the height of the hill, which we can call h.

Next, we can label the two angles given in the problem as angle 1 and angle 2. Angle 1 will be the angle formed between the ground and the line connecting the closer point to the top of the hill. Angle 2 will be the angle formed between the ground and the line connecting the further point to the top of the hill.

Using the definition of sine, we can set up the equation sin(angle 1) = h/d. Solving for h, we get h = d*sin(angle 1). Similarly, using the definition of tangent, we can set up the equation tan(angle 2) = h/d. Solving for h, we get h = d*tan(angle 2).

Since both equations equal h, we can set them equal to each other and solve for d. This gives us the formula d = h/(sin(angle 1) - tan(angle 2)). We can then rearrange the formula to solve for h, giving us the final formula h = d*(sin(angle 1) - tan(angle 2)). This formula gives us the height of the top of the hill on a roller coaster in terms of the given variables, d, angle 1, and angle 2.