Horizontal Height in a Circle in terms of Time

Question

Background: a Ferris Wheel has a diameter of 135 feet and a maximum height of 150 feet. Its ride duration is 10 minutes and it makes two complete rotations in this time. Assume that at time t = 0, a rider is at the point farthest to the right, and that the ferris wheel is rotating counterclockwise from the side that you are looking at it.

Question: What is the equation of a rider’s horizontal position x as a function of the time t in minutes. Assume that the x-position of the center of the wheel is 0 feet?

Answer 1

The base function for this would be cosine since you are concerned with the horizontal (x) component of distance.

At t=0, cos(0) =1 so by multiplying by the radius 135/2 that will give the correct distance when the ride begins:

(135/2) cos(t)

The last step is to adjust "t" to reflect the rotational speed of the ferris wheel.

The wheel will rotate 2 turns is 10 minutes, or 1 turn in 5 minutes

2π= 1 turn = 5 minutes.

So 2π per 5 minutes

It follows then that expression (2π/5)t will produce the radians at a given point in time "t"

For example at t=5 min (2π/5)(5)= 2π and at t = 2.5 min (2π/5)(2.5)= π

This can then be added to the function as such...

(135/2) cos( (2π/5)t )