Question

A Ferris wheel is 520 feet in diameter & reaches a height of 550 feet. One complete revolution in 30 minutes. Passengers get on & off at the bottom, the lowest point. What is the cosine function that models the riders position & time since boarding?

Answer 1

Given that a Ferris wheel is 520 feet in diameter & it reaches a maximum height of 550 feet. And time required for one complete revolution is 30 minutes=1800s.

So any seat at the periphery of the wheel undergoes a periodic motion of amplitude equal to the radius of the wheel i.e.`"Amplitude (A)"=520/2=260` ft.

The minimum height of any seat at its lowest position at time `t=0` is `550-520=30` ft.The distance of any seat at its lowest position from center of the wheel is `260` ft and the heigt of its center is `260+30=290` ft.
As one complete revolution of the wheel requires 30 minutes. We can say the period of circular motion of the wheel is `T=1800` sec. Hence angular velocity of the wheel will be
`omega=(2pi)/T=(2pi)/1800=pi/900`rad/s.

So the height of any seat at time `t` which starts at `t=0` from its lowest position can be written as a function of `t` as follows.

`h(t)=290-A*cos(omega*t)`

`color(magenta)(=>h(t)=290-260*cos((pit)/900))`