The diameter of the wheel is 165 ft, it rotates at 1.5 revolutions per minute, and the bottom of the wheel is 9 ft above the ground. Find an equation that gives a passenger's height above the ground at any time t during the ride. Assume the passenger starts the ride at the bottom of the wheel.

Answer 1

1.5 revs -- 60 sec
1 rev --- 40 seconds

So the period is 40 seconds

2π/k = 40
k = π/20

amplitude = 165/2 = 82.5

So let's start with
height = 82.5 sin (π/20)t, where t is in seconds
But that would give us a min of 82.5, and we want to have a min of 9
so height = 82.5 sin (π/20)t + 91.5

But we know that for t = 0 , we should get 9
so we have to apply a phase shift

let height = 82.5 sin (π/20)(t + c) + 91.5
when t = 0, height = 9
9 = 82.5 sin(π/20)(t+c) + 91.5
-1 = sin (π/20)(c)
we know sin 3π/2) = -1
so (π/20)(c) = 3π/2
c/20 = 3/2
c = 30

height = 82.5 sin (π/20)(t+30) + 91.5

checking:
at t = 0 we should get height = 9 , we do!
at t = 10 we should get 91.5 , we do!
at t= 20 , we should get a max of 174 , we do!
at t = 30, we should get 91.5 again, we do!
at t = 40 we should be back to 9, we do!

This is just one such equation, we could have used a cosine function in a similar method.