Question

A Ferris wheel has a radius of 10 meters and is revolving 6 times each minute (wheel's frequency.) The wheel's center is 12 meters from the ground. If time starts on the ground, then find a sine function that shows the height from the ground for a car on the Ferris wheel at any time.

Answer 1

6 rotations take 1 minutes
1 rotation takes 1/6 minutes
2π/k = 1/6
k = 12π
let's start using only amplitude and frequency to get

height = 10 sin ( 12πt)
We want to move the curve up 12 units

height = 10 sin (12πt) + 12

so now we have to have a horizontal shift so that when t = 0, the height will be 2
2 = 10 sin 12π(0 + d) + 12
-1 = sin 12πd
now we know that sin 3π/2 = -1
then 12πd = 3π/2
d = 1/8

so one version of our graph would be
height = 10 sin 12π(t + 1/8) + 12

testing: if t = 1/12, we should be at our highest point of 22

if t = 1/24
height = 10 sin 12π(1/12+1/8) + 12
=10 sin 12π(5/24) + 12
= 10 sin 5π/2 + 12
= 10(1) + 12
= 22

confirmation here:
http://www.wolframalpha.com/input/?i=y+%3D+10+sin+%2812π%28t+%2B+1%2F8%29%29+%2B+12++for+0+%3C+t+%3C+.166666