Trigonometry/Worked Example: Ferris Wheel Problem
"Jacob and Emily ride a Ferris wheel at a carnival in Billings, Montana. The wheel has a 16 meter diameter, and turns at three revolutions per minute, with its lowest point one meter above the ground. Assume that Jacob and Emily's height h above the ground is a sinusoidal function of time t, where t=0 represents the lowest point on the wheel and t is measured in seconds."
"Write the equation for h in terms of t."
[For those interested the picture is actually of a Ferris wheel in Vienna. We couldn't find one of a Ferris wheel in Billings Montana.]
The Khan Academy has video material that walks through this problem, which you may find easier to follow:
A 16m diameter circle has a radius of 8m.
A wheel turning at three revolutions per minute is turning
per second. Simplifying that's
At t=0 our height h is 1. At t =10 we will have turned through 180o, i.e. half a circle and will be at the top most point which has height 16 + 1= 17.
A cosine function, i.e. is 1 at and -1 at . That's almost exactly opposite to what we want as we want the most negative value at 0 and the most positive at 180. So let's start with negative cosine as our function.
At t=10 we want , so we will take . That's -1 at t=0 and +1 at t=10. Multiply by 8 and we get:
Add 9 and we get
Our required formula is
with the understanding that cosine is of an angle in degrees (not radians).