# Trigonometry/Worked Example: Ferris Wheel Problem

## The Problem

### Exam Question

"Jacob and Emily ride a Ferris wheel at a carnival in Vienna. 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.]

-Lang Gang 2016

The Khan Academy has video material that walks through this problem, which you may find easier to follow:

## Solution

 Diameter to Radius A 16m diameter circle has a radius of 8m.
 Revolutions per Minute to Degrees per Second A wheel turning at three revolutions per minute is turning ${\displaystyle \displaystyle {\frac {3\times 360^{\circ }}{60}}}$ per second. Simplifying that's ${\displaystyle \displaystyle 18^{\circ }}$ per second.
 Formula for height 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. ${\displaystyle \displaystyle \cos \theta }$ is 1 at ${\displaystyle \displaystyle \theta =0^{\circ }}$ and -1 at ${\displaystyle \displaystyle \theta =180^{\circ }}$. 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 ${\displaystyle \theta =180^{\circ }}$, so we will take ${\displaystyle \displaystyle -\cos(18t)}$. That's -1 at t=0 and +1 at t=10. Multiply by 8 and we get: ${\displaystyle \displaystyle -8\cos(18t)}$. That's -8 at t=0 and +8 at t=10 Add 9 and we get ${\displaystyle \displaystyle 9-8\cos(18t)}$. Which is 1 at t=0 and +17 at t=10 Our required formula is ${\displaystyle \displaystyle h=9-8\cos(18t)}$. with the understanding that cosine is of an angle in degrees (not radians).