# 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 ${\displaystyle {\mathit {16}}\,}$ 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 ${\displaystyle h}$ above the ground is a sinusoidal function of time ${\displaystyle t}$, where ${\displaystyle {\mathit {t=0\,}}}$ represents the lowest point on the wheel and ${\displaystyle t}$ is measured in seconds."

"Write the equation for ${\displaystyle h}$ in terms of ${\displaystyle t}$."

[For those interested, the picture is actually of a Ferris wheel in Vienna.]

-Lang Gang 2016

 Diameter to Radius A ${\displaystyle 16{\text{ m}}}$ diameter circle has a radius of ${\displaystyle 8{\text{ m}}}$.
 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 ${\displaystyle t=0}$ our height ${\displaystyle h}$ is ${\displaystyle 1}$. At ${\displaystyle t=10}$, we will have turned through ${\displaystyle 180^{\circ }=10\times 18^{\circ }}$, i.e. half a circle, and will be at the top most point of height ${\displaystyle 16+1=17}$ (because the diameter of the circle is ${\displaystyle 16}$ meters). A cosine function, i.e. ${\displaystyle \displaystyle \cos \theta }$, is ${\displaystyle 1}$ at ${\displaystyle \displaystyle \theta =0^{\circ }}$ and ${\displaystyle -1}$ at ${\displaystyle \displaystyle \theta =180^{\circ }}$. That's almost exactly opposite to what we want as we want the most negative value at ${\displaystyle 0}$ and the most positive at ${\displaystyle 180}$. Ergo, let's use the negative cosine to start our function. At ${\displaystyle t=10}$ we want ${\displaystyle \theta =180^{\circ }}$, so we will multiply ${\displaystyle t}$ by ${\displaystyle 18}$ so that we get ${\displaystyle \displaystyle -\cos(18t)}$. The formula we made is ${\displaystyle -1}$ at ${\displaystyle t=0}$ and ${\displaystyle 1}$ at ${\displaystyle t=10}$. Multiply by ${\displaystyle 8}$ and we get: ${\displaystyle \displaystyle -8\cos(18t)}$, which is ${\displaystyle -8}$ at ${\displaystyle t=0}$ and ${\displaystyle 8}$ at ${\displaystyle t=10}$ To get make sure reality is not messed up (we can't have negative height ${\displaystyle h}$), add ${\displaystyle 9}$ and we get ${\displaystyle \displaystyle 9-8\cos(18t)}$, which is ${\displaystyle 1}$ at ${\displaystyle t=0}$ and ${\displaystyle 17}$ at ${\displaystyle t=10}$ Our required formula is ${\displaystyle \displaystyle h=9-8\cos(18t)}$. with the understanding that cosine is of an angle in degrees (not radians).