# 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 ${\mathit {16}}\,$  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 $h$  above the ground is a sinusoidal function of time $t$ , where ${\mathit {t=0\,}}$  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

 Diameter to Radius A $16{\text{ m}}$ diameter circle has a radius of $8{\text{ m}}$ .
 Revolutions per Minute to Degrees per Second A wheel turning at three revolutions per minute is turning $\displaystyle {\frac {3\times 360^{\circ }}{60}}$ per second. Simplifying that's $\displaystyle 18^{\circ }$ per second.
 Formula for height At $t=0$ our height $h$ is $1$ . At $t=10$ , we will have turned through $180^{\circ }=10\times 18^{\circ }$ , i.e. half a circle, and will be at the top most point of height $16+1=17$ (because the diameter of the circle is $16$ meters). A cosine function, i.e. $\displaystyle \cos \theta$ , is $1$ at $\displaystyle \theta =0^{\circ }$ and $-1$ at $\displaystyle \theta =180^{\circ }$ . That's almost exactly opposite to what we want as we want the most negative value at $0$ and the most positive at $180$ . Ergo, let's use the negative cosine to start our function. At $t=10$ we want $\theta =180^{\circ }$ , so we will multiply $t$ by $18$ so that we get $\displaystyle -\cos(18t)$ . The formula we made is $-1$ at $t=0$ and $1$ at $t=10$ . Multiply by $8$ and we get: $\displaystyle -8\cos(18t)$ , which is $-8$ at $t=0$ and $8$ at $t=10$ To get make sure reality is not messed up (we can't have negative height $h$ ), add $9$ and we get $\displaystyle 9-8\cos(18t)$ , which is $1$ at $t=0$ and $17$ at $t=10$ Our required formula is $\displaystyle h=9-8\cos(18t)$ .with the understanding that cosine is of an angle in degrees (not radians).