# Trigonometry/Worked Example: Ferris Wheel Problem

## The ProblemEdit

### Exam QuestionEdit

"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.]

 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 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. $\displaystyle \cos \theta$ 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. So let's start with negative cosine as our function. At t=10 we want $\theta=180^\circ$, so we will take $\displaystyle -\cos( 18 t )$. That's -1 at t=0 and +1 at t=10. Multiply by 8 and we get: $\displaystyle -8\cos( 18 t )$. That's -8 at t=0 and +8 at t=10 Add 9 and we get $\displaystyle 9-8\cos( 18 t )$. Which is 1 at t=0 and +17 at t=10 Our required formula is $\displaystyle h = 9 - 8\cos( 18 t )$. with the understanding that cosine is of an angle in degrees (not radians).