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

Video Links

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

• Ferris Wheel Trig Problem
• Ferris Wheel Trig Problem (part 2)

Solution

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).