Trigonometry/Worked Example: Ferris Wheel Problem
"Jacob and Emily ride a Ferris wheel at a carnival in Vienna. The wheel has a 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 above the ground is a sinusoidal function of time , where represents the lowest point on the wheel and is measured in seconds."
"Write the equation for in terms of ."
[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:
A diameter circle has a radius of .
A wheel turning at three revolutions per minute is turning
per second. Simplifying that's
At our height is . At , we will have turned through , i.e. half a circle, and will be at the top most point of height (because the diameter of the circle is meters).
A cosine function, i.e. , is at and at . That's almost exactly opposite to what we want as we want the most negative value at and the most positive at . Ergo, let's use the negative cosine to start our function.
At we want , so we will multiply by so that we get . The formula we made is at and at . Multiply by and we get:
To get make sure reality is not messed up (we can't have negative height ), add and we get
Our required formula is
with the understanding that cosine is of an angle in degrees (not radians).