Trigonometry/Worked Example: Ferris Wheel Problem

The Problem 400px|left Exam Question "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 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 diameter circle has a radius of . Revolutions per Minute to Degrees per Second A wheel turning at three revolutions per minute is turning per second. Simplifying that's per second. Formula for height 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: , which is at and at To get make sure reality is not messed up (we can't have negative height ), add and we get , which is at and at Our required formula is . with the understanding that cosine is of an angle in degrees (not radians).