Trigonometry/Worked Example: Ferris Wheel Problem

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The Problem[edit]

Wiener Riesenrad DSC02378.JPG

Exam Question[edit]

"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[edit]

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

Solution[edit]

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).