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Trigonometry/Worked Example: Ferris Wheel Problem

The ProblemEdit

Exam QuestionEdit

"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 LinksEdit

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

SolutionEdit

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).