# Constructing sine waves from a ferris wheel? A Ferris wheel 50 feet in diameter makes one revolution every 40 seconds. The center of the wheel is 30 feet above the ground. Find a sine function that describes the height of the Ferris wheel over time.

The ferris wheel has a diameter of 50 feet, so a radius of 25 feet. If the center is 30 feet from the ground, the bottom of the wheel is 5 feet from the ground and the maximum height is 55 feet.

The midline is at 30 feet.

The period is `(1"rev")/(40"sec")` or `(3"rev")/(2"min")` which is `(2/3"rev")/"min"` .

Assuming that the ride starts at t=0, the car should be at the bottom of the ride or 5 feet above the ground at t=0. This means there is a horizontal shift (phase shift) left `pi/2` units.

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The general formula is `y=asin(b(t-h))+k` where a is the amplitude, the period is `(2pi)/b` , h is the phase shift, and k the vertical translation.

a=25 (The most the function moves from the midline)

k=30 (the midline)

`b=(2pi)/(2/3)=3pi` where t is measured in minutes

The phase shift is `pi/2` units left. To get this, we have `3pih=pi/2==>h=1/6`

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One function is `25sin(3pi(x-1/6))+30`

The graph:

Note that the ride starts at the minimum at t=0 and completes 3 revolutions in 2 minutes or 1 revolution every 40 seconds.(Time is measured in minutes and is graphed along the horizontal axis -- the vertical axis is height in feet.)

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