A ferris wheel is 45 meters in diameter and boarded from a platform that is 3 meters above the ground.

Question

The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 8 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. Write an equation for h = f(t).

Answer 1

Typically, when I am modeling motion with a periodic function, I choose the easiest one to use. In this case, a cosine function is easier because it requires no phase shift (aka horizontal shift). But you would use a negative cosine function since it starts at the bottom. So:

f(t) = -22.5cos[((2π)/8)t] + 25.5 or, simplifying:

f(t) = -22.5cos[(π/4)t] + 25.5

Answer 2

The radius is 45 meters divided by 2, or 22.5 meters.

That is the amplitude of the sine wave. It is a sine wave because we are modeling the height of a wave rather than the horIzontal distance from the center of the wave (the cosine function).

The vertical offset is the radius plus the height above ground of the ferris wheel: 22.5 meters plus 3 meters, or 25.5 meters.

To convert the speed into radians divide 2*pi by 8 minutes, and we get pi/4 cycles per minute.

The cycle starts at the 6 o'clock position, which is 1/4 cycle behind the normal start of a sine wave. Moving the wave to the right this much involves subtracting 1/4 of 2pi, or pi/2.

Putting all these pieces together,

f(t) = 22.5*[sin(((pi*t)/4)-(pi/2))] + 25.5

Amplitude*(sin(cycle time minus cycle offset)) plus center height (above ground)