The height of a certain gondola on a Ferris Wheel as it rotates can be modeled using the cosine function. The Skyview Ferris Wheel in Atlanta is 200 feet tall at its highest point and has 42 gondolas. Each ride lasts 15 minutes and makes 4 full rotations. If you are 4 feet from the ground when you enter the gondola, which of the following cosine functions would represent your height above the ground? Express the model as a function of time, t, in minutes.
Let h(t) represent your height above the ground at time t minutes.
Given that the gondola is 200 feet tall at its highest point, we can use the cosine function to model the height at any given time. The general form of a cosine function is:
h(t) = A*cos(Bt) + C
where:
A = amplitude (half the height of the ferris wheel = 100 feet)
B = 2π / period (for one full rotation, the period is 15 minutes)
C = vertical shift (initial height when t = 0)
Since the ride makes 4 full rotations, the period is 15 minutes for one rotation. So, the complete period is 4*15 = 60 minutes.
Therefore, the function representing your height above the ground would be:
h(t) = 100*cos(2π/60*t) + 104
Simplified, this becomes:
h(t) = 100*cos(π/30*t) + 104