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.
The general form of a cosine function is given by:
\[f(t) = A \cos(B(t-C)) + D\]
Where:
- A is the amplitude of the function, which is the maximum distance from the mean value
- B is the frequency, which determines how many cycles are completed in a given interval
- C is the phase shift, which determines the horizontal shift of the graph
- D is the vertical shift, which determines the mean value
Given that the Ferris Wheel has a maximum height of 200 feet, the amplitude would be 100 feet. Since the ride lasts 15 minutes with 4 full rotations, the frequency would be $\frac{4}{15}$ cycles per minute. The phase shift can be considered as 0, and the mean value would be 104 feet (100 feet above the ground plus the initial height of 4 feet before entering the gondola).
Therefore, the cosine function that represents your height above the ground would be:
\[f(t) = 100 \cos\left(\frac{4\pi}{15}t\right) + 104\]