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.
A)h(t)=-98 cos (2pi/7.5t)+102
B)h(t)=-98 cos (2pi/3.75t)+102
C)h(t)=98 cos (2pi/3.75t)+100
D)h(t)=-100 cos (2pi/15t)+104
To determine which of the following cosine functions would represent your height above the ground, we first need to identify the key components of the cosine function that relate to the given information:
1. The amplitude of the cosine function should be 98. The gondola's height variation ranges from 102 feet to 4 feet above the ground.
2. The period of the cosine function should be \(15\t) minutes. This corresponds to the duration of one full rotation of the Ferris Wheel.
3. The vertical shift of the cosine function should be 100. This accounts for the initial height of 4 feet when entering the gondola.
Therefore, the cosine function that represents your height above the ground would be:
C) \(h(t) = 98 \cos\left(\frac{2\pi}{3.75}t\right) + 100\)
This model satisfies the conditions mentioned above and accurately represents your height above the ground in the gondola on the Ferris Wheel.