Before being replaced in 2016, a large Ferris wheel on Chicago, Illinois’ Navy Pier had a diameter of 140 feet and the riders boarded from a platform 10 feet above the ground. The Ferris wheel completed one rotation every 7 minutes.

Express a rider’s height, H, in feet above the ground as a function of time, t, in minutes.

To express a rider's height, \( H \), in feet above the ground as a function of time, \( t \), in minutes, for the Ferris wheel on Chicago's Navy Pier, we can use the properties of trigonometric functions. Here are the steps to construct this function:

1. **Determine the basic components:**
   - The Ferris wheel has a diameter of 140 feet, which means the radius, \( r \), is 70 feet.
   - The platform from which riders board is 10 feet above the ground.

2. **Vertical movement due to the wheel rotation:**
   The vertical displacement of a rider from the center of the Ferris wheel can be modeled using a sine or cosine function. We will use the cosine function here for simplicity.

3. **Rotation and Period:**
   - The Ferris wheel completes one rotation every 7 minutes.
   - The period, \( T \), of the cosine function that models the height is 7 minutes.
   - The angular frequency, \( \omega \), is given by \( \omega = \frac{2\pi}{T} = \frac{2\pi}{7} \).

4. **Height Function:**
   The cosine function typically starts at its maximum value when \( t = 0 \). We want the function to start at the bottom of the Ferris wheel (the boarding platform), which is 10 feet above the ground. Therefore, we use:
   \[
   H(t) = 70 \cos\left(\frac{2\pi}{7}t\right) + 80
   \]
   Here, 70 is the amplitude (radius of the wheel), and 80 is the vertical shift that accounts for the radius plus the boarding platform height (70 + 10).

Thus, the height \( H \) of a rider above the ground as a function of time \( t \) in minutes is:
\[
H(t) = 70 \cos\left(\frac{2\pi}{7}t\right) + 80
\]

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How does changing the period affect the height function?
2. What is the effect of using a sine function instead of a cosine function in this context?
3. How to determine the maximum and minimum heights of the riders?
4. How does the function \( H(t) \) change if the platform height is different?
5. What are the implications of the angular frequency on the motion of the Ferris wheel?

**Tip:** When modeling periodic motion such as that of a Ferris wheel, always consider the amplitude, period, phase shift, and vertical shift to accurately represent the motion.

Learn how to model a rider's height on Chicago's Navy Pier Ferris wheel using trigonometric functions. This problem involves trigonometry, periodic functions, and modeling real-world motion. Suitable for high school students.

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