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.