Let's solve the problem step by step.

Part a: Finding the Amplitude, Midline, and Period of $h(t)$

Given:

• The diameter of the Ferris wheel is 21 meters.
• The platform is 1 meter above the ground.
• The Ferris wheel completes one full revolution in 8 minutes.

1. Amplitude:

The amplitude $A$ is half the height of the Ferris wheel because it represents how much the height varies from the midline. Since the diameter of the Ferris wheel is 21 meters, the radius (which is the amplitude) is: $A = \frac{21}{2} = 10.5 \text{ meters}$

2. Midline:

The midline is the average height of the person above the ground as they ride the Ferris wheel. Since the platform is 1 meter above the ground, and the radius of the Ferris wheel is 10.5 meters, the midline is the platform height plus the radius: $\text{Midline} = 1 + 10.5 = 11.5 \text{ meters}$

3. Period:

The period $P$ is the time it takes for one complete revolution, which is given as 8 minutes: $P = 8 \text{ minutes}$

So, the answers are:

• Amplitude $A = 10.5$ meters
• Midline $h = 11.5$ meters
• Period $P = 8$ minutes

Part b: Finding the Formula for the Height Function $h(t)$

We want to find a formula for the height $h(t)$, where $t$ is the time in minutes. The height function for a Ferris wheel is typically sinusoidal (either a sine or cosine function). Since the person starts at the lowest point of the Ferris wheel (at the platform), we will use a cosine function, but phase-shifted accordingly to model starting at the bottom.

The general formula for a sinusoidal function is: $h(t) = A \cos\left(\frac{2\pi}{P} t\right) + \text{Midline}$ Where:

• $A = 10.5$ meters (amplitude),
• $P = 8$ minutes (period),
• Midline = 11.5 meters.

We use cosine because at $t = 0$, the person is at the lowest point. This means we need to shift the cosine function downward. The height function is: $h(t) = 10.5 \cos\left(\frac{2\pi}{8} t - \pi\right) + 11.5$

This represents the height of a person at time $t$.

Summary of Answers:

a.

• Amplitude $A = 10.5$ meters
• Midline $h = 11.5$ meters
• Period $P = 8$ minutes

b. The height function is: $h(t) = 10.5 \cos\left(\frac{\pi}{4} t - \pi\right) + 11.5$

Would you like more details on any part of this? Here are some related questions:

1. What would the formula look like if the Ferris wheel rotated faster or slower?
2. How does the choice of sine versus cosine affect the height function?
3. How would the amplitude change if the Ferris wheel's diameter increased?
4. What if the platform were higher or lower than 1 meter above the ground?
5. How does the phase shift impact the starting point of the function?

Tip: In a sinusoidal function, the phase shift determines the starting position of the curve. For Ferris wheels, whether you start from the lowest or highest point affects whether you use sine or cosine.