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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.

A Ferris wheel is 21 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 8 minutes. The function h(t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn. Find the amplitude, midline, and period of h(t). Assume that a person has just boarded the Ferris wheel from the platform and that the Ferris wheel starts spinning at time t = 0. Find a formula for the height function h(t).

Learn how to determine the amplitude, midline, and period of a sinusoidal function describing a Ferris wheel's height. This problem covers trigonometric concepts suitable for high school students. It also provides the formula for the height function when a person boards the Ferris wheel.

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