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The problem is about modeling a Ferris wheel's height using a trigonometric function based on the given parameters. Here's a step-by-step guide to help you find the answers:

### Key Information:
- **Diameter of the Ferris wheel**: 30 meters
- **Radius of the Ferris wheel**: \( \frac{30}{2} = 15 \) meters
- **Height of the platform**: 3 meters above the ground
- **Period of one revolution**: 12 minutes
- **Position at \( t = 0 \)**: The person starts at the lowest point (6 o'clock position), which is level with the platform.

We will model the height function \( h(t) \), where \( t \) is in minutes and \( h(t) \) represents the height of the person above the ground.

### (a) Period of \( h(t) \)
The period is the time it takes for the Ferris wheel to complete one full revolution. From the problem, we know this is **12 minutes**.

### (b) Amplitude of \( h(t) \)
The amplitude is half the total vertical distance traveled by the person, which corresponds to the radius of the Ferris wheel. So, the amplitude is:
\[
\text{Amplitude} = 15 \text{ meters}
\]

### (c) Vertical shift of \( h(t) \)
The vertical shift corresponds to the average height above the ground, which is the distance from the ground to the center of the Ferris wheel. Since the radius is 15 meters and the platform is 3 meters above the ground, the vertical shift is:
\[
\text{Vertical shift} = 15 + 3 = 18 \text{ meters}
\]

### (d) Formula for \( h(t) \)
Given the amplitude, period, and vertical shift, the height function \( h(t) \) can be modeled by a cosine function. The general formula for the height is:
\[
h(t) = A \cos \left( \frac{2\pi}{T} t \right) + D
\]
Where:
- \( A \) is the amplitude
- \( T \) is the period
- \( D \) is the vertical shift

Using the values:
- Amplitude \( A = 15 \)
- Period \( T = 12 \) minutes
- Vertical shift \( D = 18 \)

Since the person starts at the lowest point, we use a cosine function but shifted by \( \pi \) radians (so that \( h(0) \) gives the lowest point):
\[
h(t) = 15 \cos \left( \frac{2\pi}{12} t - \pi \right) + 18
\]
Simplifying:
\[
h(t) = 15 \cos \left( \frac{\pi}{6} t - \pi \right) + 18
\]

### (e) How high off the ground is the person after 6 minutes?
To find the height after 6 minutes, substitute \( t = 6 \) into the formula for \( h(t) \):
\[
h(6) = 15 \cos \left( \frac{\pi}{6} \times 6 - \pi \right) + 18
\]
Simplifying:
\[
h(6) = 15 \cos(\pi - \pi) + 18 = 15 \cos(0) + 18 = 15(1) + 18 = 33 \text{ meters}
\]

### Final Answers:
1. **Period of \( h(t) \)**: 12 minutes
2. **Amplitude of \( h(t) \)**: 15 meters
3. **Vertical shift of \( h(t) \)**: 18 meters
4. **Formula for \( h(t) \)**: \( h(t) = 15 \cos \left( \frac{\pi}{6} t - \pi \right) + 18 \)
5. **Height after 6 minutes**: 33 meters

Would you like more details on any part of this explanation?

### Related Questions:
1. How would the formula change if the Ferris wheel completed one revolution in 10 minutes?
2. What would the phase shift be if the person started at the top of the Ferris wheel?
3. How does the amplitude affect the range of the function \( h(t) \)?
4. How would the height function change if the platform was 5 meters above the ground?
5. What would be the height of the person after 9 minutes?

### Tip:
Always remember that the cosine function starts at its maximum, so if a Ferris wheel begins at its lowest point, you need to shift the cosine function by \( \pi \) radians to adjust the phase.

A Ferris wheel is 30 meters in diameter and boarded from a platform that is 3 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes one full revolution in 12 minutes. The function h(t) gives a person's height above the ground t minutes after the wheel begins to turn.
Find the period of h(t), the amplitude of h(t), the vertical shift of h(t), the formula for h(t), and the height after 6 minutes.

Solve a problem involving a Ferris wheel with a diameter of 30 meters, period of 12 minutes, and boarding platform 3 meters above the ground. Learn how to calculate the period, amplitude, vertical shift, and height function using trigonometric concepts. Find out how high a person is after 6 minutes.

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