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In this problem, a ferris wheel is modeled using a sine function to describe the movement over time. The given function is:

\[
F(t) = A \sin\left(\frac{2\pi}{T}(t - t_0)\right) + c
\]

Where:
- \( A \) is the amplitude.
- \( T \) is the period.
- \( t_0 \) is the horizontal shift (phase shift).
- \( c \) is the vertical shift.

From the problem:
- The maximum height of the ferris wheel is 15 meters, and the minimum height is 0 meters.
- One cycle on the ferris wheel takes about 5 minutes.

Now, let’s determine the values for \( A \), \( T \), \( t_0 \), and \( c \):

### 1. Amplitude \( A \):
The amplitude represents half the distance between the maximum and minimum values. Since the maximum height is 15 meters and the minimum is 0 meters, the amplitude is:

\[
A = \frac{\text{Max} - \text{Min}}{2} = \frac{15 - 0}{2} = 7.5
\]

### 2. Period \( T \):
The period is the time it takes to complete one full cycle. Given that one cycle takes 5 minutes, the period \( T \) is:

\[
T = 5 \text{ minutes}
\]

### 3. Horizontal shift \( t_0 \):
The problem does not specify a particular starting point for the ride (e.g., whether you start at the bottom or the top), so we assume that \( t_0 \) can be 0 unless further information is provided. This represents no phase shift.

\[
t_0 = 0
\]

### 4. Vertical shift \( c \):
The vertical shift \( c \) is the average value between the maximum and minimum heights. This shift moves the sine wave up or down. It can be calculated as:

\[
c = \frac{\text{Max} + \text{Min}}{2} = \frac{15 + 0}{2} = 7.5
\]

### Final Function:
Now that we have all the values:

- \( A = 7.5 \)
- \( T = 5 \) minutes
- \( t_0 = 0 \)
- \( c = 7.5 \)

Thus, the function for your movement on the ferris wheel becomes:

\[
F(t) = 7.5 \sin\left(\frac{2\pi}{5} t\right) + 7.5
\]

Let me know if you'd like more details or have any questions!

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Here are five related questions to deepen your understanding:

1. How would the equation change if the ferris wheel had a minimum height of 2 meters instead of 0?
2. What effect does the phase shift \( t_0 \) have on the ride’s motion?
3. How can you determine the speed of the ferris wheel at any given moment using this function?
4. How would the function look if the cycle took 3 minutes instead of 5?
5. What would happen to the sine wave if the maximum height was 20 meters instead of 15?

**Tip**: The amplitude represents the distance from the centerline (average height) to the maximum or minimum points of the motion!

A ferris wheel has a maximum height of 15 m and a minimum of 0. One cycle on the ferris wheel is about 5 minutes. Your movement while taking a ride on the ferris wheel can be approximated by the following sine function: F(t) = A * sin(2π/T * (t - t₀)) + c. What are the values of A, T, t₀, and c in F(t)?

This problem involves determining the amplitude, period, phase shift, and vertical shift of a sine function that describes a ferris wheel ride. The ferris wheel has a maximum height of 15 meters, a minimum of 0 meters, and completes one cycle in 5 minutes. The solution covers key trigonometric concepts such as periodic motion and sine wave properties.

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