A Ferris wheel has a maximum height of 308 feet and a wheel diameter of 290 feet. Find an equation for the wheel if the center of the wheel is on the​ y-axis and y represents the height above the ground.

To find the equation for the Ferris wheel, we'll assume it models the vertical motion of a point on a circular path, which can be described as a sinusoidal function. Here are the key points we know:

- The **maximum height** of the Ferris wheel is 308 feet.
- The **diameter** of the Ferris wheel is 290 feet.
- The center of the wheel is on the y-axis, and the equation is in terms of \( y \), which represents height above the ground.

### Step 1: Center of the wheel
The radius of the wheel is half the diameter:
\[
\text{Radius} = \frac{290}{2} = 145 \text{ feet}.
\]
Since the Ferris wheel reaches a maximum height of 308 feet, the center of the wheel must be 145 feet below this maximum height. So the center is located at:
\[
\text{Center height} = 308 - 145 = 163 \text{ feet}.
\]
This tells us that the vertical center of the Ferris wheel is 163 feet above the ground.

### Step 2: Sinusoidal motion
The equation for a Ferris wheel can be modeled as a sinusoidal function. We can write the height \( y \) above the ground as a function of \( \theta \) (the angular position of the rider) in the form:
\[
y(\theta) = A \sin(\theta) + D,
\]
where:
- \( A \) is the amplitude, which equals the radius of the Ferris wheel (145 feet),
- \( D \) is the vertical shift, which equals the height of the center of the Ferris wheel (163 feet).

### Step 3: The final equation
Thus, the equation for the height \( y \) of a point on the Ferris wheel in terms of the angle \( \theta \) is:
\[
y(\theta) = 145 \sin(\theta) + 163.
\]
Here:
- \( \theta \) represents the angular position, measured in radians, and
- \( y(\theta) \) gives the height above the ground in feet.

Would you like to explore further details about Ferris wheel motion or transformations? Here are some related questions:

1. How does the period of the Ferris wheel affect the speed of a ride?
2. How can we express the position of a rider at any given time if we know the speed of rotation?
3. How does a cosine function relate to the sinusoidal model of the Ferris wheel?
4. Can we model the horizontal position of the rider as well using similar methods?
5. How does the height above the ground change with respect to time for a rotating Ferris wheel?

**Tip:** Sinusoidal functions are ideal for modeling periodic motion, like the motion of a Ferris wheel or a pendulum.

Explore how to derive the equation for a Ferris wheel with a maximum height of 308 feet and a diameter of 290 feet. Learn about sinusoidal functions, amplitude, and vertical shifts, ideal for high school students studying trigonometry. Suitable for Grades 9-12.

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