A Ferris wheel has a maximum height of 220 feet and a wheel diameter of 210 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 of the Ferris wheel, let's break down the problem:

1. **Maximum Height:** The Ferris wheel has a maximum height of 220 feet.
2. **Diameter of the Wheel:** The diameter is 210 feet.
3. **Center of the Wheel:** The center is on the y-axis.

### Step 1: Identify the radius
- The radius \( r \) of the wheel is half of the diameter:
  \[
  r = \frac{210}{2} = 105 \text{ feet}
  \]

### Step 2: Determine the center of the wheel
- The maximum height of the Ferris wheel occurs when the wheel is at its highest point, which corresponds to the top of the circle. 
- The distance from the center of the wheel to the top is the radius \( r = 105 \) feet. Therefore, if the maximum height is 220 feet, the height of the center \( C_y \) from the ground is:
  \[
  C_y = 220 - 105 = 115 \text{ feet}
  \]

- The center of the wheel is on the y-axis, so the coordinates of the center are \( (0, 115) \).

### Step 3: Write the equation of the circle
The general equation of a circle with center \((h, k)\) and radius \(r\) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]

Substituting the values for the Ferris wheel:
- \( h = 0 \) (since the center is on the y-axis)
- \( k = 115 \)
- \( r = 105 \)

The equation becomes:
\[
x^2 + (y - 115)^2 = 105^2
\]

### Step 4: Simplify
Calculate \( 105^2 \):
\[
x^2 + (y - 115)^2 = 11025
\]

So, the equation of the Ferris wheel is:
\[
\boxed{x^2 + (y - 115)^2 = 11025}
\]

This equation represents the Ferris wheel, where \( y \) is the height above the ground, and the center of the wheel is at \( (0, 115) \) with a radius of 105 feet.

---

Would you like further details or have any questions? Here are some related questions you might be interested in:

1. How would the equation change if the center was not on the y-axis?
2. What would be the equation if the maximum height was 250 feet instead?
3. How do you determine the wheel's position at any given time?
4. What if the Ferris wheel rotated counterclockwise? How would you model its motion?
5. How would you find the time it takes for one full revolution of the Ferris wheel?

**Tip:** When dealing with circles, always identify the radius and the center first—they are key to forming the equation.

Learn how to find the equation of a Ferris wheel with a maximum height of 220 feet and a diameter of 210 feet. This problem covers key geometry concepts and the equation of a circle, where the center is located on the y-axis. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation