A ferris wheel is elevated 1 m above the ground. When a car reaches the highest point on
the ferris wheel, its altitude from the ground is 31 m. How far away from the center, horizontally, is the car when it is at an altitude of 25 m?

To solve the problem, we can model the Ferris wheel as a circle centered at a certain height above the ground. Let's break it down step by step.

### 1. Understanding the Problem
- The Ferris wheel is elevated 1 meter above the ground.
- The highest point of the Ferris wheel is at an altitude of 31 meters.
- Therefore, the diameter of the Ferris wheel is \( 31 \, \text{m} - 1 \, \text{m} = 30 \, \text{m} \).
- The radius \( r \) of the Ferris wheel is \( r = \frac{30 \, \text{m}}{2} = 15 \, \text{m} \).
- The center of the Ferris wheel is at an altitude of \( 1 \, \text{m} + 15 \, \text{m} = 16 \, \text{m} \) from the ground.

### 2. Establish the Coordinate System
- Place the center of the Ferris wheel at the point \( (0, 16) \) in a coordinate system where the ground is at \( y = 0 \).
- The equation of the circle representing the Ferris wheel is:
  \[
  (x - 0)^2 + (y - 16)^2 = 15^2
  \]
  Simplifying, this becomes:
  \[
  x^2 + (y - 16)^2 = 225
  \]

### 3. Find the Horizontal Distance at \( y = 25 \)
- We are asked to find the horizontal distance \( x \) from the center when the altitude \( y = 25 \).
- Substitute \( y = 25 \) into the circle equation:
  \[
  x^2 + (25 - 16)^2 = 225
  \]
  Simplifying, we get:
  \[
  x^2 + 9^2 = 225
  \]
  \[
  x^2 + 81 = 225
  \]
  \[
  x^2 = 225 - 81
  \]
  \[
  x^2 = 144
  \]
  \[
  x = \sqrt{144} = 12
  \]

### 4. Conclusion
The horizontal distance \( x \) from the center of the Ferris wheel when the car is at an altitude of 25 meters is **12 meters**.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How would the horizontal distance change if the altitude is 20 meters?
2. What is the maximum horizontal distance from the center that the car can reach?
3. How does the altitude vary with respect to the horizontal distance from the center?
4. If the Ferris wheel spins at a constant rate, how long does it take to reach the highest point from the lowest?
5. How would the equation change if the Ferris wheel had a different diameter?

**Tip:** When dealing with circular motion problems, always consider the center as a reference point to set up your equations.

A ferris wheel is elevated 1 m above the ground. When a car reaches the highest point on the ferris wheel, its altitude from the ground is 31 m. How far away from the center, horizontally, is the car when it is at an altitude of 25 m?

Solve the Ferris wheel problem where you calculate the horizontal distance from the center when the car is at an altitude of 25 meters. This solution involves circle equations and the Pythagorean Theorem, making it suitable for students in Grades 9-10.

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