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Let's solve the problem step-by-step:

### Given Data:
- Ferris wheel has 20 carts evenly spaced around the wheel.
- The wheel rotates at a constant speed and takes 5 minutes to complete a full revolution.
- Total height of the Ferris wheel is 200 meters.
- Customers enter the carts at the lowest point, which is 2 meters above the ground.
- We need to find the distance from the ground after riding the wheel for 4 minutes and 10 seconds.

### Step 1: Determine the period and angular speed of the Ferris wheel
- The period \( T \) (time for one full revolution) is 5 minutes, which is 300 seconds.
- The angular speed \( \omega \) in radians per second:
  \[ \omega = \frac{2\pi}{T} = \frac{2\pi}{300} = \frac{\pi}{150} \text{ radians per second} \]

### Step 2: Determine the total ride time in seconds
- 4 minutes and 10 seconds is:
  \[ 4 \times 60 + 10 = 240 + 10 = 250 \text{ seconds} \]

### Step 3: Determine the position of the cart after 250 seconds
- The angle \( \theta \) in radians that the wheel has turned through:
  \[ \theta = \omega \times \text{time} = \frac{\pi}{150} \times 250 = \frac{250\pi}{150} = \frac{5\pi}{3} \]

### Step 4: Calculate the height above the ground
- The radius \( r \) of the Ferris wheel:
  \[ \text{Diameter} = 200 \text{ meters} \]
  \[ \text{Radius} = \frac{200}{2} = 100 \text{ meters} \]

- The height \( h \) of a cart on the Ferris wheel can be modeled using the sine function because the Ferris wheel rotates in a circular motion:
  \[ h(t) = \text{Radius} \times (1 - \cos(\theta)) + \text{base height} \]
  \[ h(t) = 100 \times (1 - \cos(\theta)) + 2 \]

Plugging in \( \theta = \frac{5\pi}{3} \):
\[ \cos\left(\frac{5\pi}{3}\right) = \cos\left(2\pi - \frac{\pi}{3}\right) = \cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

\[ h(t) = 100 \times (1 - \frac{1}{2}) + 2 \]
\[ h(t) = 100 \times \frac{1}{2} + 2 \]
\[ h(t) = 50 + 2 \]
\[ h(t) = 52 \text{ meters} \]

### Answer:
The distance from the ground after riding the Ferris wheel for 4 minutes and 10 seconds is \( \boxed{52} \) meters.

If you have any further questions or need more details, feel free to ask! Here are some questions you might be interested in:

1. How can I model the height of the cart as a function of time?
2. What is the significance of using trigonometric functions in this problem?
3. Can you provide a graph of the height of the cart over time?
4. How would the answer change if the ride time were different?
5. Are there alternative methods to solve this problem?

**Tip:** Understanding the periodic nature of trigonometric functions is crucial for solving problems involving circular motion and oscillations.

Learn how to solve a Ferris wheel problem involving circular motion. Find the distance from the ground after riding the Ferris wheel for 4 minutes and 10 seconds. Key concepts include trigonometry and circular motion equations. Suitable for high school students.

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