The centers of 2 wheels are horizontally 31 cm apart. The diameter of the larger wheel is 14 cm and the diameter of the smaller wheel is 5 cm. Find the actual distance between the centers of the two wheels.

The centers of two wheels are horizontally 31 cm apart. We are tasked with finding the actual distance between the centers, considering that the wheels have different radii, which will impact the vertical positioning.

### Step-by-Step Solution:

1. **Radius of the wheels:**
   - The larger wheel has a diameter of 14 cm, so its radius is:
     \[
     r_{\text{large}} = \frac{14}{2} = 7 \, \text{cm}
     \]
   - The smaller wheel has a diameter of 5 cm, so its radius is:
     \[
     r_{\text{small}} = \frac{5}{2} = 2.5 \, \text{cm}
     \]

2. **Difference in vertical alignment:**
   Since the radii of the wheels are different, the vertical distance between the centers of the wheels is the difference in their radii:
   \[
   \text{Vertical distance} = r_{\text{large}} - r_{\text{small}} = 7 - 2.5 = 4.5 \, \text{cm}
   \]

3. **Using the Pythagorean Theorem:**
   The horizontal distance between the centers is 31 cm, and the vertical distance is 4.5 cm. To find the actual distance, \( d \), between the centers of the two wheels, we apply the Pythagorean theorem:
   \[
   d = \sqrt{(\text{horizontal distance})^2 + (\text{vertical distance})^2}
   \]
   Substituting the known values:
   \[
   d = \sqrt{31^2 + 4.5^2} = \sqrt{961 + 20.25} = \sqrt{981.25}
   \]
   \[
   d \approx 31.33 \, \text{cm}
   \]

### Final Answer:
The actual distance between the centers of the two wheels is approximately **31.33 cm**.

Would you like further details or have any other questions?

Here are 5 related questions for further exploration:
1. How would the distance change if both wheels had the same radius?
2. What would happen if the wheels were vertically aligned instead of horizontally?
3. Can you apply the Pythagorean theorem to solve problems involving circular motion?
4. What is the distance between the centers if the horizontal distance increases to 40 cm?
5. How would we calculate the distance if the problem involved three wheels instead of two?

**Tip:** The Pythagorean theorem is a fundamental tool in geometry for calculating distances in 2D space when you know two perpendicular sides.

Learn how to find the actual distance between the centers of two wheels with diameters 14 cm and 5 cm, placed 31 cm apart horizontally. The solution applies geometry concepts and the Pythagorean theorem to calculate the final distance of approximately 31.33 cm. Suitable for students in Grades 8-10.

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