The two hot-air balloons in the drawing are 48.2 and 61.0 m above the ground. A person in the left balloon observes that the right balloon is 13.3° above the horizontal. What is the horizontal distance x between the two balloons?

To solve this problem, we can use trigonometry, specifically the tangent function. Here's the breakdown of the problem:

### Problem Overview:
- Height of the left balloon (observer's height): \( h_1 = 48.2 \, \text{m} \)
- Height of the right balloon: \( h_2 = 61.0 \, \text{m} \)
- Angle of elevation from the left balloon to the right balloon: \( \theta = 13.3^\circ \)

We need to find the horizontal distance \( x \) between the two balloons.

### Steps to Solve:

1. **Determine the height difference between the two balloons:**
   \[
   \Delta h = h_2 - h_1 = 61.0 \, \text{m} - 48.2 \, \text{m} = 12.8 \, \text{m}
   \]

2. **Use the tangent function:**
   The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case:
   \[
   \tan(\theta) = \frac{\Delta h}{x}
   \]
   Substituting the known values:
   \[
   \tan(13.3^\circ) = \frac{12.8 \, \text{m}}{x}
   \]

3. **Solve for \( x \):**
   \[
   x = \frac{12.8 \, \text{m}}{\tan(13.3^\circ)}
   \]
   Now, calculate the value:
   \[
   \tan(13.3^\circ) \approx 0.2366
   \]
   \[
   x = \frac{12.8 \, \text{m}}{0.2366} \approx 54.1 \, \text{m}
   \]

### Final Answer:
The horizontal distance \( x \) between the two balloons is approximately **54.1 meters**.

Would you like more details or have any questions?

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**Relative Questions:**
1. How would the horizontal distance change if the angle of elevation were larger?
2. What if the height of the left balloon was greater than the right balloon?
3. How would you calculate the distance if the angle was below the horizontal?
4. What role does the tangent function play in such trigonometry problems?
5. How can we use similar methods to determine the distance between objects at different heights?

**Tip:** When working with angles and distances in trigonometry, always ensure your calculator is set to the correct mode (degrees or radians) based on the angle's unit.

Learn how to calculate the horizontal distance between two hot-air balloons using trigonometry. This problem involves determining the distance based on the heights of the balloons and the angle of elevation observed. Suitable for high school students studying trigonometry.

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