Three people are watching a hot air balloon travel over their town. Person c stands directly below it at 369m, person b is 150m away from person c and the angle between person a and the balloon is 39 degrees. How far is person a from the hot air balloon? 

To solve this problem, let's clarify the information and calculate the distance from Person A to the hot air balloon.

### Problem Analysis

- **Person C** is directly below the hot air balloon at a vertical distance of \(369\) meters.
- **Person B** is \(150\) meters horizontally away from Person C.
- **Person A** observes the balloon at an angle of \(39^\circ\). We assume that this angle is the angle of elevation from Person A to the balloon.

We are looking to find the straight-line distance from **Person A** to the hot air balloon.

### Visualizing the Problem

Consider a right triangle where:

- The vertical leg is the distance from Person C to the balloon (\(369\) meters).
- The horizontal distance between Person C and Person B is \(150\) meters.
- The hypotenuse of this right triangle is the distance from Person A to the balloon, which we'll call \(d\).

### Solving the Problem

To solve for \(d\), we need to use the tangent of the angle \(39^\circ\) since it relates the opposite side (height of the balloon) to the adjacent side (horizontal distance from Person A to Person C).

#### Step-by-step Calculation:

1. **Set Up the Trigonometric Function:**

   \[
   \tan(39^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{369}{x}
   \]

   where:
   - Opposite is the vertical distance from Person C to the balloon (\(369\) meters).
   - Adjacent \(x\) is the horizontal distance from Person A to Person C.

2. **Solve for \(x\):**

   Rearranging for \(x\):

   \[
   x = \frac{369}{\tan(39^\circ)}
   \]

   Calculate \( \tan(39^\circ) \):

   \[
   \tan(39^\circ) \approx 0.8098
   \]

   Substitute the value:

   \[
   x = \frac{369}{0.8098} \approx 455.73 \, \text{meters}
   \]

3. **Find the Distance \(d\) from Person A to the Balloon:**

   The straight-line distance from Person A to the balloon \(d\) forms a right triangle with the vertical distance of the balloon (\(369\) meters) and the horizontal distance \(x\).

   We use the Pythagorean theorem:

   \[
   d = \sqrt{x^2 + 369^2}
   \]

   Substitute \(x = 455.73\) meters:

   \[
   d = \sqrt{(455.73)^2 + (369)^2}
   \]

   Calculate \(d\):

   \[
   d \approx \sqrt{207686.47 + 136161} \approx \sqrt{343847.47} \approx 586.36 \, \text{meters}
   \]

### Conclusion

The distance from **Person A** to the hot air balloon is approximately **586.36 meters**.

Would you like further details or have any questions about this solution?

### Related Questions for Further Study:

1. How would the distance change if the height of the balloon was different?
2. What if the angle observed by Person A was different? How would that affect the distance?
3. How can we determine the horizontal distance between Person A and Person C?
4. If the angle between Person A and the balloon was an angle of depression instead, how would this change the calculations?
5. How would you calculate the distances if Person B was positioned at a different distance from Person C?

**Tip:** Practice using trigonometric functions and the Pythagorean theorem to solve similar problems involving angles and distances.

Learn how to calculate the distance from Person A to a hot air balloon using trigonometric principles and the Pythagorean theorem. This problem involves finding the height of the balloon, horizontal distances, and angle of elevation. Suitable for students in Grades 10-12.

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