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 an and the balloon is 39 degrees. How far is person a from the hot air balloon?

To solve this problem, we will use trigonometry to determine the distance from person A to the hot air balloon. 

### Step-by-step Solution:

1. **Understand the problem setup:**

   - Person C is directly below the hot air balloon at a vertical height of \( 369 \) meters.
   - Person B is \( 150 \) meters horizontally away from Person C.
   - Person A is somewhere else such that the angle between Person A and the balloon, as viewed from the horizontal line connecting Person B and Person C, is \( 39^\circ \).

2. **Visualize the Geometry:**

   We can imagine a right triangle where:
   
   - The vertical line from Person C to the hot air balloon is one leg of the triangle (\( 369 \) m).
   - The horizontal distance from Person B to Person C (\( 150 \) m) forms the base.
   - The distance from Person A to the hot air balloon is the hypotenuse of a right triangle.

3. **Apply Trigonometry:**

   From the given data, we need to find the distance from Person A to the hot air balloon. Let’s call this distance \( d \).

   To use the angle given, we identify that the tangent function relates the opposite side (the height of the balloon) to the adjacent side (the horizontal distance from Person B to Person C). However, we are given the angle from Person A to the balloon and need to find \( d \).

   Given the angle \( 39^\circ \) and the height \( 369 \) meters, the opposite side in the context of angle \( 39^\circ \) and the hypotenuse \( d \):

   \[
   \sin(39^\circ) = \frac{\text{opposite side}}{\text{hypotenuse}}
   \]

   Here, the opposite side is \( 369 \) meters, and the hypotenuse \( d \) is what we want to find. 

   \[
   \sin(39^\circ) = \frac{369}{d}
   \]

   Rearranging to solve for \( d \):

   \[
   d = \frac{369}{\sin(39^\circ)}
   \]

4. **Calculate the distance \( d \):**

   First, compute \( \sin(39^\circ) \):

   \[
   \sin(39^\circ) \approx 0.6293
   \]

   Then, solve for \( d \):

   \[
   d = \frac{369}{0.6293} \approx 586.34 \, \text{meters}
   \]

### Conclusion:

Person A is approximately **586.34 meters** away from the hot air balloon.

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

### Related Questions for Further Study:

1. How would the calculation change if the angle given was instead the angle of elevation from Person A to the balloon?
2. What if Person B was positioned at a different angle to the balloon? How would this affect the calculation?
3. How would you determine the horizontal distance from Person A to Person C?
4. Can you calculate the area of the triangle formed by Persons A, B, and C?
5. What would be the height of the hot air balloon if the angle of elevation from Person A was \(45^\circ\) and Person A was at the same distance?

**Tip:** Knowing how to visualize geometric setups and apply trigonometric identities is crucial for solving such problems efficiently.

Learn how to calculate the distance from a hot air balloon using trigonometry with a detailed step-by-step solution. This problem involves finding the distance based on given angles and distances between observers and the balloon. Suitable for students in Grades 10-12.

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