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This problem involves calculating the height \( h \) of a hot air balloon (at point \( C \)) above the ground, given angles and distances. Here's the breakdown:

- The distance between the two points \( A \) and \( B \) on the ground is 2900 meters.
- From the hot air balloon, the angle \( \alpha = 65^\circ \) is the angle to point \( A \), and the angle \( \beta = 22^\circ \) is the angle to point \( B \).

We can solve this problem using trigonometry by recognizing that the hot air balloon is above point \( G \) on the ground, and we need to find the vertical distance \( h \).

### Solution Steps:

1. **Label the triangle:**
   - \( A \), \( B \), and \( C \) form a triangle.
   - We have the angles \( \alpha \) and \( \beta \), as well as the distance between \( A \) and \( B \), which is 2900 meters.

2. **Use the tangent function:**
   - The tangent of an angle in a right triangle is the ratio of the opposite side (height \( h \)) to the adjacent side (the distance on the ground).
   - For \( \alpha = 65^\circ \):
     \[
     \tan(65^\circ) = \frac{h}{GA}
     \]
   - For \( \beta = 22^\circ \):
     \[
     \tan(22^\circ) = \frac{h}{GB}
     \]

3. **Relationship between \( GA \), \( GB \), and the total distance \( AB \):**
   \[
   GA + GB = 2900 \text{ meters}
   \]

Now, let's calculate \( h \). I'll proceed with the calculation.The height \( h \) of the hot air balloon above point \( G \) is approximately \( 985.93 \, \text{meters} \).

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How would the height change if the angle \( \alpha \) were smaller?
2. What trigonometric relationships apply to right triangles and why do they work here?
3. How can the distance between the points \( A \) and \( B \) affect the height?
4. What if we only had one angle and distance, how would the calculation differ?
5. How could we solve this problem using the law of sines?

**Tip:** Always ensure angles are converted to radians when using trigonometric functions in most calculators or programming languages.

Vom Heißluftballon der Universität kann man die beiden Orte A und B sehen. Die beiden Orte liegen 2900 m voneinander entfernt und vom Heißluftballon kann man sie unter den Winkelweiten α = 65° und β = 22° anpeilen. In welcher Höhe h schwebt der Ballon (Punkt C) über dem Punkt G?

Calculate the height of a hot air balloon above the ground using trigonometric relationships. The problem involves angles of 65° and 22° with a distance of 2900 meters between two points on the ground.

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