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To solve this problem, we can use trigonometry, specifically the tangent function, which relates the opposite side and the adjacent side of a right triangle.

### Problem Recap:
- Mary Grace sees a hot air balloon with an angle of elevation of \(13^\circ\).
- The horizontal distance from Mary Grace to the point directly below the hot air balloon is 8 meters.
- We need to find the height of the hot air balloon.

### Solution:

Let's denote:
- \(h\) as the height of the hot air balloon.
- \(d = 8\) meters as the horizontal distance from the point on the ground directly below the balloon.

The tangent of an angle in a right triangle is given by:

\[
\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}
\]

Here, the angle \(\theta = 13^\circ\), the opposite side is the height \(h\), and the adjacent side is the distance \(d = 8\).

\[
\tan(13^\circ) = \frac{h}{8}
\]

To find the height \(h\), we rearrange the equation:

\[
h = 8 \times \tan(13^\circ)
\]

Now, let's compute the value.The height of the hot air balloon is approximately **1.847 meters** when rounded to three decimal places.

Would you like more details or have any other questions?

### Related Questions:
1. How would the height change if the angle of elevation increased to 20 degrees?
2. What is the formula for calculating the distance to the hot air balloon if only the height and angle are known?
3. How does changing the distance from Mary Grace to the hot air balloon affect the angle of elevation?
4. How can trigonometric functions be used to calculate distances and heights in other real-life scenarios?
5. What are the sine and cosine functions, and how could they be applied to similar problems?

**Tip:** Understanding the relationships between angles and sides in right triangles can help solve many real-world problems involving height and distance.

While visiting Salt Lake City, Mary Grace sees a hot air balloon. The angle of elevation to from the ground to the hot air balloon is 13°. If the hot air balloon is directly above a point 8 meters away, what is the height of the hot air balloon?

Learn how to calculate the height of a hot air balloon using trigonometry. This problem involves the tangent function and right triangles, perfect for students in Grades 8-10.

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