https://file.aiquickdraw.com/mathbot/file/172725731883102gnm101.jpg

To find the height \( h \), we can use trigonometry based on the right triangle provided in the diagram.

### Analysis:
1. We are given two angles: \( 21.5^\circ \) and \( 46.3^\circ \), and the adjacent side to both angles (which is the horizontal distance of \( 466 \)).
2. We are asked to find the height \( h \), which is the opposite side to angle \( 46.3^\circ \).

### Applying Trigonometric Ratios:
We can use the tangent function, which relates the opposite side to the adjacent side in a right triangle:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
For \( \theta = 46.3^\circ \) and adjacent = 466, we have:
\[
\tan(46.3^\circ) = \frac{h}{466}
\]
Solving for \( h \):
\[
h = 466 \times \tan(46.3^\circ)
\]

Let me calculate this value for you.The height \( h \) is approximately 488 when rounded to the nearest integer.

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. How would you find the height if only the angle \( 21.5^\circ \) was given?
2. Can you use the sine or cosine function to find the height \( h \)? How?
3. How does the tangent function change as the angle increases or decreases?
4. How would you find the length of the hypotenuse using the given information?
5. What would change in the calculation if the base were a different length?

**Tip:** Trigonometric ratios are powerful tools for solving right triangles when you know one angle and one side.

Find h as indicated in the figure with two angles 21.5° and 46.3°, and a base length of 466.

Learn how to calculate the height (h) in a right triangle using trigonometry with angles 21.5° and 46.3°, and a base length of 466. This step-by-step solution uses the tangent function to find the height.

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