We are given two trigonometric equations involving tangent functions:

$\tan(50.6^\circ) = \frac{h}{x}$ $\tan(29.4^\circ) = \frac{h}{x + 410}$

We need to solve for $h$.

Step 1: Express $h$ in terms of $x$ from both equations.

From the first equation: $h = x \cdot \tan(50.6^\circ)$

From the second equation: $h = (x + 410) \cdot \tan(29.4^\circ)$

Step 2: Set the two expressions for $h$ equal to each other.

$x \cdot \tan(50.6^\circ) = (x + 410) \cdot \tan(29.4^\circ)$

Step 3: Expand and solve for $x$.

$x \cdot \tan(50.6^\circ) = x \cdot \tan(29.4^\circ) + 410 \cdot \tan(29.4^\circ)$

Move the $x$-terms to one side: $x \cdot (\tan(50.6^\circ) - \tan(29.4^\circ)) = 410 \cdot \tan(29.4^\circ)$

Solve for $x$: $x = \frac{410 \cdot \tan(29.4^\circ)}{\tan(50.6^\circ) - \tan(29.4^\circ)}$

Step 4: Plug in the values for the tangents.

Using the calculator values: $\tan(50.6^\circ) \approx 1.2103 \quad \text{and} \quad \tan(29.4^\circ) \approx 0.5637$

Now substitute: $x = \frac{410 \cdot 0.5637}{1.2103 - 0.5637}$ $x = \frac{231.117}{0.6466} \approx 357.52$

Step 5: Use the value of $x$ to find $h$.

Now that we have $x \approx 357.52$, substitute this back into the equation $h = x \cdot \tan(50.6^\circ)$:

$h = 357.52 \cdot 1.2103 \approx 432.57$

Final Answer:

The height $h$ is approximately 432.57 units.

Would you like further details or clarifications on any step?

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Here are 5 questions related to this problem:

1. How would the problem change if the angles were different?
2. What if the distance 410 units were instead a variable? How would you solve for both $h$ and $x$?
3. Can this method be applied to non-right triangles?
4. How do tangent functions relate to other trigonometric ratios in this context?
5. How would the result change if the angle measurements were in radians?

Tip: Always check if your calculator is set to degrees or radians when working with trigonometric functions, depending on the problem context.