Tan(50.6degree)=h/x, tan(29.4degree)=h/(x+410). find h.

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.

Learn how to solve for the height (h) using tangent functions and algebraic manipulation with the angles 50.6° and 29.4°. This step-by-step solution includes key trigonometric concepts, including the tangent formula and algebraic manipulation to find unknown values. Suitable for students in Grades 9-12.

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