To solve for $x$ in this triangle, let's analyze the given information:

1. Known Measurements:

   • The length of the base is $30$.
   • The angle adjacent to this base is split into $20^\circ$ and $28^\circ$, giving a total angle of $48^\circ$ at the base.

2. Goal:

   • We want to find the vertical height $x$ from the top of the triangle to the base.

3. Approach: This can be solved using trigonometric ratios in right triangles.

Let's break down the problem step-by-step.

Step 1: Calculate the third angle in the larger triangle.

The larger triangle has: $180^\circ - 90^\circ - 48^\circ = 42^\circ$

Step 2: Apply trigonometric ratios.

In the larger triangle:

• The base (adjacent to $48^\circ$) is $30$.
• The height $x$ is opposite the $42^\circ$ angle.

Using the tangent function: $\tan(42^\circ) = \frac{x}{30}$

Step 3: Solve for $x$.

Rearrange the equation to find $x$: $x = 30 \cdot \tan(42^\circ)$

Using a calculator: $x \approx 30 \cdot 0.9004 \approx 27.012$

Answer:

The height $x \approx 27.01$.

Would you like more details, or have any questions?

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Related Questions:

1. What trigonometric function would be used if we needed to find the hypotenuse?
2. How would you solve for $x$ if the angle were different?
3. How can this setup be used in real-world applications?
4. What are the relationships among all three angles in a right triangle?
5. How would the answer change if the base length were different?

Tip: When working with right triangles, remember that the sum of the angles is always $180^\circ$.