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To solve problem 11, we'll use trigonometric ratios. The diagram shows a right triangle, where:

- The angle is \(64^\circ\),
- The side adjacent to the angle is 11 units,
- The side opposite the angle is \(x\), which we need to find.

### Step-by-Step Solution:

1. **Identify the Trigonometric Function:**
   Since we know the angle and the adjacent side, and we're looking for the opposite side, we use the tangent function:
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
   \]

2. **Set Up the Equation:**
   \[
   \tan(64^\circ) = \frac{x}{11}
   \]

3. **Solve for \(x\):**
   \[
   x = 11 \times \tan(64^\circ)
   \]

4. **Calculate the Value:**
   \[
   x \approx 11 \times 2.0503 = 22.5533
   \]

5. **Round to the Nearest Hundredth:**
   \[
   x \approx 22.55
   \]

### Final Answer:
The value of \(x\) is approximately **22.55**.

Would you like any further details on this solution, or do you have any other questions?

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**Relative Questions:**

1. How would you solve for \(x\) if the angle were given as \(30^\circ\)?
2. What if the length of the adjacent side was changed to 15 units, how would that affect the value of \(x\)?
3. Can you describe the relationship between the angle and the ratio of the opposite to adjacent sides in a right triangle?
4. How would you find the hypotenuse of the triangle in problem 11 if you know \(x\) and the adjacent side?
5. What is the sine of a \(64^\circ\) angle, and how could you use it to find the hypotenuse?

**Tip:** Remember, rounding early in multi-step calculations can lead to inaccuracies. Always round off only at the final step unless instructed otherwise.

Learn how to solve for the opposite side in a right triangle with a 64° angle and adjacent side of 11 units using trigonometric principles. Step-by-step solution provided.

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