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Solve for x to the nearest tenth of a metre solving by the cosine law. Provide the great calculations for this problem and the exact answer for the x should be 11.5 m.

To solve for \( x \) in this problem, we can use the **Cosine Law**, which is applicable for non-right triangles when we know two sides and the included angle, or all three sides of the triangle.

Here’s the setup based on the provided triangle information:

### Given Data:
- Side \( BC = 6.3 \, \text{m} \)
- Side \( CD = x \, \text{m} \) (the value we want to find)
- Side \( BD = 12.5 \, \text{m} \)
- Angle \( \angle BCD = 46^\circ \)

Since we're solving for side \( x = CD \) in \(\triangle BCD\), we can apply the Cosine Law:
\[
x^2 = BC^2 + BD^2 - 2 \cdot BC \cdot BD \cdot \cos(\angle BCD)
\]

### Step-by-Step Solution:

1. **Substitute the known values** into the formula:
   \[
   x^2 = (6.3)^2 + (12.5)^2 - 2 \cdot 6.3 \cdot 12.5 \cdot \cos(46^\circ)
   \]

2. **Calculate each term**:
   - \( (6.3)^2 = 39.69 \)
   - \( (12.5)^2 = 156.25 \)
   - \( 2 \cdot 6.3 \cdot 12.5 = 157.5 \)
   - \( \cos(46^\circ) \approx 0.6947 \)

3. **Substitute and simplify**:
   \[
   x^2 = 39.69 + 156.25 - 157.5 \cdot 0.6947
   \]
   \[
   x^2 = 39.69 + 156.25 - 109.89675
   \]
   \[
   x^2 = 196.94 - 109.89675
   \]
   \[
   x^2 \approx 87.04325
   \]

4. **Take the square root** to find \( x \):
   \[
   x \approx \sqrt{87.04325}
   \]
   \[
   x \approx 9.33 \, \text{m}
   \]

### Conclusion:
The answer appears to be approximately \( x = 9.3 \, \text{m} \), not 11.5 m as stated. If further clarification or a different approach is needed, let me know.

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Would you like more details or have any further questions?

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Here are related questions for further exploration:

1. How would you solve this problem using the Sine Law?
2. How does the accuracy of \( \cos(46^\circ) \) affect the solution?
3. What if angle \( \angle BCD \) was different; how would that impact \( x \)?
4. How can we use trigonometry to find angles if we know all three sides?
5. How does changing the length of side \( BD \) impact the value of \( x \)?

**Tip:** When using the Cosine Law, ensure you have precise angle values for accurate results.

Solve for x to the nearest tenth of a metre using the cosine law for the triangle setup.

Learn how to solve for the unknown side x in a triangle using the cosine law, with provided measurements and angle. This problem covers key trigonometric concepts and demonstrates detailed steps for accurate calculation.

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