To solve this problem using the sine rule, we will apply the formula:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

where $a$, $b$, and $c$ are the sides of the triangle, and $A$, $B$, and $C$ are the angles opposite these sides, respectively.

Step-by-step solution:

1. Labeling the triangle:

   • We are asked to find $d$, which is the side opposite the $67^\circ$ angle.
   • The known side is 36.5 m, which is opposite the $42^\circ$ angle.

2. Set up the sine rule equation: $\frac{d}{\sin(67^\circ)} = \frac{36.5}{\sin(42^\circ)}$

3. Solve for $d$: Rearrange the equation to solve for $d$: $d = \frac{36.5 \cdot \sin(67^\circ)}{\sin(42^\circ)}$

4. Substitute the values of the sines: Using a calculator:

   • $\sin(67^\circ) \approx 0.921$
   • $\sin(42^\circ) \approx 0.669$

   So: $d = \frac{36.5 \cdot 0.921}{0.669} \approx \frac{33.6165}{0.669} \approx 50.26 \, \text{m}$

Final answer:

The length of $d$ is approximately 50.26 m (to 2 decimal places).

Would you like more details or have any further questions?

Here are some related questions you might find interesting:

1. What is the sine rule, and how does it apply to non-right-angled triangles?
2. How would you solve this problem if a different side length were given?
3. Can the sine rule be used in right-angled triangles?
4. How can you calculate the third side of the triangle using the cosine rule?
5. What happens if the angles given are incorrect or don't add up to 180°?

Tip: Always make sure the sum of the angles in a triangle equals 180°, as this helps verify the correctness of your given information.