To solve for $x$ in this right triangle, we can use trigonometric ratios. The given angle is $20^\circ$, and we are provided with the side opposite to this angle, which is 15 m.

The relationship between the opposite side, the hypotenuse, and the angle is given by the sine function:

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

In this case:

$\sin(20^\circ) = \frac{15}{x}$

Now we solve for $x$:

$x = \frac{15}{\sin(20^\circ)}$

I'll calculate the value of $x$.The value of $x$, the hypotenuse of the triangle, is approximately $43.86 \, \text{m}$, rounded to the nearest hundredth.

Would you like more details on the calculation? Here are a few related questions:

1. How would you calculate the length of the adjacent side in this triangle?
2. What is the value of $\cos(20^\circ)$ in this triangle, and how would you use it?
3. Can you determine the area of this triangle using the given information?
4. How would the hypotenuse change if the angle were increased to $30^\circ$?
5. How can we verify the calculation using another trigonometric function?

Tip: The sine function relates the angle and the opposite side, making it especially useful in right triangles when these are known.