The problem involves finding the length of the side opposite to a 30-degree angle in a right-angled triangle, given that the hypotenuse is 6 km long. The length of the opposite side is denoted as $x$.

1. Problem Description

The problem involves finding the length of the side opposite to a 30-degree angle in a right-angled triangle, given that the hypotenuse is 6 km long. The length of the opposite side is denoted as $x$.

2. Solution Steps

The trigonometric ratio that relates the opposite side and the hypotenuse is the sine function. We are given that

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

In this case, $\theta = 30^\circ$, the opposite side is $x$, and the hypotenuse is 6 km. Thus, we can write:

$\sin(30^\circ) = \frac{x}{6 \text{ km}}$

Multiplying both sides by 6 km, we get:

$x = 6 \text{ km} \cdot \sin(30^\circ)$

We know that $\sin(30^\circ) = \frac{1}{2}$.

$x = 6 \text{ km} \cdot \frac{1}{2}$

$x = 3 \text{ km}$

3. Final Answer

$x = 3 \text{ km}$