We are given a right triangle with the following information: the hypotenuse is $6$ km, the opposite side is $3$ km, and one of the angles is $30^\circ$. We are asked to find the length of the adjacent side, correct to $3$ decimal places.

1. Problem Description

We are given a right triangle with the following information: the hypotenuse is $6$ km, the opposite side is $3$ km, and one of the angles is $30^\circ$. We are asked to find the length of the adjacent side, correct to $3$ decimal places.

2. Solution Steps

Let the adjacent side be $a$. We can use the Pythagorean theorem or trigonometric ratios to find the adjacent side. Since we are given the hypotenuse and the opposite side, we can use $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. However, in this case, we are already given the opposite and hypotenuse lengths.

Using the Pythagorean theorem:

$a^2 + x^2 = h^2$, where $a$ is the adjacent side, $x$ is the opposite side and $h$ is the hypotenuse.

We have $x = 3$ km and $h = 6$ km.

Therefore, $a^2 + 3^2 = 6^2$,

$a^2 + 9 = 36$,

$a^2 = 36 - 9$,

$a^2 = 27$,

$a = \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$.

To approximate $a$ to three decimal places, we can use the approximation $\sqrt{3} \approx 1.732$.

$a = 3 \times 1.732 = 5.196$.

Alternatively, we can use the cosine ratio:

$\cos(30^{\circ}) = \frac{a}{6}$

$a = 6 \cos(30^{\circ}) = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3}$.

We know that $\sqrt{3} \approx 1.7320508$.

$3\sqrt{3} \approx 3 \times 1.7320508 = 5.1961524$.

Rounded to three decimal places, we get $5.196$.

3. Final Answer

$3\sqrt{3} \approx 5.196$