A ship is sailing due north. At a certain point, the bearing of a lighthouse $8.3 \mathrm{~km}$ away is $\mathrm{N} 36.3^{\circ} \mathrm{E}$. Later on, the captain notices that the bearing of the lighthouse has become $\mathrm{S} 31.7^{\circ} \mathrm{E}$. How far did the ship travel between the two observations of the lighthouse?

The ship travelled $\square$ km between the two observations. (Do not round until the final answer. Then round to the nearest tenth as needed.)

Step 1 ：Understand the problem. The ship is sailing due north. At a certain point, the bearing of a lighthouse 8.3 km away is N 36.3° E. This means that the angle between the north direction and the line connecting the ship and the lighthouse is 36.3°. Later on, the bearing of the lighthouse has become S 31.7° E. This means that the angle between the north direction (which is now opposite to the ship's direction) and the line connecting the ship and the lighthouse is 31.7°.

Step 2 ：Draw a triangle with the ship's path as one side, the line from the ship's starting point to the lighthouse as another side, and the line from the ship's ending point to the lighthouse as the third side. The angle at the ship's starting point is 36.3°, the angle at the ship's ending point is 31.7°, and the angle at the lighthouse is \(180° - 36.3° - 31.7° = 112°\).

Step 3 ：Denote the distance the ship travelled as d. Use the law of sines to find d: \(\frac{\sin(112°)}{8.3} = \frac{\sin(36.3°)}{d}\)

Step 4 ：Solve for d: \(d = \frac{\sin(36.3°) * 8.3}{\sin(112°)}\)

Step 5 ：Calculate the above expression: \(\boxed{d ≈ 5.6 km}\)