The bearing of a ship from a lighthouse was found to be $\mathrm{N} 17^{\circ} \mathrm{E}$. After the ship sailed 6.7 miles due south, the new bearing was $\mathrm{N} 30^{\circ} \mathrm{E}$. Find the distance between the ship and the lighthouse at each location. The ship began miles from the lighthouse. (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth if needed.)

Step 1 ：The problem involves trigonometry. We can form two right triangles with the lighthouse as one of the vertices. The first triangle is formed by the initial position of the ship, the lighthouse, and the point directly south of the lighthouse. The second triangle is formed by the final position of the ship, the lighthouse, and the point directly south of the lighthouse.

Step 2 ：In the first triangle, the angle at the lighthouse is 17 degrees and the opposite side is the initial distance of the ship from the lighthouse. In the second triangle, the angle at the lighthouse is 30 degrees and the opposite side is the final distance of the ship from the lighthouse.

Step 3 ：We can use the tangent of the angles to find the distances. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the adjacent side is the same for both triangles and is the distance the ship sailed south, which is 6.7 miles.

Step 4 ：So, we have the following equations: \(\tan(17) = \frac{\text{initial distance}}{6.7}\) and \(\tan(30) = \frac{\text{final distance}}{6.7}\).

Step 5 ：We can solve these equations to find the initial and final distances. The initial distance is approximately 2.0 miles and the final distance is approximately 3.9 miles. These distances are reasonable given the information in the problem.

Step 6 ：Final Answer: The initial distance of the ship from the lighthouse is approximately \(\boxed{2.0}\) miles and the final distance is approximately \(\boxed{3.9}\) miles.