Question

The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.

Accepted Answer

Draw a diagram to visualize the problem: mark the lighthouse as a fixed point, and the two ship positions. The first position has a bearing of N 37° E to the lighthouse, and after sailing 2.5 miles due south, the second position has a bearing of N 25° E to the lighthouse. Assign variables to the distances from the lighthouse to the ship at the first and second positions, say $$d_1$$ and $$d_2$$ respectively. The bearings give the angles between the north direction and the line from the ship to the lighthouse. Using the bearings, express the coordinates of the ship relative to the lighthouse. For the first position, the ship lies along a line making an angle of 37° east of north from the lighthouse. For the second position, after moving 2.5 miles south, the ship lies along a line making an angle of 25° east of north. Set up equations using trigonometric relationships. For example, use the sine and cosine of the bearing angles to relate the east-west and north-south components of the distances. The vertical displacement between the two ship positions is 2.5 miles south, so use this to relate $$d_1$$ and $$d_2. $$Solve the system of equations to find $$d_1$$ and $$d_2$$, which represent the distances from the lighthouse to the ship at the first and second positions respectively.

The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.

Verified video answer for a similar problem:

Key Concepts

Understanding Bearings

Using Trigonometric Ratios in Right Triangles

Applying the Law of Cosines