The bearing of a lighthouse from a ship was found to be n 37 degrees e. after the ship sailed 2.5 miles due south, the new bearing was n 25 degrees east. find the distance between the ship and the lighthouse at each location.

The distance between the lighthouse and the ship

from the start position A = 5.08 miles

from the Final point B = 7.23 miles

Explanation:

Note: Refer the figure

Let the position of the lighthouse be 'L'  

Given:

When the ship is at the position A, ∠DAL=37°

Now, when the ship sails through a distance of 2.5 i.e at position B

mathematically,

AB=2.5 miles

∠ABL=25°

Now,

∠DAL + ∠LAB = 180°

or

37° +  ∠LAB = 180°

or

∠LAB = 180° - 37° = 143°

Also, In ΔLAB

∠LAB + ∠ABL + ∠ALB = 180°

or

143° + 25° + ∠ALB = 180°

or

∠ALB = 180° - 143° - 25° = 12°

Now using the concept of the sin law

In ΔLAB

or

AL = 5.08 miles

and,

or

BL = 7.23 miles

hence,

The distance between the lighthouse and the ship

from the start position A = 5.08 miles

from the Final point B = 7.23 miles