Question

A boat is sailing due east parallel to the shoreline at speed of 10 miles per hour. At a given time, the bearing to a lighthouse is S 72° E, and 15 minutes later the bearing is S 66°. How do you find the distance from the boat to the lighthouse?

Answer 1

Preliminary Calculations

Explanation:

Since the boat is traveling at a rate of 10 miles per hour (60 minutes), that same boat travels 2.5 miles in 15 minutes.

Draw a diagram. [On the diagram shown, all angles are in degrees.] This diagram should show two triangles -- one with a `72^o` angle to the lighthouse, and another with a `66^o` angle to the lighthouse. Find the complementary angles of `18^o` and `24^o`.
The angle immediately under the boat's present location measures `66^o + 90^o = 156^o`.
For the angle with the smallest measure in the diagram, I have used the fact that `6^o = 24^o - 18^o`, but you may also subtract the sum of 156 and 18 from `180^o`.

This gives us an oblique triangle whose angles measure `156^o, 18^o, and 6^o` and one of whose sides measures 2.5 miles.

You may now use the Law of Sines to find the direct distance to the lighthouse.

`(sin6^o)/2.5 = (sin18^o)/x`

This gives a direct distance of approximately 7.4 miles.

If you want the perpendicular distance to the shore, you may now use basic trigonometry. If y is the perpendicular distance, then
`y/7.4 = sin23^o`
`y = 7.4sin23^o`.
This is approximately 2.9 miles.