An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of depression to the ships are 𝛽 = 6° and 𝜃 = 8.5° (see figure). How far apart are the ships?

Question

(Round your answer to one decimal place.)

______ ft

Answer 1

From the angles of depression we can calculate the angle opposite the 350' height.

Let Ship A be the closest ship and Ship B be the furthest ship.

So let a be the distance from the lighthouse to Ship A and likewise for b.

We can determine the angle opposite the 350 height for ship A to be 8.5° and the angle for Ship B to be 6°

The angle from the lighthouse to ship A is 180 - (90 + 8.5) = 81.5°

The angle from the lighthouse to ship B is 180 - (90 + 6) = 84°

Since we have the necessary angles, we can use the law of sines to calculate the missing a and b distances.

a/sinA = b/sinB = c/sinC

350/sin(8.5) = a/sin(81.5) ⇒ a = (350)sin(81.5)/sin(8.5) = 2341.9

350/sin(6) = b/sin(84) ⇒ b = (350)sin(84)/sin(6) = 3330.0

So the distance between the two ships is 3330 - 2341.9 = 988.1 feet apart.