Question

A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is 16°18'. When the boat stops, the angle of depression is 48°51'. The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.

Answer 1

To solve this problem, we can use the tan function to find for the distances covered.

tan θ = o / a

Where,

θ = angle = 90° - angle of depression

o = side opposite to the angle = distance of boat from lighthouse

a = side adjacent to the angle = height of lighthouse = 200 ft

When the angle of depression is 16°18', the initial distance from the lighthouse is:

o = 200 tan (90° - 16°18')

o = 683.95 ft

When the angle of depression is 48°51', the final distance from the lighthouse is:

o = 200 tan (90° - 48°51')

o = 174.78 ft

 

Therefore the total distance the boat travelled is:

d = 683.95 ft - 174.78 ft

d = 509.17 ft