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 18°33'. When the boat stops, the angle of depression is 51°33'. 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

Answer: 437.21 ft

Explanation:

First off, change all minutes to degrees.

18°33’ = 18.55°

51°33’ = 51.55°

When the boat is first spotted:

Tan(x)=opp/adj

Tan(18.55°)=200/x

x=200/tan(18.55)

x=596.01ft

When the boat stops:

Tan(51.55)=200/x

x=158.80ft

Find the difference:

596.01-158.0=437.21

Yay! It’s very simple once you understand the parttern :)

Answer 2

If we let
x as the distance traveled by the boat
y as the distance between the boat and the lighthouse.

Then, we have:
tan 18°33' = 200 / (x + y)
and
tan 51°33' = 200 / y

Solving for y in the second equation:
y = 200 / tan 51°33'

Rearranging the first equation and substituting y
x = 200 / tan 18°33' - 200 / tan 55°33'
x = 458.81 ft

Therefore, the boat traveled 458.81 ft before it stopped.