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 13°7'. When the boat stops, the angle of depression is 50°42' . 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

The distance from the original boat position to the bottom of the lighthouse is a. The boat travels x distance and ends up b distance from the bottom of the lighthouse.

tan 76.88333 = a/200

a = 200 tan 76.88333 = 858.3177

tan 39.3 = b/200

b = 200 tan 39.3 = 163.6981

x = a - b = 858.3177 - 163.6981 = 694.6196

Answer: 694.62 ft

Answer 2

Explanation:

We know that from the boat to the top of the lighthouse, it makes a right triangle. The angle between the horizontal line and the diagonal line will make the angle of depression that is 13.116666..

Now, tangent of that angle (tan is opposite over adjacent) is=
`(200')/(d_(1))`, therefore

tan(13.1166)=
`(200')/(d_(1))` (1)

When the boat is at the second position, similarly doing the same way we get: tan(50.7)=
`(200')/(d_(2))`

Now, the travel distance is given by:
`d_(1)- d_(2)`

`d_(1)=(200)/(tan(13.1166))` and
`d_(2)=(200)/(tan(50.7))`

Thus, travel distance=
`(200)/(tan(13.1166))-(200)/(tan(50.7))`

=858.32 - 163.70 = 694.62'