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.. . Is the answer -706.44 ft?

Answer 1

There are several information's already given in the question. Based on those given information's the answer can be easily deduced. 
First angle of depression = 18°33'
                                        = [18 + (33/60)] degree
                                        = 18.55 degrees
Second angle of depression = 51°33'
                                              = [51 + (33/60)] degrees
                                              = 51.55 degrees
Height of the lighthouse = 200 feet
Then
Distance traveled by the boat from when it
was first noticed until it stopped = (200/tan 18.55) - (200/tan 51.55)
                                                   = (200/0.3356) - (200/1.2594)
                                                   = 594.95 - 158.81
                                                   = 436.14 feet
I hope the procedure is clear enough for you to understand.

Answer 2

The answer is 437.21 after rounding