Aperson 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?

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.

Ke reqd to move from 0 to 50 will be proportional to 50x50 or 2500. this is equal to w.  to change vel from 50 to 150 you need energy prop to 22500 - 2500 = 20,000. this corresponds to 8w.  best answer is a) . work reqd is prop to velocity squared.