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 14°52'. when the boat stops, the angle of depression is 45°10'. the lighthouse is 200 feet tall. how far did the boat travel from when it was first noticed until it stopped?

The angle of depression from the lighthouse is the angle of elevation from the boat. The line of sight from the top of the lighthouse, when depressed, becomes an alternate interior angle of two parallel lines, the line of sight from the top and the boat's movement.

Step-by-step explanation:

From the boat's standpoint, which is what we want anyway, there is a right triangle, with the distance from the lighthouse the adjacent side, the opposite is 200,' and the boat the point desired. The tangent is needed.  

tangent 18.33 degrees (33 minutes is 33/60 of a degree)=200/x

x=200/tan 18.33=592.58 feet

nearer, the angle of elevation increases as the adjacent side gets smaller relative to the opposite side.

tangent 51.65 degrees (33 minutes is 33/60 of a degree)=200/x

x=200/tan 51.65=158.23 feet

That difference is 434.35 feet