:( 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 17°31'. when the boat stops, the angle of depression is 46°41'. 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.

The tangent of the angle gives the ratio of the height of the lighthouse to the boat's distance
.. tan(angle of depression) = (lighthouse height)/(boat distance)
Then the distance to the boat is
.. boat distance = (lighthouse height)/tan(angle of depression)

We want to find the difference between two distances, so
.. boat travel = (lighthouse height)*(1/tan(first angle) -1/(tan(second angle))
.. = (200 ft)*(1/tan(17°31') -1/tan(46°41'))
.. ≈ 445.10 ft