30 pointsexplain too! 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.

Initial: 200 feet = opp. side
Angle = 17.31, adj. Side = a
So tan (17.31) = opp. Side / adj. Side
tan (17.31) = 200 / a
--> a = 200/tan17.31 = 641.73
When stopped: adj. Still = a
Opp. Side still = 200, angle = 46.41
tan46.41 = 200 / a
--> a = 200/tan46.41 = 190.39
Now subtract those two distances to get the distance traveled from initial to when it stopped:
641.73 - 190.39 = 451.34 feet

445.10 feet to the nearest hundredth.

Step-by-step explanation:

First work out the distance of the boat from the lighthouse  when the angle of depression was 17°31':

tan 17°31'  = 200 / x     where x is the distance.

x = 200 / tan 17°31'

= 633.676 feet.

Now do the a similar calculation to find the distance at the second reading:

tan 46°41' = 200 / x

x = 200 / tan 46°41'

=  188.580 feet

Therefor the distance the boat travelled

= 633.676 -  188.580

= 445.10 feet.

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