Explanation:
I assume the boat is going straight out on the same horizontal line.
we have 2 right-angled triangles.
at point A
there is the ground distance from A to the lighthouse. which is 1390 ft.
then there is the height of the lighthouse.
and the line of sight from A to the lighthouse light.
the angle at A is 7°.
the angle at the bottom of the lighthouse is 90°.
therefore, the angle at the lighthouse light = 180 - 90 - 7 = 83°.
at point B
the ground distance from B to the lighthouse.
the height of the lighthouse again.
and the line of sight of B to the lighthouse light.
the angle at B = 4°.
the angle at the lighthouse bottom is again 90°.
therefore, the Angie at the lighthouse light is 180 - 90 - 4 = 86°.
we need the law of sine to solve this and find the missing side lengths.
a/sin(A) = b/sin(B) = c/sin(C)
with the sides and the correlated angles being opposite.
the first we need to do is finding the height of the lighthouse via triangle A, so that we can use it then also in triangle B.
1390/sin(83) = height/sin(7)
lighthouse height = 1390×sin(7)/sin(83) = 170.6705397... ft
now, we can use that in triangle B to get the ground distance to the lighthouse :
height/sin(4) = ground distance/sin(86)
ground distance = height×sin(86)/sin(4) =
= 2,440.702427... ft
the distance from A to B is the difference between that ground distance and the original distance of 1390 ft :
2,440.702427... - 1390 = 1,050.702427... ft ≈
≈ 1,051 ft