Question

From the top of a 90 m lighthouse, an operator sees a capsized boat and determines an angle of depression of
12.5° to the boat. A patrol boat is also spotted at an angle of depression of 9°.
If the two boats are on the opposite side of the lighthouse, how far apart are the two boats?

Answer 1

Let's assume that the distance between the two boats is x meters.

From the top of the lighthouse, the operator sees the capsized boat and determines an angle of depression of 12.5°. This means that the angle between the horizontal line and the line connecting the operator and the capsized boat is 12.5°.

Using trigonometry, we can find the height of the capsized boat as follows:
tan(12.5°) = height of capsized boat / 90 m
height of capsized boat = tan(12.5°) * 90 m

Similarly, the angle of depression of the patrol boat is 9°. This means that the angle between the horizontal line and the line connecting the operator and the patrol boat is 9°.

Using trigonometry, we can find the height of the patrol boat as follows:
tan(9°) = height of patrol boat / 90 m
height of patrol boat = tan(9°) * 90 m

Since the two boats are on the opposite sides of the lighthouse, the total distance between them is equal to the sum of their individual heights:
Distance between the two boats = height of capsized boat + height of patrol boat
Distance between the two boats = tan(12.5°) * 90 m + tan(9°) * 90 m
Distance between the two boats = (tan(12.5°) + tan(9°)) * 90 m

Calculating this expression gives the distance between the two boats.