From the top of a 90 m lighthouse, an operator sees a capsized boat and determines an angle of depression of

Question

An oceanic scene with a 90-meter tall lighthouse standing sturdily on an island. The lighthouse is in the center. To the left, there's a boat flipped upside down representing a capsized boat on a relatively calm sea. On the right side, another boat is visible, this one upright and seeming more like a patrol boat. The ocean extends indefinitely behind the boats. The angles mentioned in the text are subtly indicated through the directions of the lighthouse beam towards the position of boats but without using numerical representations. There's a clear sky overseeing the scene with light clouds.

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°.
How far from the lighthouse is the patrol boat?
How far from the lighthouse is the capsized boat?
If the two boats are on the opposite side of the lighthouse, how far apart are the two boats?

Answer 1

To solve this problem, we can use trigonometry.

Let's label the distance from the lighthouse to the patrol boat as "x" and the distance from the lighthouse to the capsized boat as "y".

From the given information, we can apply the tangent function to find the distances.

For the patrol boat:
tan(9°) = x/90
x = 90 * tan(9°)
x ≈ 14.99 meters

For the capsized boat:
tan(12.5°) = y/90
y = 90 * tan(12.5°)
y ≈ 19.65 meters

To find the distance between the two boats, we can subtract the distances from the lighthouse to each boat.
Distance between the two boats = y - x
Distance between the two boats ≈ 19.65 - 14.99
Distance between the two boats ≈ 4.66 meters