Geometry

Question Analysis

The main focus is to use the angle of depression from the top of the lighthouse to the boat and the height of the lighthouse to find the horizontal distance between the boat and the base of the lighthouse. We will apply trigonometric relations in the right - triangle formed by the lighthouse, the water surface, and the line of sight to the boat.

Key Concept Explanation

1. Angle of Depression: It is the angle between the horizontal line of sight from an elevated point (top of the lighthouse) and the line of sight to an object below (the boat). In a right - triangle context, this angle helps in establishing relationships between the sides of the triangle.

2. Trigonometric Functions in Right - Triangles: For an acute angle  in a right - triangle, , , . Given the height of the lighthouse (opposite side) and wanting to find the adjacent side (distance of the boat from the base of the lighthouse), the tangent function is the correct one to use.

Step - by - Step Solution

1. Let the distance of the boat from the base of the lighthouse be  meters.

The height of the lighthouse (opposite side)  meters and the angle of depression .

Since the angle of depression is equal to the angle of elevation from the boat to the top of the lighthouse, and using the tangent function

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