A spotlight on a boat is 2.5 m above the water, and the...

A spotlight on a boat is $2.5 \mathrm{~m}$ above the water, and the light strikes the water at a point that is $8.0 \mathrm{~m}$ horizontally displaced from the spotlight (see the drawing). The depth of the water is $4.0 \mathrm{~m}$. Determine the distance $d$, which locates the point where the light strikes the bottom.

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Key Concepts

Trigonometric Relationships

Trigonometry, specifically the tangent function, is key in relating the vertical height differences to the horizontal displacements of the ray. By calculating the tangent of the angle formed by the light’s path in the air, one can apply the same relationship to the underwater segment, assuming a straight-line propagation, to determine the additional horizontal distance traveled.

Coordinate Geometry

Using a coordinate system helps in visualizing and solving the problem by placing the light source, the water surface, and the bottom at specific coordinate points. This approach allows one to derive the equation of the line that represents the light ray’s trajectory and subsequently determine where it intersects the bottom.

Similar Triangles

When a light ray passes from one level to another that has a proportional vertical and horizontal displacement, the triangles formed by the source, the surface, and the point of incidence are similar to those formed between the surface and the bottom. This similarity allows one to relate known distances above the surface to unknown distances below by maintaining the ratio of vertical to horizontal components.