An opaque cylindrical tank with an open top has a diameter...

An opaque cylindrical tank with an open top has a diameter of $3.00 \mathrm{~m}$ and is completely filled with water. When the setting Sun reaches an angle of $28.0^{\circ}$ above the horizon, sunlight ceases to illuminate any part of the bottom of the tank. How deep is the tank?

Key Concepts

Parallel Rays Assumption

Because the sun is extremely far away, the rays of sunlight striking the tank can be considered parallel. This simplification lets us use straightforward geometric and trigonometric reasoning to analyze how the light is blocked by the tank’s side walls without having to account for diverging angles or curvature in the paths of the rays.

Geometry of Shadows in Enclosures

The problem relies on understanding how an object – in this case the curved walls of a cylindrical container – casts a shadow on a surface (the bottom) when illuminated by parallel light rays. The critical condition is determined by the ray that just grazes the edge of the bottom, so that no point remains directly exposed to sunlight. Analyzing this geometric constraint provides the basis for finding the required depth of the tank.

Similar Triangles

Similar triangles arise when relating the vertical height of the tank to the horizontal displacement of the light ray from the top edge to the bottom. By identifying triangles that share the same angle (specifically the sun’s altitude) and have proportional sides, one can set up a relation that links the dimensions of the tank with the sun’s angle, leading directly to the solution.

Trigonometric Relationships

The use of trigonometric functions—particularly the tangent function—is central to this problem. The tangent of the sun’s altitude relates the vertical drop (the tank’s depth) to the horizontal distance that the ray travels inside the tank. This trigonometric relationship allows one to translate the geometric condition of the ray barely reaching the bottom into an equation that can be solved for the tank’s depth.

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