Light Beam Speed Along a Straight Coastline

This problem describes a classic rotating-beam setup, with a lighthouse located a fixed distance inland from a straight coastline and its light sweeping across the shore at a constant angular speed. The candidate is asked to translate the rotation of the light into the motion of the illuminated point where the beam hits the shoreline, and then to evaluate that speed both in general and at a specific distance along the coast. Variants of this style of question are common in calculus interviews and exams, especially where geometric intuition and physical interpretation are important.

Solving it leans on related rates, trigonometric relationships in a right triangle, and careful handling of angular versus linear velocity. The core step is expressing the position of the light spot along the coast as a function of the rotation angle, then differentiating with respect to time. An interviewer is watching for correct setup of the triangle, appropriate use of tangent or similar trigonometric functions, and clean manipulation of derivatives. They also look for awareness that the coastal speed can grow large far from the lighthouse, and the ability to articulate why that happens.

What it tests

This class of problems is governed by related rates in right triangle geometry, where a fixed point emits a rotating beam or line, and you track how the intersection point with a constraint (like a line or curve) moves. The key is that the angular speed of the rotating line translates nonlinearly into the linear speed of its intersection point, depending on the geometry. Specifically, as the angle increases, the tangent point moves faster along the constraint because the same angular increment sweeps out a longer arc at greater distances from the pivot. The rate at which the intersection moves is found by differentiating the geometric relationship (often involving tangent or another trigonometric function) between the angle and the position. This non-uniform translation from angular to linear speed is why the intersection can move arbitrarily fast far from the pivot, even if the angular speed is constant.