On a clear night a lighthouse sits on a cliff above sea level and from a straight…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

On a clear night a lighthouse sits on a cliff 100 m100\,\text{m} above sea level and 270 m270\,\text{m} from a straight shoreline that extends infinitely in both directions. The lighthouse beam rotates horizontally at a constant rate of 0.020 rad s−10.020\,\text{rad s}^{-1}, sweeping the shoreline. Let x(t)x(t) meters be the shortest horizontal distance along the shoreline from the foot of the cliff to the point illuminated by the beam at time tt. The angle α(t)\alpha(t) (in radians) is measured between the line from the lighthouse to the foot of the cliff and the line from the lighthouse to the illuminated point on the shoreline. (a) Express xx in terms of α\alpha. (b) Determine dxdt\frac{dx}{dt} when α=π/3\alpha = \pi/3. Include units and state whether the light spot is moving toward or away from the foot of the cliff. (c) Find the speed of the light spot along the shoreline when α=π/3\alpha = \pi/3. (Note: speed is the magnitude of velocity.) (d) Describe how dxdt\frac{dx}{dt} would change as α\alpha approaches π/2\pi/2, and provide a mathematical justification for your description.

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