From the lantern room of a –tall lighthouse

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

Problem

From the lantern room of a

30\,\text{m}

–tall lighthouse, a coast-guard officer sights a small boat on the ocean surface. The officer measures the angle of depression

\alpha(t)

from the horizontal to the boat. At the moment when

\alpha = 0.40\,\text{rad}

, the angle is decreasing at a rate of

0.015\,\text{rad}\,\text{s}^{-1}

. Assume the boat is moving directly away from the base of the lighthouse along a straight line. (a) Express the horizontal distance

x

(in metres) between the boat and the base of the lighthouse as a function of

\alpha

. (b) Determine the speed of the boat at the instant described. Give both the value and the units.