A patrol vessel passes a fixed lighthouse at noon

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A patrol vessel passes a fixed lighthouse at noon, sailing on a straight course that will carry it directly away from the lighthouse. The vessel’s speed is 18 km/h18\text{ km/h}. At 12:20 p.m. the captain notes that the bearing angle θ\theta between the vessel’s course and the line of sight back to the lighthouse is π6\dfrac{\pi}{6} radians. Let d(t)d(t) be the vessel’s distance (in kilometers) from the lighthouse at tt hours after noon, and let θ(t)\theta(t) be the corresponding bearing angle (in radians). (a) Establish a relationship between dd and θ\theta. (b) Determine dθdt\dfrac{d\theta}{dt} (in rad/h) at 12:20 p.m. (c) Compute the time after noon when θ\theta reaches π4\dfrac{\pi}{4} radians. Round your answer to the nearest minute.

Related theory

Browse the problem library