Submarine A departs point O and travels due east along a straight course
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Submarine A departs point O and travels due east along a straight course. Its speed in knots (nautical miles per hour) is given by vA(t)=6+0.10t, where t is measured in hours after departure. At the same instant, Submarine B is located 30 nmi east and 40 nmi south of point O and travels due north with speed vB(t)=8+0.05t knots.
Use a coordinate system with origin at O, x-axis pointing east, and y-axis pointing north.
(a) Determine the position of each submarine as a function of time for t ≥ 0 h.
(b) Obtain an expression for the rate of change of the squared distance between the submarines, dtd[D(t)2], in terms of t.
(c) Find the time t (0 ≤ t ≤ 0.5 h) at which the submarines are closest to each other. Show that this time yields an extremum and classify it.
(d) At the time found in part (c), compute the rate at which the actual distance D(t) is changing and interpret the result in context.