Boat A departs a harbor at noon and travels due east at a constant speed of 20 km h
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Boat A departs a harbor at noon and travels due east at a constant speed of 20 km h−1.
Boat B departs the same harbor three hours later and travels due north at a constant speed of 25 km h−1. Let t be the number of hours since Boat A departed (so Boat B begins its motion at t=3). East is the positive x–direction and north is the positive y–direction.
(a) Write an explicit formula for the distance D(t) between the two boats for all t≥0. State any domain restrictions clearly.
(b) Derive an expression for dtdD for all t in the domain found in part (a). Do not simplify numerical coefficients at this stage.
(c) At the moment Boat B has traveled exactly 40 km, determine the rate at which the distance between the boats is changing. Include correct units and indicate whether the distance is increasing or decreasing.
(d) Determine the time after noon when the two boats are closest together, and compute that minimum distance. Justify that your answer indeed corresponds to a minimum.