A tugboat departs a dock at the origin at time h and travels due east
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A tugboat departs a dock at the origin at time t=0 h and travels due east. Its speed in knots (nautical miles per hour) after t hours is given by vT(t)=5+0.2t. At the same moment, a sailboat is located 2 nautical miles due north of the dock and travels due south at a constant 7 knots.
Let t denote hours since the tugboat departed.
(a) Express the squared distance Q(t) between the two vessels as a function of t. Explain your coordinate assignments.
(b) Derive a simplified expression for dtdQ and hence for dtdQ, the instantaneous rate of change of the actual distance.
(c) Determine all times t>0 (if any) at which the distance between the vessels is instantaneously not changing. Show all algebraic steps leading to your conclusion.
(d) For each time found in part (c), decide whether it corresponds to a minimum distance between the vessels and compute that minimum distance to three decimal places.