An airplane is flying due north at an altitude of 2 mi directly above a straight…

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

Problem

An airplane is flying due north at an altitude of 2 mi directly above a straight east–west highway. At noon the plane passes vertically above a point

P

on the highway and maintains a speed of 300 mi h

^{-1}

relative to the ground. A car travels due east along the highway at 60 mi h

^{-1}

and passes point

P

at the same moment the airplane is overhead. Let

t

be the number of hours after noon. (a) Express the straight–line distance

S(t)

between the airplane and the car in terms of

t

. Clearly state any geometric relationships you use. (b) Calculate

\dfrac{dS}{dt}

when the horizontal ground distance between the airplane and point

P

is 5 mi. Give units with your answer. (c) Interpret the sign of

\dfrac{dS}{dt}

obtained in part (b) within the context of the problem.