At noon a fire truck leaves a station and drives due east along a straight road at a…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
At noon a fire truck leaves a station and drives due east along a straight road at a constant speed of 40 mi/h. Five minutes later, an ambulance leaves the same station and drives due north along a road that is perpendicular to the truck’s route. The ambulance maintains a constant speed of 60 mi/h. Let t be the time (in hours) after the ambulance departs, and let D(t) be the distance (in miles) between the two vehicles at time t (for t≥0).
(a) Express D(t) in terms of t and any necessary constants that represent the initial separation of the vehicles when the ambulance departs.
(b) Compute dtdD for general t.
(c) Determine the time after the ambulance departs at which the distance between the two vehicles is a minimum. Show that your result indeed corresponds to a minimum.
(d) Find the minimum distance between the fire truck and the ambulance.