During a winter storm

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

Problem

During a winter storm, the volume

V

(in cubic meters) of a conical snow pile in a parking lot is empirically related to its vertical height

h

(in meters) by the model

V(h)=12h^{2.1}.

At a particular instant the snow pile is

1.50\text{ m}

tall, and its height is increasing at a constant rate of

0.030\text{ m h}^{-1}

. (a) Compute

V'(h)

, including correct units. (b) Determine the instantaneous rate of change of the pile’s volume with respect to time when

h=1.50\text{ m}

. (c) Using a differential or linear approximation, estimate how long (in hours) it will take for the volume of the pile to increase by

1.0\text{ m}^{3}

from its value at

h=1.50\text{ m}

. (d) Critically discuss one limitation of relying on this linear approximation over extended time intervals.