Sand spills from a conveyor belt onto a conical pile so that the height of the pile…

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

Problem

Sand spills from a conveyor belt onto a conical pile so that the height of the pile is always one-half its radius. Sand is added at a constant rate of

9\text{ m}^3\!\!\big/\!\!\text{min}

. At time

t

, let

h

be the height (in meters) of the pile and

V

its volume. Recall that the volume of a right circular cone of radius

r

and height

h

V=\dfrac{1}{3}\pi r^{2}h

. (a) Express the volume

V

solely in terms of

h

. (b) Differentiate your expression from part (a) with respect to time to find

\dfrac{dh}{dt}

when

h=1\text{ m}

. (c) At the same instant, determine the rate at which the radius of the pile is increasing.

Related theory

Motion
Related Rates
Rates of change
Rate of change of quantities

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