Water is being pumped into an inverted right circular conical tank at a rate of

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

Problem

5\,\text{ft}^{3}\!/\text{min}

. Simultaneously, water leaks from a small hole at the bottom at

0.2\,h

cubic feet per minute, where

h(t)

is the depth of the water in feet at time

t

. The tank is

12\,\text{ft}

deep and

10\,\text{ft}

across at the top, and the water surface is always circular. (a) Express the radius of the water surface in terms of the water depth

h

. (b) Write the volume of water in the tank solely as a function of

h

. (c) Formulate a differential equation for

\dfrac{dh}{dt}

that incorporates both the pumping rate and the leakage rate. (d) When

h=6\,\text{ft}

, compute

\dfrac{dh}{dt}

and state whether the water level is rising or falling at that instant. (e) Determine all water depths (if any) for which the water level remains constant over time, and classify the stability of such a depth based on the sign of

\dfrac{dh}{dt}

nearby.

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