Water flows into a right circular conical tank whose pointed end is downward

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

Problem

Water flows into a right circular conical tank whose pointed end is downward. The tank has a height of

12\text{ m}

and a base radius of

4\text{ m}

. Beginning at

t=0

minutes, water is pumped into the tank at a rate of

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

, but simultaneously leaks out through a small hole in the bottom at a rate of

6t\text{ m}^3\text{/min}

. Let

h(t)

be the depth of the water (in meters) inside the cone at time

t

minutes, and assume the water always forms a similar smaller cone. (a) Express the volume

V

of water in the tank as a function of the depth

h

. (Do not substitute numerical values yet.) (b) Using similar triangles, rewrite

V

solely in terms of

h

, then differentiate to obtain

\dfrac{dh}{dt}

in terms of

h

and

t

. Eliminate

h

so that

\dfrac{dh}{dt}

is expressed only in terms of

t

. (c) Determine the time

t_\text{max}

at which the water depth attains its greatest value. Justify that your result indeed corresponds to a maximum of

h(t)

. (d) Find the maximum water depth

h(t_\text{max})

, giving your answer to three significant figures and including units.