Water flows into an inverted right circular conical tank at a constant rate of

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

Problem

3\text{ m}^3\!\, /\!\, \text{min}

. The tank has a height of

10\text{ m}

and a top radius of

5\text{ m}

. The apex of the cone is at the bottom. (a) Let

h(t)

be the depth of the water at time

t

. Express the volume

V

of water in the tank as a function of

h

. (b) Derive a formula for

\dfrac{dh}{dt}

in terms of

h

using your result from part (a) and the given inflow rate. (c) Calculate

\dfrac{dh}{dt}

when the water depth is

6\text{ m}

. Include units. (d) Find the instantaneous rate of change of the radius of the water’s surface when

h=6\text{ m}

, and interpret your answer within the context of the filling process.