Water with a dissolved pollutant flows into an inverted right circular cone whose tip…

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

Problem

Water with a dissolved pollutant flows into an inverted right circular cone whose tip is down. The cone has a height of 6 m and a top radius of 4 m. The height

h

(in meters) of the liquid is increasing at a constant rate of

\dfrac{dh}{dt}=0.05

m/min at the instant when

h=2

m. The pollutant concentration (in mg/L) is modeled by

C(V)=\dfrac{0.8}{\sqrt{V}}

, where

V

is the volume of liquid in cubic meters. The volume of liquid in the cone and the liquid height are related by

V=\dfrac{\pi h^{3}}{12}

. Assume all functions are differentiable. (a) Derive an expression for

\dfrac{dV}{dh}

in terms of

h

. (b) Obtain a formula for

\dfrac{dC}{dt}

in terms of

h

and

\dfrac{dh}{dt}

by differentiating the appropriate composition of functions. (c) Evaluate

\dfrac{dC}{dt}

numerically when

h=2

m. Include units and interpret the sign of your answer with respect to pollutant concentration.

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