A vertical rain barrel is a right circular cylinder with an interior radius of

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

Problem

0.40\,\text{m}

. • Water enters the barrel through a hose at a constant rate of

0.002\,\text{m}^3\,\text{s}^{-1}

. • A crack near the base allows water to leak out at a rate (in

\text{m}^3\,\text{s}^{-1}

) given by

0.0005h

, where

h

is the instantaneous depth of the water (in

\text{m}

). At time

t

seconds after the hose is turned on, the water depth is

h(t)

. (a) Write an expression for the net rate of change of the barrel’s water volume,

\dfrac{dV}{dt}

, as a function of the current depth

h

. Include appropriate units. (b) Using the geometric relationship between the volume of water in the cylindrical barrel and its depth, derive a differential equation for

\dfrac{dh}{dt}

in terms of

h

. Show all algebraic steps. (c) Determine whether the depth is increasing or decreasing when

h = 0.60\,\text{m}

. Justify your conclusion using the differential equation from part (b) and interpret the sign of

\dfrac{dh}{dt}

in the context of the barrel.

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