A storage vessel has a variable cross-sectional area given by A(h)=0.25h^ 2 +0.80 (m…

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

Problem

A storage vessel has a variable cross-sectional area given by

A(h)=0.25h^{2}+0.80 \quad \text{(m}^2\text{)}

where

h

(in meters) is the depth of water measured from the bottom. Water enters the vessel at a time-dependent rate

R(t)=0.05e^{0.02t}\quad \text{(m}^3\text{ min}^{-1})

for

t

measured in minutes since the inflow began. Let

h(t)

denote the water depth at time

t

. (a) Formulate a differential equation expressing

\dfrac{dh}{dt}

in terms of

t

and

h

by equating the inflow rate to the rate of volume increase in the vessel. (b) Evaluate

\dfrac{dh}{dt}

at the instant

t=20

min when the measured depth is

h=3.0

m. Include units and interpret whether the water level is rising quickly or slowly at that moment.