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\vskip -2.75in \vtop{\hsize=4.25in \noindent {\bf Problem statement} A
waste holding tank for an industrial process is constructed as shown
to the right. The cross-sectional area of the holding tank, a
cylinder, is 5 square feet, and the tank's height is 15 feet.  Assume
that the effluent is entering the top of the tank using the sluiceway
shown and the rate of fluid entering the tank is modeled by the
periodic function $f(t)= 3.5+\sin\left({{2\pi}\over{24}}t\right)$.
$f(t)$ is measured in cubic feet per hour and $t$ is measured in
hours, with $t=0$ being midnight.

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\noindent Each of the pipes which empty the tank has a carrying
capacity of 2 cubic feet per hour. One pipe is always open. The other
pipe is open from $t=12$ (noon) until $t=24$ (the next midnight). You
may assume that when a pipe is open, its carrying capacity is fully
used.

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\noindent The tank at time $t=0$ contains 10 cubic feet of fluid, so
the fluid depth is 2 feet. What is the depth at the next
midnight, when $t=24$? Does the fluid overflow the tank during the
first 24 hours, where $0\le t\le 24$?} 

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\noindent If this model is accurate, does the fluid ever overflow the
tank?




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