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\begin{center} 

{\large \bf Extra Credit}

{\bf Math 305}

{\bf Equals one homework set}

\end{center}

A chemical spill has polluted a pond. Your company,
Clean Ponds Inc., has been contracted to clean the 
pond. Federal regulations require that 90 \% of the 
pollutant be removed within one month (31 days). 
The pond has 600,000 gallons of water in it.

The following pump/filter systems are available:
\vspace{.14in}

\begin{center}
\begin{tabular}{|r||c|c|c|}
\hline
name & Econo-Pump & Super Filter     & Pump Master          \\
     & and Filter &   System         & with Filter          \\
\hline
\hline
cost &          \$150,000   &         \$250,000        &       \$350,000          \\
\hline
pump rate &   4000 gal/hr   &       3000 gal/hr        &        5000 gal/hr      \\
\hline
efficiency  &   40 \%       &        75 \%             &        65 \%            \\
\hline
\end{tabular}
\end{center}
\begin{quote}{\small
Notes to Table: The cost listed is the minimum rental charge per month.
Filter efficiency is the percentage of pollutant removed on each 
pass through the filter.
}\end{quote}


A) Which pump/filter system should you get.

Note: Since the pump sends the water back into the pond after it is filtered,
the water coming in is part new water and part filtered water. Assume the 
water in the pond is well mixed. (See figure)



% @@@@@@@@@@@@@@@@@@@@@@@@@@@@ FIGURE @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
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		\epsfxsize=2in
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		\epsfbox{pond.eps}  
		}
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	%\label{}
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% @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

B) Becauise of a lawsiut by an environmental group you must
get 98 \%  of the polluntant out in one month (31 days). Now what 
do you do?
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{\sf
\begin{center} Solution \end{center}

Step 1: Show that the system is modeled by the differential equaltion:
\[
\frac{dP}{dt} = ER(1-P)/V,
\]
where $P$ is the percent of the pollutant removed at time $t$, $E$ is the
efficiency, $R$ is the pump rate, and $V$ is the total volume of water 
in the pond.

Do this by setting up a difference equation and taking limits.

Step 2: Solve this to get $P=1 -  \exp(-ERt/V)$ or $t=-V (\ln 1-P)/ER.$

Step 3: In part A it is easy to check the SFS is the best choice.

Step 4: In part B if you said PMF you're fired. It is cheaper by 
\$50,000 to get two EPFs!
}
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