% (c) 1996 Gavin LaRose (glarose@NebrWesleyan.edu)
% permission is granted to use and distribute for free
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     \lower 1.5ex \vbox{\headerfont Chemproc, Inc.} }
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\centerline{\bf Math 105---Calculus I\quad Project 2}
\centerline{\bf \dots living in an industrial world}
\centerline{\sevenrm \copyright 1995 Gavin LaRose
(glarose@NebrWesleyan.edu), Nebraska Wesleyan University}
\centerline{\sevenrm permission granted to use and distribute for 
free in an academic setting.}
\bigskip
As highly successful independent mathematical and scientific technical
contractors, celebrating the successful completion of your largest
contract to date, you have just received a new contract---clearly a
direct result of your past success.  The letter you received from
Chemproc, Inc. (recently purchased by the MPC, Inc.~conglomerate)
on the subject of their recently initiated chemical processing factory
follows.
\vfill\eject

\logo
\medskip
\longindent 20000 Ryan-ears Blvd.\par
\longindent Lonlinc, SK\par
\longindent 04685\par
\medskip
\longindent 8 October, 1995\par

\bigskip
\bigskip

\noindent Independent Mathematical Contractors, Inc.\par
\noindent Suite 2, Strawmarket Business Plaza\par
\noindent Lonlinc, SK 04685\par
\bigskip
\bigskip

\noindent Dear IMC:
\medskip
\noindent
As you know, Chemproc, Inc.\ has recently opened a chemical
reprocessing plant in East Lonlinc.  After an immensely successful
start, we are beginning calibration and fine-tuning of the second of
our four reprocessing lines.  This line is concerned with the
reclamation of gold from waste-chemicals produced by other
companies---which then contract with us to recover the gold.  (The
gold is used as a catalyst for the reactions with which they are
concerned, and can be reused if properly extracted.)  However, there
are two technical issues about which we are contacting you that
must be resolved before we are able to begin operation of the line. 

\medskip
\noindent
First, we are in the process of purchasing the chemicals that we use
in this reclamation and need to determine which of these is the most
effective.  We obtained for
demonstration purposes a sample of the chemicals being offered by each
of three different suppliers, and ran our reclamation line using these
samples to determine which produced the best results.

\medskip
\noindent 
The results of this test run are given in the table which
follows\footnote{$^1$}{If it should prove useful, Dr.\ LaRose can
provide you with a soft-copy of this data.};
this was compiled by our consulting scientist, Dr.\ Gavin 
LaRose (recently hired from the private art sector by way of the
enticement of a large industrial salary).  It gives the amount of gold
reclaimed as a function of time.  To facilitate the run, we did not
restart the line with each different chemical sample---instead, we ran
it with the first sample, waited briefly for the chemical to clear,
and then added the second, etc.  The first chemical was added at time
0, the second at the 5 minute mark, and the third after 10 minutes.

\medskip
\noindent
In order to determine which of the suppliers we should deal with, we
need to know several things based on this test.  We need to know which
supplier's chemical was the most effective, and we need to know which
produced the greatest reclamation rate when working at its peak
efficiency---for, as you can tell from the data, the reclamation rate
varies with time even while a single chemical is being used.  The
results that you obtain should be quantitative (numerically
substantiated).

\medskip
\noindent
Second, Dr.\ LaRose has determined that the amount of gold reclaimed
as a function of time is approximately given by the formula
$$ R(t) = a\;(\,1 - .003\,(b - t)\,e^{-c\,t}\,), $$
where $a$ is the total amount of gold reclaimed using the chemical in
question, $b$ and $c$ are 
constants determined by the chemicals used in the reaction, and $t$ is
given in seconds.  Based on the data with which we have supplied you,
we need to know the correct values for these constants,
and then, based on the resulting model $R(t)$, how long it will take
for 50\%, 90\% and 99\% of the reclamation to take place.  We need
this information for the two chemicals giving the best results
(as determined by your earlier investigation).

\medskip
\noindent
As specified in your contract, your report should be submitted by the
27$^{\rm th}$ of this month so that we are able to begin full
operation of the 
line by November (and therefore impress our stock holders).  If you
should find in the course of your investigation that you have
questions regarding this project, you are to contact Dr.~LaRose with
the other members of your investigative team; he will be more than
willing to assist you.  You should further contact him (also as a
team) by the 17$^{\rm th}$ to report on your progress.

\medskip
\noindent 
We look forward to seeing your finished report, which should conform
to the enclosed technical report requirements.
\bigskip

\longindent Yours sincerely,
\bigskip
\bigskip
\bigskip
\longindent E. Idu Pont, President\par
\longindent Chemproc, Inc.
\bigskip
\bigskip
\bigskip
\noindent Encl: technical report requirements

\vskip .75in
\hrule
\bigskip
\centerline{{\bf Gold Recovery Data for Test Run} ($t$ in seconds)}
\medskip
\hrule
\medskip
\settabs 19 \columns
\+& $t$ & $R(t)$ && $t$ & $R(t)$ && $t$ & $R(t)$&& $t$ & $R(t)$&& $t$
& $R(t)$&& $t$ & $R(t)$\cr 
\smallskip
\hrule
\smallskip

\+& 0 &       0 &&            155 &     0.272371 &&   305 &
0.294328 &&   455 &     0.576046 &&   605 &     0.629450 &&  755 &
0.768905 \cr
 
\+& 5 &	0.0383053 &&    160 &     0.272737 &&	310 &     0.312679 &&
460 &     0.579587 &&   610 &     0.638378 &&  760 &      0.769586 \cr

\+& 10 &	0.071275 &&     165 &     0.273052 &&	315 &
0.330099 &&   465 &     0.582926 &&   615 &     0.651469 &&  765 &
0.770197 \cr
 
\+& 15 &	0.0996523 &&    170 &     0.273323 &&	320 &
0.346633 &&   470 &     0.586073 &&   620 &     0.66334 &&   770 &
0.770745 \cr
 
\+& 20 &	0.124077 &&     175 &     0.273557 &&	325 &
0.362324 &&   475 &     0.589038 &&   625 &     0.674103 &&  775 &
0.771235 \cr
 
\+& 25 &	0.145099 &&     180 &     0.273758 &&	330 &
0.377212 &&   480 &     0.59183 &&    630 &     0.683858 &&  780 &
0.771675 \cr
 
\+& 30 &	0.163193 &&     185 &     0.273931 &&	335 &
0.391337 &&   485 &     0.594459 &&   635 &     0.692698 &&  785 &
0.772068 \cr
 
\+& 35 &	0.178767 &&     190 &     0.27408 && 	340 &
0.404735 &&   490 &     0.596932 &&   640 &     0.700707 &&  790 &
0.77242 \cr
  
\+& 40 &	0.192172 &&     195 &     0.274208 &&	345 &
0.417441 &&   495 &     0.599258 &&   645 &     0.707962 &&  795 &
0.772734 \cr
 
\+& 45 &	0.203709 &&     200 &     0.274318 &&	350 &
0.42949 &&    500 &     0.601445 &&   650 &     0.714531 &&  800 &
0.773015 \cr
 
\+& 50 &	0.213639 &&     205 &     0.274413 &&	355 &
0.440913 &&   505 &     0.603501 &&   655 &     0.720479 &&  805 &
0.773265 \cr
 
\+& 55 &	0.222186 &&     210 &     0.274495 &&	360 &
0.451741 &&   510 &     0.605431 &&   660 &     0.725862 &&  810 &
0.773487 \cr
 
\+& 60 &	0.229543 &&     215 &     0.274565 &&	365 &
0.462002 &&   515 &     0.607243 &&   665 &     0.730732 &&  815 &
0.773686 \cr
 
\+& 65 &	0.235875 &&     220 &     0.274626 &&	370 &
0.471726 &&   520 &     0.608942 &&   670 &     0.735138 &&  820 &
0.773862 \cr
 
\+& 70 &	0.241324 &&     225 &     0.274678 &&	375 &
0.480937 &&   525 &     0.610536 &&   675 &     0.739123 &&  825 &
0.774018 \cr
 
\+& 75 &	0.246015 &&     230 &     0.274723 &&	380 &
0.489662 &&   530 &     0.61203 &&    680 &     0.742725 &&  830 &
0.774157 \cr
 
\+& 80 &	0.250053 &&     235 &     0.274761 &&	385 &
0.497924 &&   535 &     0.613429 &&   685 &     0.74598 &&   835 &
0.77428 \cr
  
\+& 85 &	0.253528 &&     240 &     0.274795 &&	390 &
0.505746 &&   540 &     0.614738 &&   690 &     0.748922 &&  840 &
0.774388 \cr
 
\+& 90 &	0.256518 &&     245 &     0.274823 &&	395 &
0.51315 &&    545 &     0.615962 &&   695 &     0.751578 &&  845 &
0.774484 \cr
 
\+& 95 &	0.259093 &&     250 &     0.274848 &&	400 &
0.520157 &&   550 &     0.617105 &&   700 &     0.753976 &&  850 &
0.774569 \cr
 
\+& 100 &	0.261309 &&     255 &     0.274869 &&	405 &
0.526786 &&   555 &     0.618173 &&   705 &     0.756141 &&  855 &
0.774643 \cr
 
\+& 105 &	0.263216 &&     260 &     0.274887 &&	410 &
0.533056 &&   560 &     0.61917 &&    710 &     0.758094 &&  860 &
0.774708 \cr
 
\+& 110 &	0.264857 &&     265 &     0.274903 &&	415 &
0.538986 &&   565 &     0.620099 &&   715 &     0.759855 &&  865 &
0.774765 \cr
 
\+& 115 &	0.26627 &&	     270 &     0.274917 &&	420 &
0.544592 &&   570 &     0.620964 &&   720 &     0.761442 &&  870 &
0.774815 \cr
 
\+& 120 &	0.267486 &&     275 &     0.274928 &&	425 &
0.549891 &&   575 &     0.621768 &&   725 &     0.762873 &&  875 &
0.774858 \cr
 
\+& 125 &	0.268533 &&     280 &     0.274938 &&	430 &
0.554898 &&   580 &     0.622516 &&   730 &     0.764161 &&  880 &
0.774895 \cr
 
\+& 130 &	0.269433 &&     285 &     0.274947 &&	435 &
0.559628 &&   585 &     0.62321 &&    735 &     0.765321 &&  885 &
0.774928 \cr
 
\+& 135 &	0.270209 &&     290 &     0.274954 &&	440 &
0.564094 &&   590 &     0.623853 &&   740 &     0.766365 &&  890 &
0.774956 \cr
 
\+& 140 &	0.270876 &&     295 &     0.274961 &&	445 &
0.568311 &&   595 &     0.624449 &&   745 &     0.767304 &&  895 &
0.77498 \cr
  
\+& 145 &	0.271451 &&     300 &     0.274966 &&	450 &
0.572291 &&   600 &     0.625 &&      750 &     0.768147 &&  900 &
0.775 \cr
    
\+& 150 &	0.271945\cr
\medskip
\hrule

\vfill
\eject

\logo
\bigskip
\centerline{\bf Technical Report Requirements}
\bigskip
\noindent
It is of paramount importance that the reports submitted to Chemproc,
Inc. be written so that all members of the board of directors can
understand and apply the information contained therein.  These board
members are all experts in their respective areas of chemical
processing, but for their mathematical background should be assumed
only to remember algebra and precalculus.
\bigskip
\noindent Reports should further:
\bigskip
\item{$\bullet$} Be written in the first person plural.
\medskip
\item{$\bullet$} Include mathematical formulas in the report as
appropriate to describe the methods used and results obtained.
\medskip
\item{$\bullet$} Clearly explain how the mathematical formulas that
are included bear on the problem being solved, and how the solution(s) 
obtained answer(s) the question(s) that the report is to answer.
\medskip
\item{$\bullet$} Consist of 
\smallskip
\itemitem{$\bullet$} An {\sl Introduction\/}, indicating generally the
problem(s) to be solved, and giving an explanation of the organization
of the remainder of the report.
\smallskip
\itemitem{$\bullet$} A section for each of the problems considered,
each having
\smallskip
{\narrower
\itemitem{$\bullet$} An {\sl Introduction\/}, specifically
describing the 
problem to be solved, and an indication of the mathematical method used to
solve it.
\smallskip
\itemitem{\qquad $\bullet$} A {\sl Body\/}, describing the mathematical
problem that was solved to answer the question(s) posed in the
introduction, and the solution to it.
\smallskip
\itemitem{\qquad $\bullet$} A {\sl Conclusion\/}, summarizing the
results 
obtained from the solution described in the body and clearly stating
their relevance to the original problem as described in the
introduction. 
\medskip
}
\item{$\bullet$} Be 2$1\over 2$--5 pages in length.

\bye
