% (c) 1996 Gavin LaRose (glarose@NebrWesleyan.edu)
% permission is granted to use and distribute for free
% in an academic setting
% 
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\font\headerfont=pnb10 at 15 truept
\font\sevenrm=cmr7
\def\longindent{\hskip 4in}

\font\logofont=cmti12 at 15 truept
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\def\longindent{\hskip 4in}
\def\logo{
        \longindent
        {\logofont Rocket Tech Division}
        \smallskip
        \hrule
    }
\def\address{
     \longindent Utoff A.F.\ Base\par
     \longindent 1 Piecemeal Dr.\par
     \longindent Haoma, SK\par
     \longindent 13681-0050\par
    }


\settabs\+\indent & \hskip .5in & \hskip 1.5in & \cr
 
\nopagenumbers
 
\centerline{\bf Math 105---Calculus I\quad Project 2}
\centerline{\bf constructing freedom, or something like that\dots }
\centerline{\sevenrm \copyright 1996 Gavin LaRose
(glarose@NebrWesleyan.edu), Nebraska Wesleyan University}
\centerline{\sevenrm permission granted to use and distribute for 
free in an academic setting.}

\bigskip
After your resoundingly successful debut at IMC, Inc., the company is
clearly eclipsing the number of contracts obtained in its
first year of operation.  The next contract you have been assigned
comes from the famous Rocket Tech Division of Utoff A.F.\ Base, having
to do, apparently, with the development of the new space station.  The
letter you received on the matter follows.
 
\vfill\eject
 
\logo
\smallskip
\address
\medskip
\longindent 18 October, 1996\par
 
\bigskip
\bigskip
 
\noindent Independent Mathematical Contractors, Inc.\par
\noindent Suite 2, Strawmarket Business Plaza\par
\noindent Lonlinc, SK 04685
 
\bigskip
\bigskip
\noindent
Dear IMC:
\medskip
\noindent
As you undoubtedly know, the National Space Agency (NSA) is in the
process of developing a space station, to be built in orbit in the
foreseeable future.  After having in the last year capitalized on our
success in the development and use of such defensive works as the
Otpatri missile system to carefully assess the success of a system to
protect the space station from space asteroids and debris, we are
faced now with another aspect of the planning for the construction of
the space station Omfreed.

\medskip
\noindent
In the course of the construction of the space station, it will be
necessary to send manned and unmanned Extra-Vehicular Craft (EVC)
between different units of the station being assembled.  It is
initially planned that these units be initially positioned some 500 feet 
apart.  The EVC will move in a straight path between the units, and it is
proposed that it move so that its distance from the first unit be
given by one of the two formulas
$$\eqalign{d_1(t) =& 30t(1 - e^{-.0458t}) - .339t^2 \quad\hbox{or}\cr
           d_2(t) =& 20t(1 - e^{.02t}) + .887t^2,\cr}$$ 
where distances are in feet and time $t$ in minutes.  The thrust
(force) generated by the engines of the EVC is given to be $T(r) =
.15\,r^{8/3},$ where $r$ is the rate at which its fuel is burned.  As
you also know, the great-great-grandfather of the space program, Isaac
Newton, reported that the force on an object is proportional to its
mass, $m$, times its acceleration, $a$.

\medbreak
\noindent
We need from your company:\par
{\rightskip\parindent
\item{1.} An analysis of the rate at which the fuel in the EVC is
burned for the two movement proposals,
\item{2.} An analysis of where the maximum and minimum fuel
consumption is taking place for each of $d_1$ and $d_2$,
\item{3.} An analysis of the total amount of fuel burned for both, and
\item{4.} Based on your answers to (1)--(3), a recommendation for
which of $d_1$ or $d_2$ is the preferable movement proposal.\par
}
\medbreak
\noindent
We must submit a the results of our investigation to the NSA by the
15th of November, and therefore need your finished report by the 13th
of that month.  To facilitate your work on the project, we have
arranged that you be able to contact Dr.\ Gavin LaRose, a technical
expert in many fields, with whom we have an affiliation.  Please note
that you should in any event contact him by the 25th of October with
an indication of your initial work on the project and again by the 1st
of November with further evidence of your progress.  Failure to meet
any of these deadlines will, of course, be grounds for reducing your
payment. 

\medbreak
\noindent
We look forward to receiving your report, which should be typewritten
and 3--5 pages in length.  Equations explaining your solution should
be included in the report, and it is permissible to hand-write these
in blank lines between your typewritten explanation if this greatly
facilitates the production of the report.
\bigskip

\longindent
Sincerely,
\bigskip
\bigskip
\medskip
\longindent
Lieutenant General Rick N. Backer\par
\longindent
Commander, Rocket Tech Division

\bye
