% (c) 1996 Gavin LaRose (glarose@NebrWesleyan.edu)
% permission is granted to use and distribute for free
% in an academic setting
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\def\logo{\line{\hfil
%        \longindent
        {\logofont Satellite Allocation Division}\hfil}
        \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 3}
\centerline{\bf \dots visions spinning around and around}
\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
Your project solution for the Rocket Tech Division housed at Utoff
A.F.\ Base was so successful that IMC, Inc. has received another
contract from Utoff, this one from the Satellite Allocation Division
(SAD) of the Base.  In light of your work on the previous contract,
you have been assigned this one---as the pi\`ece de r\'esistance of your
mathematical consulting career at IMC, as it were.  The background
information from Utoff follows.
 
\vfill\eject
 
\logo
\medskip
\address
\medskip
\longindent 21 November, 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 the nation moves steadily towards the establishment of a space
station, there is a necessity to more completely explore and chart the
nature of the area of space closest to the Sun.  To this end, we are
investigating the possibility of putting a near-space satellite
(tentatively christened Sidney) in an elliptical orbit about the Sun
that would extend in its longest dimension somewhat past the orbit of
Mars and in its shortest slightly outside that of Mercury.

\medskip
\noindent
A significant consideration for such a satellite is of course the
energy, $E(t)$, required to transmit the signal which it returns to
Earth.  Aspects of this are proportional to the distance from the
satellite to Earth.  Sidney's elliptical orbit will be given, relative
to the Earth's position, by $(x_s(t)/a)^2 - (x_s(t)/a) + (y_s(t)/b)^2
= 1$.  Here $x_s(t)$ and $y_s(t)$ are the $x$ and $y$ positions of the
satellite as functions of time, and $a$ and $b$ are the major- and
minor-axes of the ellipse.  A special case which you may find useful
to consider is that in which the major- and minor-axes of the
elliptical orbit are given to be $a= 7.5\times10^{10}$ meters and
$b=2.5\times10^{11}$ meters, but we do not require that this special
case be explicitly considered in your project report.  We need from you
some sense of what the maximum and minimum values of $E(t)$ are and,
more importantly, where they occur.

\medbreak
\noindent
In order to submit our analysis of the satellite Sidney to the
National Space Agency by the close of the year, we must have your
report by the 13th of December.  Owing to our close association with
the Rocket Tech Division of Utoff, we have commissioned the services
of Dr.\ Gavin LaRose, (with whom we understand you are already
somewhat aquainted), whom you may contact with any questions you may 
have in the process of completing this project.  You should also
contact him by the 26th of November with an indication of your initial
thoughts on the project and by the 6th of December with some evidence
of your work towards its completion.

\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
Dr.\ Vern R.\ Von Brawn\par
\longindent
Head Scientist, SAD

\bye
