\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\input epsf
\def\R{{\Bbb R}}
\font\srm=cmr8
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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 27\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 27} 
\bigskip

\itemitem{1.\hskip12pt a)}  In polar coordinates, the orbit of Mercury is
given by the equation
$$
r={3.442\times10^7\over1-0.206\cos\theta}
$$
where $r$ is the distance in miles from the sun.  Find the distance of
Mercury's closest and farthest approach to the sun.

\medskip
\itemitem{b)}  The orbit of Saturn is given by the equation
$$
r={1.4228\over1-0.0543\cos\theta}
$$
where $r$ is the distance in billions of kilometers from the sun.
Find the disance of Saturn's closest and farthest approach to the sun.

\medskip
\itemitem{c)}  Sketch the orbits of Mercury and Saturn.


\medskip
\item{2.}  {\bf Involute of a circle.}  If a string wound around a
fixed circle is unwound while held taut in the plane of the circle,
its end traces an {\it involute} of the circle.  Let the fixed circle
be located with its center at the origin $O$ and have radius $a$.  Let
the initial position of the tracing point $P$ be $A=(a,0)$ and let the
unwound portion of the string $PT$ be tangent to the circle at $T$.
Derive parametric equations for the involute, using angle $AOT$ as the
parameter $t$.

\medskip
\centerline{\epsfysize=2.5in\epsfbox{/home/oehrtman/m210s96/involute.eps}}


\eject
\item{3.}  In the Odd Galaxy, there is a planet called Id.  The
Idiots, the inhabitants of Id, have noticed that their planet goes in
a large slow circular orbit given by the path ${\bf I}(t)=(\sin t,\cos
t,0)$.  The unit of time is one million earth years, and the unit of
length is one Idian Astronomical Unit (IAU).  Their astronomers notice
a very large asteroid with path given by ${\bf R}(t)=(\csc t,0,\cot t)$
for $0