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\font\medrm=cmr10 scaled\magstep2

\centerline{\bigrm Isolating Perry Metric in a Quality}
\smallskip
\centerline{\bigrm Cell}
\bigskip

\noindent
{\bf Purpose:}
The purpose of this project is to introduce you to a type of
optimization problem and solution procedure which is based on
multivariable calculus.
\bigskip

\noindent
{\bf Procedure:}  You are to follow the steps below.  You will
probably need to
do the steps in the order indicated since the results of one step are
often used in the next step.
After completing the outline below you should write your
results and calculations {\bf neatly}.  Be sure to include relevant
graphs and diagrams
in what you turn in. Your grade will be determined
in part by your presentation.

Perry Metric was caught stealing a balloon from a toy store. Perry's
lawyer plea bargained, and was able to reduce Perry's sentence to two
years in a cell designed by Perry.  The judge told Perry that he could
design the cell to be any shape that Perry wanted as long as the
perimeter was
no more than 40 meters.  Perry's goal is to enclose as much area as
possible and have no corner  in his cell wall.
Perry is pretty sure what shape the  wall
should be in order to accomplish his goal.  After thinking about the
problem he decides that he had better be safe and see if there is
justification for his conjecture concerning the shape of the wall. Perry
calls you for help in proving his conjecture.  The judge is only giving
Perry two weeks to design the cell, so he needs a proof within two weeks
or else the judge will impose the shape of the cell (which will probably
be a 1 meter by 19 meter rectangle).
Before calling you, Perry looked through
some mathematics books and found a few theorems which may be of use to
you.
\medskip
\noindent
{\bf Theorem 1:} Suppose that $F(x,y,t)$ is a smooth function. Then
               ${\partial\over\partial x}\int_a^b F(x,y,t) dt
                = \int_a^b{\partial\over\partial x}F(x,y,t) dt$.
\medskip

Actually, the above theorem is true if $F$ is a function of more than
three variable as well.  You should interpret $\int_a^b F(x,y,t) dt$ as
being a function of two variables, $x$ and $y$.  So it makes sense to
take the partial derivative of this function with respect to $x$, or $y$
for that matter.

The next theorem is a special case of Green's
theorem which we may cover in class later in the semester.
\medskip
\noindent
{\bf Theorem 2:} Suppose that a planar curve is given parametrically
               by smooth functions
               $x(t)$ and $y(t)$ for $0\leq t\leq 1$. Furthermore,
               suppose that $x(0)=x(1)$ and $y(0)=y(1)$, but otherwise
               the curve does not intersect itself (a curve meeting
               the condition that it starts and ends at the same point is
               called a closed curve).
               Then the area
               enclosed by the curve is given by
               ${1\over 2}\int_0^1 xy'-yx' dt$ as long as the curve goes
               around the area in a counter--clockwise direction.

\medskip

The third theorem is useful at times when you want to be sneaky and
show that a function is 0 for every point in the domain.

\medskip
\noindent
{\bf Theorem 3:} If $g(x)$ is continuous and $\int_0^1 g(x)\eta(x) dx = 0$
                 for every smooth function $\eta(x)$ having domain
                 $[0,1]$ with $\eta(0)=\eta(1)=0$
                 then $g(x)=0$ for every $0\leq x\leq 1$.
\medskip

It may not be obvious how to solve Perry Metric's problem using the
three theorems listed, but hopefully with the outline below you will be
able to do it.  The first three parts are to help you understand the
theorems which you will need to solve Perry's problem.

\item{1.}Do a few examples of Theorem~1.  Make up your own functions
         for $F$ where you can do the indicated integrals (polynomials
         work well) and check that both sides are the same.
\item{2.}Next write the parametric equations for an ellipse in standard
         form and use Theorem~2 to compute the area inside the ellipse.
         Compute the area of the ellipse in another way to make sure you
         obtain the correct answer.  What happens if the curve goes
         around the area in a clockwise direction?
\item{3.}Draw a picture and explain why you think Theorem~3 should be
         true.  (Hint: suppose that $g(x)>0$ at some point.  Draw a
         picture of a function $\eta(x)$ which would make
         $\int_0^1g(x)\eta(x) dx >0$.)
\medskip

Now that you know what the theorems are saying and perhaps have a
feeling about why they are true you may use the theorems to complete
the steps below.
\medskip

\item{4.}Suppose that the wall of Perry's cell is given by a pair of
         parametric equations which are parameterized by
         $t$ with $0\leq t\leq 1$. (The parametric equations give
         the curve on the floor where Perry's wall touches the floor.)
         We may as well assume that $x(0)=y(0)=0=x(1)=y(1)$, that is, the curve
         starts and ends at the origin since the area and arc length is not changed 
         if the curve is translated to a different position in the plane.
         Write formulas which give the arc
         length and the area enclosed by Perry's walls.  Keep in mind
         that $x$, $y$, $x'$, and $y'$ are functions of $t$, and not
         simply variable.
%         Denote the function whose integral gives you
%         the area as $F(x,y,x',y')$ and the function whose integral
%         gives the length as $G(x,y,x',y')$.
\item{5.}Suppose that $x(t)$ and $y(t)$ define a parametric curve
         giving Perry's desired wall.  Then for an arbitrary (but fixed)
         pair of
         smooth functions $\eta_1(t)$ and $\eta_2(t)$ and arbitrary
         values  of the variables $z$ and $w$, write conditions under
         which the curve parameterized by  $z\eta_1(t)+x(t)$ as the
         $x$-value and $w\eta_2(t)+y(t)$ as the $y$-value is a closed
         curve of length 40 which starts and ends at the origin. 
         Also write the formula for the area
         enclosed by this curve.  
\item{6.}Now, if $(x(t),y(t))$ give the desired wall for Perry then the
         values $w=0$ and $z=0$ should yield a maximum for the area formula
         you found in  part~5 subject to the conditions you gave in
         part~5.  Write the Lagrange multiplier formula for this
         situation. (You will need to use
         Theorem~1 and the chain rule.)
\item{7.}Simplify your answer in part~6 as much as you can.  Try to use
         Theorem~3 to conclude that a certain function is zero. (Hint:
         it may be useful to use integration by parts on two of the
         integrals in each formula you derive and then combine the
         integral formulas in order to use Theorem~3.)
\item{8.}Can you now conclude that the curve is what you suspect it
         should be?  (With some work, the answer should be yes.)
\item{9.}Think carefully about what the above calculations show and give
         a clear statement of what you proved.
\bye
