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\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/27/06}\\
{\it Closed book \hfill Van Dommelen \hfill 11:50-12:40 pm}
\end{center}

Solutions should be fully {\em derived} showing all intermediate
results, using class procedures.  Show all reasoning.  Bare answers are
absolutely not acceptable, because I will assume they come from your
calculator (or the math handbook, sometimes,) instead of from you.
You must state what result answers what part of the question if there
is any ambiguity.  Answer exactly what is asked; you do not get any
credit for making up your own questions and answering those.

One book of mathematical tables, such as Schaum's Mathematical
Handbook, may be used, as well as a calculator, and a handwritten
letter-size formula sheet.

\begin{enumerate}
  
\item {\bf Background:} The curved surface of a barrel will be more
  expensive to make than the top and bottom.  This question intends to
  find the most economical dimensions if the cost per square feet of
  the curved surface is twice that of the flat bottom and top
  surfaces.

  {\bf Question:} If for a cylindrical container, the cost of the
  cylindrical surface is \$4/ft$^2$ and that of the top and bottom
  circles is \$2/ft$^2$, then what is the lowest possible cost
  of a barrel that is to hold 10 ft$^3$?

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}

\item {\bf Background:} In the Lagrangian description of an established
  unsteady separation point, the position of the particles at the
  separation position can be written as $y=\pm x\sqrt{1+x}$.

  {\bf Question:} Analyze and graph {\em very neatly} the curve
  \begin{displaymath}
    y = x\sqrt{1+x}
  \end{displaymath}
  Discuss presence or absence of $x$-and $y$- intercepts, local maxima
  and minima, global maxima and minima, vertical, oblique, and
  horizontal asymptotes, vertical slopes, cusps, and kinks, and give
  $x$ and $y$ extents.  Features that are unclear or ambiguous in
  the graph count as failed.  Note: you do {\em not} have to discuss
  inflection points.

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Unlike you might think from the book, finding
  centroids and moments of inertia of plates is not always just a
  matter of integrating powers of $x$ and $y$.  If the thickness or
  material density vary, any integral may show up.

  {\bf Question:} Integrate
  \begin{displaymath}
    \int \frac{\exp\big(\arctan(y/x)\big)}{x}\;{\rm d}A
  \end{displaymath}
  over the surface bounded by $y=x$, $y=-x$, and $x^2+y^2 - 2 x=0$.
  Hint: Use polar coordinates.

\begin{rawhtml}
<A HREF="figures/3.jpg">Solution.</A>
\end{rawhtml}

\end{enumerate}

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