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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/16/98}\\
{\it Closed book \hfill Van Dommelen \hfill 11:00-11:50am}
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used.

\begin{enumerate}

\item A graduate student will probably spent about 12.5 hours
of study outside class for each 3 credit hour class.  But that
does not necessarily mean that you spent 12.5 hours a week
on Continuum Mechanics and 12.5 hours on Analysis in Mechanical
Engineering!  Assume that your grade in Analysis can be computed as
\begin{equation}
    g_a = 1 + 3 \sqrt{t_a\over 25},
\end{equation}
where $t_a$ is the time you spent studying on Analysis each
week. So, you will get 1 grade point, a D, with no study and
4 grade points, an A, if you study 25 hours per week
on Analysis alone.
Also assume that your grade in Continuum Mechanics is
\begin{equation}
    g_c = 2 + 2 \sqrt{t_c\over 25},
\end{equation}
where $t_c$ is the time you spent studing on Continuum Mechanics each
week.  Find the maximum grade point average $GPA$,
\begin{equation}
    GPA = {g_a + g_c\over 2}
\end{equation}
that you can get if $t_a + t_c = 25$, i.e. if you study 25 hours on
Analysis and Continuum Mechanics {\em combined}. How much of the 25
hours would you spent on Analysis?
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<A HREF="figures/1.gif">Solution.</A>
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\item Oblique shock waves in supersonic flows are described by
the ``shock polar"
\begin{equation}
    (2-x) y^2 = (1-x)^2 (x - {\textstyle\frac12})
\end{equation}
Draw this curve as accurately as possible.  Derive all symmetries,
intercepts, extents, asymptotes, and behavior at large $|x|$.
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<A HREF="figures/2.gif">Solution.</A>
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\item Find the centroid of half a sphere by multiple integration.
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<A HREF="figures/3.gif">Solution.</A>
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\end{enumerate}

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