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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/17/96 }\\
{\it Closed book \hfill Van Dommelen \hfill 9:00-9:50am }
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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.

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\item
(33 points).
For a cylindrical barrel, assume that the cost of the flat ends is $f$ dollar
per square feet, while the cylindrical surface costs $2f$ dollar per
square feet. Find the ratio of diameter $D$ to height $h$ of the
barrel that gives the lowest total cost for a given barrel volume $V$.
(It seems that the people who invented barrels may not have known
much about cost reduction?)
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<A HREF="figures/1.gif">Solution.</A>
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\item
(33 points).
A vehicle moves along the hyperbolic curve
$$y^2 - x^2 = 1$$
with a constant, unit speed.  Find the time to travel from $x=0$ and $y=1$
to $x=0.5$.  Note: you will need to make an approximation.  Include the
first three nontrivial terms in this approximation.
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<A HREF="figures/2.gif">Solution.</A>
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\item
(34 points).
A force field is described by
$$ F_x = \sqrt{x+y+z} + y^2 + z^2 $$
$$ F_y = \sqrt{x+y+z} + 2 xy $$
$$ F_z = \sqrt{x+y+z} + 2 xz $$
Find the work that this force does on a body travelling from
the origin along the half circle
$$ (x-{\textstyle\frac12})^2 + y^2 +z^2 = {\textstyle\frac14}
\qquad y=z\ge 0$$
to the point $(x,y,z)=(1,0,0)$ on the $x$-axis.  Hint: a brute force
approach may be a fun challenge, but more than you can do in the
allotted time. If you can simplify this problem, explain why,
and do it.
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<A HREF="figures/3.gif">Solution.</A>
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