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\section*{Quiz 7}
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{\bf Quiz7 Problem 1}. {\bf Heat Transfer} and the Mean Value Property.

Consider the cross section of a long rectangular dam on a river, represented
in the figure. 

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The boundaries of the dam are subject to three factors:
the temperature in degrees Celsius of the air (20),  the water (25), and
the ground at its base (30). 

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An analysis of the heat transfer from the three sources will be done from the
equilibrium temperature, which is found by the Mean Value Property below.

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{\bf The Mean Value Property}\\[3ex]

If a plate is at thermal equilibrium, and $C$ is a circle contained in
the plate with center $P$, then the temperature at $P$ is the average
value of the temperature function over $C$.

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A version of the Mean Value Property says that the temperature at center $P$
of circle $C$ is the average of the temperatures at four equally-spaced points on $C$.
We construct a grid as in the figure below, 
label the unknown temperatures at interior grid points as $x_1,x_2,x_3,x_4$,
then use the property to obtain four equations.


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{\bf Four-Point} \\
{\bf Temperature Averages} 
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\begin{center}\large\rm
$x_1 = \dd\frac14(20+25+x_2+x_3)$ \\[1ex]
$x_2 = \dd\frac14(20+20+x_1+x_4)$ \\[1ex]
$x_3 = \dd\frac14(25+30+x_1+x_4)$ \\[1ex]
$x_4 = \dd\frac14(20+30+x_2+x_3)$ 
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{\bf References}. 


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{\bf Quiz7 Problem 2}. {\bf Separation train}:  Xylene, Styrene, Toluene, and Benzene are separated in 3 distillation columns. Symbols $Ft, D, B, D1, B1, D2, B2$ are molar flow rates in mol/min.

{\bf References}. Edwards-Penney Sections 3.1, 3.2, 3.3. Course manuscript
\href{http://www.math.utah.edu/~gustafso/s2014/2250/linearequDRAFT.pdf}{Linear Algebraic Equations, No Matrices}.


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{\bf Quiz7 Problem 3}. {\bf Traffic Flow}\newline

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