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\leftline{\it A First Mathematical Modeling Problem}
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A small house is to be built on flat ground in the shape of a rectangular
solid with exterior walls that are 9 meters high.  The region 
where the house will be built has extremely severe winters, during 
which cold winds blow primarily from due north (think Minnesota or
North Dakota in the U.S. or Norway or Sweden in Europe).  
One wall of the house will face directly north (it has 
``northern exposure'') and bear the full force of the 
cold winds.  The south-facing wall will be sheltered 
from the winds, while the other two, east- and west-facing
walls will be somewhat less sheltered.

Annual heating  costs  for  the  house satisfy the following: 
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\item{$\bullet$} Each square meter of exterior wall with 
the northern exposure adds $\$ 50$ Dollars to the annual 
heating cost.  
\item{$\bullet$} Each square meter of exterior wall 
with  an  eastern  or  western  exposure  adds  $\$ 25$  
to  the  annual  heating cost.  
\item{$\bullet$} Each  square  
meter of exterior wall with a southern exposure 
adds   $\$ 12$   to   the   annual   heating cost.

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\item{(A)} Denote by $L$ the length of the north- and 
south-facing walls.  (Note these are equal
because the house is a rectangle.) If the length of the 
east- and west-facing walls are constant (for instance at 20 meters), 
what can you say about the total annual heating cost as
a function of $L$.
\item{(B)} Now suppose that the length of the north- and
south-facing walls is constant (for instance at 30 meters)
and $W$ represents the length of each of  the  east-  and  
west-facing  walls.  What can you say about the total 
annual heating cost as a function of $W$.  
\item{(C)} If $L$ and $W$ are both free to vary, what is the 
total annual heating cost as function of $L$ and $W$ together? 
\item{(D)} Now assume the total area of the ``footprint'' of the 
house is to be exactly 200 square meters.  Express the total annual cost of
heating the house in terms of $L$ (the length of the 
north- and south-facing walls). 
This is a mathematical model(!)
\item{(E)}  Give examples of two different 
real-world questions that could be addressed
using this model.  
\bye
