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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Discussion 4 -- {\it A Mathematical Crime Story}}
\centerline{\it April 12, 2006}

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{\it Apologies in advance to any sensitive Italian-Americans 
or fans of ``The Sopranos''(!)}
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{\it Background}

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There's a ``war'' going on in the Bigtown, New Jersey Mafia.  
Carmine ``Calamari'' Squidiarini is the longtime head of the
``family,'' but one of his lieutenants, Biagio ``Bacala'' 
Bacilieri, has been wanting to declare independence and 
set up on his own, taking over most of Squidiarini's territory
and business in garbage removal, trucking, and ``other things.''
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{\it The Situation}
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Shortly before midday, ``Bacala's'' dead body is found in a back
room of the Bella Napoli Pizzeria and Social Club (the 
unofficial headquarters of Squidiarini's crime family).  The room
is air-conditioned to a constant $70^\circ$ F.  He has
been ``whacked'' mob-style with one bullet in the head.  
Right after ``Bacala's'' body is discovered, at 12 Noon, 
its temperature is $80^\circ$ F.  By 1 PM, it has not 
been moved, and its temperature
has decreased to $75^\circ$ F.   

You are the county coroner's office who is called in to 
make the temperature measurements and determine a time 
of death (which is certainly the same as the time ``Bacala'' was ``whacked,''
given the nature of his gunshot wound!).
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{\it The Questions}
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\item{A)}  Write down the differential equation giving
Newton's Law of Cooling in this setting.  
\item{B)}  Assuming ``Bacala'' had a normal temperature
of $98.6^\circ$ F when he died, when did the ``hit'' take
place?
\item{C)}  Carmine Squidiarini has an airtight alibi
for the period 10:30 am - 12 noon since he was in court testifying
in a RICO (racketeering) trial.   Could he have made the 
``hit'' personally if it takes 12 minutes to get from 
the Bella Napoli Pizzeria and Social Club to the courthouse?
\item{D)}  But wait! New evidence has come to light that
``Bacala'' had a terrible cold (wadded up Kleenex were found
next to the body) and his wife says he was running a $104^\circ$ F
fever the day of the ``hit.''    What does this new knowledge
say about the time of death?  Could Carmine have done the ``hit''
given this information?
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{\it Assignment}
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Write up your group's answers and hand in at end of class.
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