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\centerline{MATH 136 -- Calculus 2}
\centerline{Group Discussion Day 5 -- Applications of differential Equations}
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\noindent
I.  A rare mineral deposit was formed in the collision of a meteorite with the earth 
some time long in the past.  The deposit originally contained some amount of the 
Uranium isotope $U^{238}$, which is radioactive with half-life $4.51 \times 10^9$ years,
but none of the lead isotope $Pb^{207}$, which is the end decay product of $U^{238}$.
The ratio of the number of $U^{238}$ atoms to the number of $Pb^{207}$ 
in the mineral deposit today is $0.9$.  When did the collision with the meteorite
occur?  
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\noindent
II.  Just before midday, the body of an apparent homicide victim is found
in a room that is kept at a constant temperature of $70$ degrees F.  At 12 noon
the temperature of the body is measured to be $80$ degrees F and at 1 pm, the temperature
has decreased to $75$ degrees F.  Assume that the 
temperature of the body at the time of death was $98.6$ degrees F and that the 
body cooled according to Newton's law.  When did the murder occur?  
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\noindent
III.  Suppose that a water tank has a hole at the bottom and water is draining out.
If $y(t)$ denotes the depth of the water at time $t$, then {\it Torricelli's Law}
implies that $y(t)$ will satisfy the differential equation 
$$A(y){dy \over dt} = -a \sqrt{2 g y}$$
where 
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\item{$\bullet$} $g$ is the acceleration of gravity near the surface of the earth ($\doteq 32$ in 
feet per seconds squared),
\item{$\bullet$} $A$ is the cross-section area of the tank at depth $y$, and
\item{$\bullet$} $a$ is a constant (it represents the area of the 
hole).  
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\noindent
\item{A)}  If the tank has the shape of the solid obtained by rotating some curve $x = g(y)$
about the $y$-axis, determine $g(y)$ so that {\it the depth of the water in the tank will fall
at a constant rate.}   
\item{B)}  For what would such a tank be useful?  (Hint:  Look up the term {\it clepsydra} 
on Wikipedia.  An ``outflow'' clepsydra formed from a tank of this shape would have
a special property.)  
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