\magnification=\magstep1
\centerline{MONT 104N -- Modeling the Environment}
\centerline{Exponential Modeling Examples}
\centerline{October 22, 2018}
\bigskip
\noindent
(A) The Syn 3.0 synthetic life organism mentioned in the background
for the essay on the upcoming midterm exam is a bacterium whose 
genetic information (DNA) was synthesized by humans in a lab and
then inserted into the cells of an existing strain of bacteria (after
the ``host's'' DNA was removed).  The resulting synthetic bacteria
was placed into a nutrient bath and it successfully divided (reproduced) 
in such a way that the number of bacteria was {\it doubling every three 
hours}.  
\bigskip
\item{1)} Is this situation explained by a linear model or an exponential model?
Why?
\item{2)} Starting from a single bacterial cell, write down a formula
for the number of Syn 3.0 cells present $x$ hours after the start of
the experiment.
\item{3)}  How many hours would go by before the number of bacteria
reached 10,000?
\vglue 3.0truein
\noindent
(B)  The radioactive isotope strontium-90 decays in such a way that
if a sample contains $c$ grams at any time $t$, then after 1 year, 
the mass of the remaining strontium-90 is $97.6\%$ of the original amount
(i.e. there are $.976 c$ grams left).  
\bigskip
\item{1)}  Is this a situation for a linear model or an exponential model?
Why?
\item{2)}  If a sample of strontium-90 consists of $100$g of the material
at time $t = 0$, how much will be left after 20 years?
\item{3)}  How long does it take for the amount to decay to $50$g (half
the original amount?  (This length of time is called the {\it half-life}.)  
\bye
