%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                  %
%  Exponential Growth and Decay                    %
%                                                  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemgrouptitle{Exponential Growth and Decay}

\item A certain type of bacteria grows exponentially and doubles every 5 hours.
\begin{enumerate}
\item Find the growth constant.
\item How many bacteria will there be after 20 hours assuming the initial population is 200 bacteria? 
\end{enumerate}
\answer{
  \begin{enumerate}
  \item $\displaystyle k= \frac {\ln 2} 5$
  \item $\displaystyle P(20)= 3200$ bacteria
  \end{enumerate}
}

\item The initial deer population in a forest was 25 deers. After one year the number of deers had increased to 75. If the number of deers grows exponentially, how many deers will there be at the end of two years?
\answer{
$\displaystyle P(2)= 225$ deer
}

\item In 2016, the population of a small island was estimated to be 30,000 people with an annual rate of increase of 2.5\%.
  \begin{enumerate}
  \item Find the growth constant.
  \item Estimate the population of the island in 2025.  
  \end{enumerate}
\answer{
  \begin{enumerate}      
  \item $\displaystyle k= 0.02469$
  \item $\displaystyle P(9)= 73465$
  \end{enumerate}
}

\item Magnesium-27 decays exponentially and has a half-life of 9.45 minutes.  
  \begin{enumerate}
  \item Find the decay constant for Magnesium-27.
  \item When will an initiall mass of 20 mg be reduced to 5 mg? 
  \end{enumerate}
\answer{
  \begin{enumerate}
  \item $\displaystyle k=- \frac {\ln 2} {9.45}=-0.07334890$
  \item $\displaystyle t=18.9$ minutes
  \end{enumerate}
}

\item A patient is given a 400 mg dose of medicine that degrades by 20\%  every 3 hours. What is the remaining drug concentration after a day?
\answer{
$\displaystyle k=-0.0743811,  m(24)=67.10$  mg 
}

\item A certain radiactive isotope decays exponentially according to the formula $m=m_0 e^{-0.3 t}$ where $m$ is the mass (in grams) of the isotope at the end of $t$ days and $m_0$ is the initial mass. Find the half-life of this isotope.
\answer{
The half-life is $\displaystyle 2.31$  days.
}

\textbook{\addtoanswerkey{\clearpage}}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../master"
%%% End:
