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\textbf{College of the Holy Cross, Fall Semester, 2012} \\

\textbf{MONT 104N -- Modeling the Environment} \\

\textbf{Thursday December~13, 8~AM}

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\noindent\textbf{Your Name}: \underline{\phantom{Name Name Name Name Name}}

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\paragraph{Instructions} Please write your answers in the spaces
provided on the following pages, and show work on the test itself. \textbf{For possible
partial 
credit, you must show work}. Use the back of the preceding
page if you need 
more space for scratch work.



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\noindent\textbf{Please do not write in the space below}


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\begin{tabular}{|l||r|}

\hline

Problem & Points/Poss    \strut \\

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I  & \hfil/\  50\hfil \strut \\

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II  & \hfil/\  35\hfil \strut \\

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III  & \hfil/\  25\hfil \strut \\

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IV  & \hfil/\  30\hfil \strut \\

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Essay  & \hfil/\  60\hfil \strut \\


\hline\hline

Total   & \hfil/200\hfil \strut \\

\hline

\end{tabular}

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\centerline{\it Have a peaceful and joyous holiday season!}
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\vskip 10pt 
\noindent
I.  Wind power has emerged as the fastest growing source of
energy for electrical power generation in recent years.
In 2004, the generating capacity of all wind turbines in use was about
$47,600$ megawatts and the generating capacity was increasing at about $26.8\%$ per
year.
\begin{enumerate}
\item[A.] (10) The typical English unit of power is the horsepower.
$1 {\rm\ horsepower} = .0007457$ megawatts.  Convert $47,600$ megawatts
to the equivalent number of horsepower.
\vskip 2.0truein
\item[B.] (10) Using the information above, construct an 
exponential model for $WP = $ wind power generation as a function of
$t = $ years since 2004.  \emph{Use units of $10^4$ megawatts for WP}.
\vskip 2.0truein
\item[C.]  (15) Fill in the table of values for $WP$ below with
values predicted by
your model for the years $2004 - 2011$.   Round to 2 decimal places.
In what year did $WP$ reach approximately double the 2004 level?
\vskip 10pt
$$
\begin{array}{c|c|c|c|c|c|c|c|c|}
Year & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 & 2011\\
\hline
WP  &\qquad\quad  &\qquad\quad &\qquad\quad & \qquad\quad &\qquad\quad &\quad\qquad  &\quad\qquad 
& \qquad\quad \\ \hline
\end{array}
$$
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\item[D.] (10) How many years will it take for wind power generation
to reach $320,000$ megawatts according to your model?
\vskip 2.0truein
\item[E.] (5)  The following graph (produced by the Global Wind Energy 
Council -- GWEC) shows the actual 
global wind electrical power generation capacity (estimated via surveys
of electrical power producers).  How do the 
actual figures compare with your model values?  Note:  The vertical 
scale of the graph is in \emph{gigawatts}.  $1$ gigawatt $ = 1000$ megawatts.  
%\begin{figure}
%\includegraphics[width=5in]{GlobalWindPowerCumulativeCapacity.png}
%\end{figure}
\end{enumerate}
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\noindent
II.  According to the United Nations Food and Agriculture
Organization, in 2000, forest area covered $4.038\times 10^9$ hectares of the Earth's surface.
The forest area in 2010 was $4.033 \times 10^9$ hectares.   Assuming
that the decrease in forest area is
linear, and that it will continue at the same rate into the future,  in
this problem you will develop a linear model for the forest area
$FA = $ (in units of $10^9$ hectares) remaining as a function of $t = $ years since 2000.
\vskip 10pt
\begin{enumerate}
\item[A.] (10)  Determine the slope for the linear model of the forest area.
\vskip 1.5truein
\item[B.] (10)  What is the linear equation modeling the forest
area as a function of $t = $ years since 2000.
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\item[C.] (10) Use your equation to predict the amount of forest area
that will remain in 2020.
\vskip 2.0truein
\end{enumerate}
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\begin{enumerate}
\item[D.]  (5)  According to your model, in what year will
the forest area reach $4.0 \times 10^9$ hectares?
\vskip 3.0truein
\end{enumerate}
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\noindent
III.  Suppose that a population of fast-reproducing insects in an area
has a natural growth rate of $7\%$ per month from births and deaths, and
that there is a net migration \emph{loss} of 100 individuals per month.
\vskip 10pt
\begin{enumerate}
\item[A.] (5)  Which of the following difference equation models for $P(n) = $
population in month $n$ fits the description
above? (Place a check next to the correct one.)
\begin{enumerate}
\item[1)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 7 P(n - 1) - 100$
\item[2)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 1.07 P(n - 100)$
\item[3)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 1.07 P(n - 1) - 100$
\item[4)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 1.07 P(n - 1) + 100$
\end{enumerate}
\vskip 10pt
\item[B.] (10)  Using an initial value $P(0) = 500$, determine the populations
in months $1,2,3,4,5$ according to the model you picked in part A and
record the values in the following table (round any decimal values to the nearest
whole number)
\vskip 10pt
$$
\begin{array}{c|c|c|c|c|c|c|}
n & 0 & 1 & 2 & 3 & 4 & 5  \\
\hline
P(n)  & 500 &\qquad\qquad &\qquad\qquad & \qquad\qquad &\qquad\qquad &\qquad\qquad  \\
\hline
\end{array}
$$
\vskip 3.0truein
\item[C.] (10)  What happens to the population in the long run as $n$ increase?  Does
it tend to a definite value?  What is that value?
\end{enumerate}
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\noindent
IV.  Answer any three of the following four questions (only the best three will be counted
if you answer more than three).
\begin{enumerate}
\item[A.]  (10) What does the correlation coefficient $r$ (or its square $r^2$) measure?  How
did we use it?  Explain what it would mean, for instance if $r^2 = 1$ or $r^2 = 0$.
\vskip 2truein
\item[B.]  (10) If you are fitting a power law model to a data set $(x_i,y_i)$ ``by hand,''
you would start by transforming the data to $(X,Y) = (\log(x_i),\log(y_i))$.  If the best fit
regression line for the transformed data is $Y = m X + b$, what is the corresponding
power law model?  (Assume the logarithms have base 10 as we discussed in class.)
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\item[C.]  (10)  A population has unrestricted growth rate $r_{max} = .03$ and carrying
capacity $K = 1000$.  What is the corresponding logistic model?
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\item[D.] (10)  The following graph shows $W = $ the world production of photovoltaic arrays
(used for solar power generation) in units of ``peak megawatts'' (the power capacity they have).  
 Between 1998 and 2007, what type of model would be most appropriate for describing how
$W$ is growing.  Explain.  \emph{Look at the vertical axis scale carefully!}
\end{enumerate}
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\noindent
V.  Essay.  (60)  In general terms, what is a mathematical model?  Describe what they are, 
how they are constructed, and how they are used.  Give examples of two 
different types of mathematical models we have studied in this course.  
Next, {\it why} do we try to build mathematical models of aspects 
of the real world?  Can {\it any} mathematical model be a 
completely accurate representation of some aspect of the natural world?  
As an example, why do scientists think it is important to 
understand how much $CO_2$ is present in the atmosphere?  What
tends to happen when $CO_2$ levels rise?  
Describe a key piece of evidence that suggests human activities might 
have changed atmospheric $CO_2$ levels over the past 50-200 years. 
Explain the case for saying the evidence points to that conclusion,
and relate your answer to the results of modeling 
exercises we did in this class.  
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\noindent
Essay (continued)
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