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\begin{center}
\textbf{College of the Holy Cross, Fall Semester, 2017} \\
\textbf{MONT 100N -- Modeling the Environment, Final Exam} \\
\textbf{Friday, December 15}
\end{center}

\vskip 40pt
\noindent\textbf{Your Name}: \Blank
\vskip 70pt

\paragraph{Instructions:} Please show all work necessary to justify your answers, and
write your answers in the spaces provided. Use the back of the
preceding page if you need more space for scratch work.  There are
100 possible points distributed as below.

\vskip 2.0truein

\begin{center}

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

\hline

Problem & Points/Poss    \strut \\

\hline\hline

I  & \hfil/\  20\hfil \strut \\

\hline

II  & \hfil/\  25\hfil \strut \\

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III  & \hfil/\  15\hfil \strut \\

\hline

IV  & \hfil/\  10\hfil \strut \\

\hline

Essay  & \hfil/\  30\hfil \strut \\


\hline\hline

Total   & \hfil/100\hfil \strut \\

\hline

\end{tabular}

\vskip 1.0truein
\emph{Have a peaceful and joyous holiday season!}

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\vfill\eject
\noindent
I.  In 1990, forests covered $4.047 \times 10^9$ hectares of the Earth's surface.
By 2000, forest area had decreased to $4.038\times 10^9$ 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
remaining as a function of $t = $ years since 1990.
\vskip 10pt
\begin{enumerate}
\item[A.]  (5)  Determine the slope for the linear model of the forest area.
\vskip 1.5truein
\item[B.]  (5)  What is the linear equation modeling the forest
area as a function of $t = $ years since 1990.
\vskip 1.75truein
\item[C.]  (5)  According to your model, in what year will
the forest area reach $4.0 \times 10^9$ hectares?
\vskip 1.75truein
\item[D.]  (5)  According to the United Nations Food and Agriculture
Organization, the actual forest area remaining in 2010 was $4.033 \times 10^9$
hectares.  How close is the prediction your model from part B gives for
the forest area in 2010?
\end{enumerate}
\vfill\eject
\noindent
II.  Wind power has emerged as a fast growing source of
energy for electrical power generation in recent years.
In 2016, the generating power of wind turbines installed around
the world was about
$301$ gigawatts and it was increasing at about $33.2\%$ per
year.
\begin{enumerate}
\item[A.] (5) The typical English unit of power is the horsepower.
$1 {\rm\ horsepower} = 7.457 \times 10^{-7}$ gigawatts.  Convert $301$ gigawatts
to the equivalent number of horsepower.
\vskip 1.5truein
\item[B.] (5) Construct an exponential model for $WP = $
wind power generation as a function of
$t = $ years after 2016.  \emph{Use units of $10^2$ gigawatts for WP} -- see
the entry for 2016 in the table below.
\vskip 1.5truein
\item[C.]  (10) Fill in the table of values for $WP$ below with
values predicted by
your model for the years $2017 - 2022$.   Round to 2 decimal places.
About how many years will it take for $WP$ to reach approximately 
double the 2016 level?
\vskip 10pt
$$
\begin{array}{c|c|c|c|c|c|c|c|}
Year & 2016 & 2017 & 2018 & 2019 & 2020 & 2021 \\
\hline
WP  & 3.01 &\qquad\qquad &\qquad\qquad & \qquad\qquad &\qquad\qquad &\qquad\qquad  \\
\hline
\end{array}
$$
\vskip 10pt
\item[D.] (5) How many years will it take for wind power generation
to reach $20 \times 10^2$ gigawatts according to your model?
\vskip 2.0truein
\end{enumerate}
\vfill\eject
\noindent
III.  Suppose that a population of fast-reproducing insects in an area
has a natural growth rate of $5\%$ per month from births and deaths, and
that there is a net migration \emph{loss} of 20 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)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 5 P(n - 1) + 20$
\item[2)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 1.05 P(n - 20)$
\item[3)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 1.05 P(n - 1) - 20$
\item[4)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 1.05 P(n - 1) + 20$
\end{enumerate}
\vskip 10pt
\item[B.] (5)  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.] (5)  What happens to the population in the long run as $n$ increase?  Does
it tend to a definite value?
\end{enumerate}
\vfill\eject
\noindent
IV.  Answer \emph{any two} of the following three questions (only the best two will be counted
if you answer more than two).
\begin{enumerate}
\item[A.]  (5) If you are fitting an exponential model to a data set $(x_i,y_i)$ ``by hand,''
you start by transforming the data to $(X,Y) = (x_i,\log(y_i))$.  If the best fit
regression line for the transformed data is $Y = m X + b$, what is the corresponding
exponential model?  (Assume the logarithms have base 10 as we discussed in class.)
\vskip 2truein
\item[B.]  (5) What does the $R^2$ statistic in measure in linear regression?  How
did we use it?  Explain what it would mean, for instance if $R^2 = 1$.
\vskip 2truein
\item[C.]  (5)  What type of chart (scatterplot, pie chart, bar chart, etc.) would be
most useful to describe the composition of a forest if there 5 different types of trees
present in different concentrations per acre?   Explain, and illustrate your answer
with a chart if a typical acre of forest contains 10 oaks, 12 maples, 5 pines, 2 hemlocks,
and 1 chestnut.
\end{enumerate}
\vfill\eject
\noindent
{\it Essay} (30) You have the choice of answering either one of the
following prompts.  Please clearly indicate which prompt you are choosing.
\bigskip   
\begin{enumerate}
\item[Topic A] 
In general terms, what is a mathematical model?  
Describe in general terms 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.  
Even if mathematical models don't capture every feature of a real world
situation, why is it still important to develop them and understand the
information we get from them?  As an example, why is it important to understand 
how radioactive substances decay?  What types of models that we discussed would 
apply to describe that process?  What are three of the main issues discussed 
in the film {\it Containment} that we saw with the rest of the Natural World Cluster?
How do those issues connect with other topics we discussed semester like the
properties of different sources of energy used in the U.S. and their 
effects on the environment?  How are mathematical models important in
understanding our choices of which energy sources to use?
\item[Topic B] 
What does Jared Diamond mean by a ``collapse'' in his book with
that name?  Discuss the five-point framework for understanding
causes of collapses that Diamond lays out in the Prologue.  
Which of those factors does he say are {\it always} present?  
Discuss how the framework applies to {\it either} the collapse 
of the Easter Island civilization or the Classic Maya civilization
in Central America.  ``The past is a foreign country; they do things 
differently there,'' is the famous opening line of a novel called 
{\it The Go-Between} by L. P. Hartley.  How does Diamond's point of view 
in {\it Collapse} relate to that attitude about the past?  For instance, 
would he say the past is not really that foreign after all,
or would he argue that there is some key difference that sets us apart 
from previous civilizations and lessens the causes for concern about
our ultimate fate?  What do {\it you} think about this question, after having
read several sections of {\it Collapse}?
\end{enumerate}
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