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\textbf{College of the Holy Cross, Fall Semester 2018} \\
\textbf{MONT 104N -- Modeling the Environment} \\
\textbf{Final Exam, December 13}

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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|}

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Problem & Points/Poss    \strut \\

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

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II  & \hfil/\  20\hfil \strut \\

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

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IV  & \hfil/\  15\hfil \strut \\

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


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

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Do all work on these sheets.  You may use the back sides if necessary. 
There are 100 possible points (distributed as indicated in the questions).

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\centerline{\it Have a peaceful and joyous holiday season!}
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\noindent
I.  The table below shows estimates collected in 
the EDGAR database created by the European Commission and
the Netherlands Environmental Assessment Agency of the amounts of \emph{carbon
dioxide} emissions from burning of fossil fuels in 2015 and 2016 by country.  
The units are megatonnes $= 10^6$ metric tonnes.  The populations are in units of 
millions $ = 10^6$ of people:
$$\matrix{ {\bf Country} & {\bf CO2\ 2015} & {\bf CO2\ 2016} & {\bf 2015\ Population}\cr
           {\rm China} & 10.642 & 10.433 & 1367\cr
           {\rm United\ States} & 5.712 & 5.012 & 321 \cr
           {\rm India} & 2.455 & 2.533 & 1252 \cr
           {\rm Russia} & 1.761 & 1.662 & 142 \cr
           {\rm Japan} & 1.253 & 1.240 & 127 \cr
           {\rm Germany} & .778 & .776 & 81\cr
           {\rm Whole\ world} & 36.061 & 35.753 & 7256 \cr}$$  
\begin{enumerate}
\item[A.]  (5)  1 metric tonne is $1000$ kg and $1\ {\rm kg} \doteq 2.205$ lb.  What was the 
equivalent amount of carbon dioxide emissions for the U.S. in 2015, in units of pounds.  
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\item[B.]  (5)  Suppose that China's carbon dioxide emissions were decreasing \emph{exponentially}.
What would be the exponential model fitting the two data points you have exactly?  Take
$t = 0$ to correspond to the year 2015.  What would your model predict for China's
emissions in the year 2020? 
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\item[C.]  (10) Now, let's look at this data from another perspective.  What 
are the {\it per capita} carbon dioxide emissions in units of metric tonnes per person for
each of these countries?  Construct a chart or charts (your choice of types) showing how
the 2015 total emissions and the 2015 emissions per capita compare for these countries.
Pay special attention to any changes in the relative orderings when you go from
the total emissions to the per capita emissions.  
\end{enumerate}
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\noindent
II.  The following graph, constructed by Makiko Sato and James Hansen at the Columbia
University Climate Science, Awareness and Solutions project, 
shows the change in global average sea level over the period 1900 to the present.  
It's also a nice example of how the basic modeling techniques we have discussed
can be adapted to deal with dynamic, changing situations.
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\includegraphics[width=4.5in,height=3in]{GlobalMeanSeaLevel.jpg}
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The vertical axis is the \emph{sea level anomaly}---the \emph{difference} between
the observed levels in later years and the average level in 1900.  
\begin{enumerate}
\item[A.] (12) The information included in the graph shows that Sato and Hansen have
constructed three separate linear models over three different time periods: one covering the 
period 1900 - 1930, the second covering the period 1930 - 1992 and the last one 
covering the period 1993 to 2017. The estimated slopes are given in units of mm/yr.
 Find the equations of the three linear models
(the equations of the three black lines in the graph) for average mean sea level $L(t)$, using
$t = $ actual year in each case.  Use the values $0$ for $L(1900)$, $18$ for $L(1930)$, and 
$104.8$ for $L(1993)$.  
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\item[B.] (3) The legend indicates that the first two models were found from adjusted
tide gauge data, while the third was obtained from satellite radar measurements.  Which
of these two methods would you guess is more accurate?  How does that relate to the 
appearance of the graph(s)?  Discuss briefly.\footnote{Note:  The lighter blue region
around the blue line plots shows a series of ``error bars'' indicating a reasonable range
of values consistent with the measured data. There's something like that too around
the red portion of the plot, but it's much less visible.}   

\end{enumerate}

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\noindent
III.  Suppose that a endangered population of spotted owls in a protected forest is
\emph{decreasing} at a net rate of $10\%$ per year from births and deaths, but
human-reared owls are being reintroduced into the habitat at a rate
of 16 individuals per year.
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\begin{enumerate}
\item[A.] (5)   Write a difference equation that models the owl population.
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\item[B.] (5)  Using an initial value $P(0) = 40$, determine the populations
in years $1,2,3,4,5$ according to the model you stated in part A and
record the values in the following table (round any decimal values to the nearest
whole number)
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$$
\begin{array}{c|c|c|c|c|c|c|}
n & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
P(n)  & 40 &\qquad\qquad &\qquad\qquad & \qquad\qquad &\qquad\qquad &\qquad\qquad  \\
\hline
\end{array}
$$
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\item[C.]  (5) What happens to the population in the long run?  Does
it tend to a definite value as $n$ increases?  If so, what is it?  If not, why not?
\end{enumerate}

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\noindent
IV.  Answer {\it any three} of the following briefly.  If you submit answers for
all four, only the best three will be counted.
\begin{enumerate}
\item[A.]  (5) If you are fitting a \emph{power function}
model to a data set $(x_i,y_i)$ ``by hand,''
you would start by transforming the data to a new form $(X_i,Y_i)$.  What is that form
in terms of the $x_i$ and $y_i$?.  If the best fit
regression line for the transformed data is $Y = m X + b$, what is the corresponding
power function model?  (You may use logarithms to base 10 as we discussed in class.)
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\item[B.] (5) What difference equation would model a population undergoing logistic growth
if the population was growing at about $3\%$ per year when the population is much smaller than
the carrying capacity $M = 1000$ of the habitat?  
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\item[C.] (5) Describe the SIR model for infectious disease outbreaks and give the corresponding
difference equations. 
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\item[D.] (5) What feature of the solutions of the Lotka-Volterra equations is considered to be a
confirmation that this model is capturing an important aspect of real-world predator-prey
interactions?
\end{enumerate}
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\noindent
{\it Essay} (30)
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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 three
different types of mathematical models we have studied this semester.  
Even if mathematical models don't capture {\it every feature} of a real world
situation, why is it still important to develop them and understand the
information we get from them?  For instance, what conclusions about use
of natural resources did we derive by looking at logistic models with various 
types of harvesting in the Chapter Project from Chapter 7?  As another example, 
how are mathematical models important in understanding our choices of which 
energy sources to use?  Why is it important to understand 
how radioactive substances decay?  What type(s) of model(s) that we discussed would 
apply to describe that process?  What are some of the issues involved with
using radioactive decay to generate electricity--that is, why is this not
a ``no-brainer'' as a solution to the problem of $CO_2$ buildup in the atmosphere
from fossil fuel burning?  
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