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MONT 102N -- Modeling The Environment\\
Final Examination -- December 14, 2011\\
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\noindent
{\it Directions:}  Do all work on these sheets.  You may use the back if necessary, but please
alert me with a note if you do.  In a case where a later part of a question depends on a previous part, I will give credit for the later part as long as you have a correct approach, 
even without the correct value from the previous part.  There
are 200 points, distributed as indicated in the questions.
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\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.
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\begin{enumerate}
\item[A.] (10)  Determine the slope for the linear model of the forest area.
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\item[B.] (5)  What is the linear equation modeling the forest
area as a function of $t = $ years since 1990.
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\item[C.] (10) Use your equation to predict the amount of forest area
that will remain in 2010.
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\end{enumerate}
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\begin{enumerate}
\item[D.]  (10)  According to your model, in what year will
the forest area reach $4.0 \times 10^9$ hectares?
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\item[E.]  (5)  According to the United Nations Food and Agriculture
Organization, the actual forest area remaining in 2010 was $4.033 \times 10^9$
hectares.  Discuss the relation between your prediction in part C and 
the actual figure.  What is happening?
\end{enumerate}
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\noindent
II.  Wind power has emerged as the fastest growing source of
energy for electrical power generation in recent years.
In 2004, the generating power of wind turbines was about
$40,000$ megawatts and it was increasing at about $25.99\%$ per
year.
\begin{enumerate}
\item[A.] (10) The typical English unit of power is the horsepower.
$1 {\rm\ horsepower} = .0007457$ megawatts.  Convert $40,000$ megawatts
to the equivalent number of horsepower.
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\item[B.] (10) 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}.
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\item[C.]  (15) Fill in the table of values for $WP$ below with
values predicted by
your model for the years $2004 - 2010$.   Round to 2 decimal places.
In what year did $WP$ reach approximately double the 2004 level?
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$$
\begin{array}{c|c|c|c|c|c|c|c|}
Year & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\
\hline
WP  &\qquad\qquad  &\qquad\qquad &\qquad\qquad & \qquad\qquad &\qquad\qquad &\qquad\qquad  &\qquad\qquad  \\
\hline
\end{array}
$$
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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?
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\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 $5\%$ per month from births and deaths, and
that there is a net migration \emph{loss} of 20 individuals per month.
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\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}
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\item[B.] (15)  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)
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$$
\begin{array}{c|c|c|c|c|c|c|}
n & 0 & 1 & 2 & 3 & 4 & 5  \\
\hline
P(n)  & 500 &\qquad\qquad &\qquad\qquad & \qquad\qquad &\qquad\qquad &\qquad\qquad  \\
\hline
\end{array}
$$
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\item[C.] (5)  What happens to the population in the long run as $n$ increase?  Does
it tend to a definite value?
\end{enumerate}
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\noindent
IV.  Answer any two of the following three questions (only the best two will be counted
if you answer more than two).
\begin{enumerate}
\item[A.]  (15) If you are fitting a power law model to a data set $(x_i,y_i)$ ``by hand,''
you 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[B.]  (15) 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 $r = 1$ or $r = -1$.
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\item[C.]  (15)  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}
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\noindent
V.  Essay.  (60)  What are the major reservoirs of carbon in the Earth's short-term
carbon cycle?  What are the major flows (``fluxes'') of carbon between those
reservoirs.  Describe them in words and in a diagram.  In our final project, we
looked at a simple model of this system.  What features of the real-world
carbon cycle did that model include?  What are three possibly important features
of the real-world carbon cycle that were left out?
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