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MONT 100N -- Modeling The Environment\\
Solutions for Final Examination -- December 15, 2017\\
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\vskip 10pt
\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.] (10)  Determine the slope for the linear model of the forest area.
\item[] {\it Solution:}  The slope is
$$m = \frac{4.038 - 4.047}{2000 - 1990} = -.0009$$
(in units of $10^9$ hectares per year; this is equivalent to a net loss of 
900,000 hectares, or 9000 square kilometers per year).
\item[B.] (5)  What is the linear equation modeling the forest
area as a function of $t = $ years since 1990.
\item[] {\it Solution:}  Writing $FA$ for the forest area, the model would be
$$FA = 4.047 - .0009 t$$
\item[C.] (10) Use your equation to predict the amount of forest area
that will remain in 2010.
\item[] {\it Solution:}  $2010$ is $t = 20$ years after 1990, so the 
model predicts
$$FA = 4.047 - (.0009)(20) = 4.029\ (\times 10^9 {\rm\ hectares}) $$
\item[D.]  (10)  According to your model, in what year will
the forest area reach $4.0 \times 10^6$ hectares?
\item[] {\it Solution: } We solve for $t$ from the equation:
$$4.0 = 4.047 - .0009 t,$$
so
$$t = \frac{4.047 - 4.0}{.0009} \doteq 52$$
This corresponds to the year $2042$.  
\item[E.]  (5)  According to the United Nations Food and Agriculture
Organization, the actual forest area remaining in 2010 was $4.033 \times 10^6$
hectares.  Discuss the relation between your prediction in part C and 
the actual figure.  What is apparently happening?
\item[] {\it Solution:}  The rate of deforestation was slower between 2000
and 2010 than the rate between 1990 and 2010.  So the model predicted a
lower forest area remaining than the actual figure. 
\end{enumerate}
\vskip 10pt
\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.
\item[] {\it Solution:}  Thinking about the unit conversion, we see
$${\rm horsepower} = {\rm megawatt} \times \frac{{\rm horsepower}}{{\rm megawatt}}$$
That is there are 
$$\frac{1}{.0007457} \doteq 1341$$
horsepower in one megawatt.  Hence $40,000$ megawatts is 
$$40,000 \times \frac{1}{.0007457} = 5.364 \times 10^7{\rm\ horsepower} $$
\item[B.] (10) Construct an exponential model for $WP = $
wind power generation in units of $10^4$ megawatts, as a function of
$t = $ years since 2004.
\item[] {\it Solution:}  In units of $10^4$ megawatts, the 
model equation is:  
$$WP = (4.0)(1.2599)^t$$
\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?
\vskip 10pt
$$
\begin{array}{c|c|c|c|c|c|c|c|}
Year & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\
\hline
WP  & 4.00  & 5.04 & 6.35 & 8.00 & 10.08 & 12.70 & 16.00  \\
\hline
\end{array}
$$
The wind power generation doubled by 2007 (and again by 2010).
\item[D.] (10) How many years will it take for wind power generation
to reach $320,000$ megawatts according to your model?
\item[] {\it Solution:}  $320000 = 32 \times 10^4$, so the equation 
we want to solve is $32 = 4(1.2599)^t$.  This is true when $t = \frac{log(8)}{\log(1.2599)} \doteq 9$,
or in the year 2013.
(This can also be seen without the calculation since the table shows that
the multiplier $M = 1.2599$ gives $M^3 \doteq 2$.  So the next doubling
to $320000$ will occur in another 3 years after 2010.)
\end{enumerate}
\vskip 10pt
\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)] $\underline{\ \ \  X \ \ \ \ }$ $P(n) = 1.05 P(n - 1) - 20$
\item[4)] $\underline{\ \ \ \ \ \ \ \ }$ $P(n) = 1.05 P(n - 1) + 20$
\end{enumerate}
\vskip 10pt
\item[B.] (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)
\item[] {\it Solution:}
$$
\begin{array}{c|c|c|c|c|c|c|}
n & 0 & 1 & 2 & 3 & 4 & 5  \\
\hline
P(n)  & 500 & 505 &510 & 516 &522 & 528  \\
\hline
\end{array}
$$
\item[C.] (5)  What happens to the population in the long run as $n$ increase?  Does
it tend to a definite value?
\item[] {\it Solution:}  It seems that the population will continue
increasing at an increasing rate as $n$ increases.  Note that if
$P(n-1) > 400$, then $P(n) = (1.05)P(n-1) - 20 > P(n-1)$ and the difference
$P(n) - P(n-1) = (.05)P(n-1) - 20$ is bigger the larger $P(n-1)$ is.
\end{enumerate}
\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 are log with base 10 as we discussed in class).
\item[] {\it Solution:}  The linear equation is equivalent to $\log(y) = m\log(x) + b$
in terms of the original variables.  So exponentiating both sides we get
$y = 10^b\cdot x^m$.  In other words, $m$ is the power in the power law, and 
$10^b$ is the constant multiplier.
\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 $r = -1$.
\item[] {\it Solution:}  The correlation coefficient $r$ measures the degree of 
linearity in a scatter plot (in other words, how close the data points come to 
lying on a single straight line).  If $r = 1$, then all the points are on a 
single line of positive slope.  If $r = -1$, all the points are on a single
line of negative slope.  We used this to measure the goodness
of fit even for exponential or power law models.  When we
did this, we were looking at the correlation coefficient
for the \emph{transformed} data (the $(\log(x_i),\log(y_i))$
in the power law case, and the $(x_i,\log(y_i))$ in
the exponential case).  
\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.
\item[] {\it Solution:}  For a chart indicating the composition of a whole made up 
of several parts, either a pie chart or a bar chart could be used.   But a \emph{pie chart} would
be a slightly superior choice to show the composition.  For a pie chart,
we would compute the percentages of the whole represented by each species:
$10 + 12 + 5 + 2 + 1 = 30$ trees.  So oaks account for $10/30 \times 100\% = 33.3\%$, 
maples account for $12/30\times 100\%$, or $40\%$, pines account for $5/30 \times 100\%$,
or $16.7\%$, hemlocks account for $6.7\%$, and chestnuts account for the remaining
$3.3\%$.  These would be shown as fractions of a whole circle in the pie chart.   
\end{enumerate}
\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?
\vskip 10pt
\noindent
{\it Solution:}
A complete answer should include the following reservoirs:  the atmosphere, 
the surface ocean, the deep ocean, the terrestrial biosphere (all living
plants and animals on land), and soil/debris on land.  The major fluxes
are gas exchange between the atmosphere and surface ocean, downwelling,
and the ``biopump'' from the surface ocean to the deep ocean, and
upwelling from the deep ocean to the surface ocean. 
Up- and down-welling are physical mixing of the
different layers of ocean water. In the ``biopump''
carbon from ocean plants and animals, including
mostly plankton, sinks to sediments in the ocean floor.  Then there are
fluxes of photosynthesis from the atmosphere to the terrestrial biosphere,
and respiration going the other way.  Photosynthesis is
the process by which plants take carbon dioxide out
of the atmosphere and incorporate the carbon into
their tissues.  Oxygen is released in the process, 
which is breathed in by animals.  Animal respiration
releases carbon dioxide into the atmosphere.  
Death takes carbon from the 
biosphere to soil/debris, and decomposition takes carbon from the soil/debris reservoir to the atmosphere.  Finally, human activity
from fossil fuel burning puts carbon into the atmosphere, and deforestation
takes carbon out the terrestrial biosphere and adds it to the atmosphere.

See the class notes for the associated diagram.

Some obvious things our model did not take into account were:
\begin{enumerate}
\item[1.]  It did not account for the way increases in 
global temperature might affect the carbon fluxes in 
the cycle.  Those rates were constant or constant proportions
(multipliers).
\item[2.]  As a climate model, it did not take the effects of other atmospheric
gases such as water vapor, methane, etc. into account.  Some of 
these (especially methane) might be even stronger greenhouse gases
than carbon dioxide.
\item[3.]  It did not take into account the fact that 
the total energy received by the Earth from solar radiation
actually varies over time.  This can cause variations in photosynthesis,
carbon levels, and in climate as well.
\item[4.]  Volcanoes add some $CO_2$ into
the atmosphere when they erupt.  
\end{enumerate}
On the other hand, some human activities such as decreasing
our rate of fossil fuel burning or deforestation \emph{can}
be accounted for in the model, and we investigated 
their effects in one the questions on the project.  But
no mathematical model can include \emph{everything} that might
affect the behavior of the real-world system that is 
being modeled.
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