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\textbf{College of the Holy Cross, Fall Semester 2018} \\
\textbf{MONT 104N -- Modeling the Environment} \\
\textbf{Solutions for Midterm Exam, October 26}
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\noindent
I.  In discussing amounts of water needed for irrigation of farmland,
a common unit of volume is the acre-foot.  One acre-foot of water is
the amount of water necessary to cover a perfectly flat field one acre in 
area, to a depth of one foot.  Using the information below, answer questions A and B.
\begin{itemize}
\item 1 yard $=$ 3 feet
\item 1 acre $=$ 43560 square feet
\item 1 meter $\doteq$ 3.28 feet
\end{itemize}
\begin{enumerate}
\item[A.] (10)  How many cubic yards (${\rm yd}^3$) of water are there in one acre-foot?
\item[] {\it Solution:}  The best way to do this is probably to express the answer in 
cubic feet first, then convert to cubic yards:  1 acre-foot = (1 acre) $\times$ (1 foot) $=$ 
43560 cubic feet.  Then 1 cubic yard $= 3^3 {\rm\ cubic\ feet} = 27 {\rm\ cubic\ feet}$, so 
$1 {\rm\ cubic\ foot} = \frac{1}{27} {\rm\ cubic\ yard}$.  Hence:
$$43560 {\rm\ cubic\ feet} \times \frac{1}{27} \frac{{\rm cubic\ yard}}{{\rm cubic\ foot}} \doteq 1613.3 {\rm\ cubic\ yards}.$$
\item[B.] (10)  How many cubic meters (${\rm m}^3$) of water are there in one acre-foot?
\item[] {\it Solution:}  To convert from cubic feet to cubic meters, use the last given 
conversion:  1 cubic meter $\doteq 3.28^3$ cubic feet $\doteq 35.29$ cubic feet.  Hence
$$43560 {\rm\ cubic\ feet} \times \frac{1}{35.29} \frac{{\rm cubic\ meter}}{{\rm cubic\ foot}} \doteq 1234.3 {\rm\ cubic\ meters}.$$
\end{enumerate}
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\noindent
II.  According to the 2014 Advancing Sustainable Materials Management factsheet produced by 
the U.S. Environmental Protection Agency, the total amounts of various types of materials 
produced, recycled, and disposed of in landfills in the U.S. were as follows.  
All figures are in units of millions of tons.
$$\matrix{ {\bf Category} & {\bf Produced} & {\bf Recycled} &{\bf Landfilled}\cr
           {\rm Paper\ and\ paperboard} & 68.61 & 44.40 & 19.47\cr
           {\rm Metals} & 23.26 & 7.90 & 12.82\cr
           {\rm Plastics} & 33.25 & 3.17 & 25.10\cr
           {\rm Glass} & 11.48 & 2.99 & 7.04 \cr
           {\rm Others} & 4.44 & 1.29 & 2.58 \cr}$$  
\begin{enumerate}
\item[A.]  (15)  Construct one bar chart showing the \emph{percentages} of
the materials produced that are recycled and a second chart showing the percentages of
these materials that are disposed of in landfills.  
\item[] {\it Solution:}  Computing percents we have  
$$\matrix{ {\bf Category} & \% {\bf Recycled} & \% {\bf Landfilled}\cr
           {\rm Paper\ and\ paperboard} & 64.7\% & 28.4\%\cr
           {\rm Metals} & 34.0\% & 55.1\%\cr
           {\rm Plastics} & 9.5\% & 75.5\% \cr
           {\rm Glass}  & 26.0\% & 61.3\% \cr
           {\rm Others} & 29.1\% & 55.9\% \cr}$$ 
Note: these don't sum to 100\% since some of the materials remain in use or are disposed
of by incineration and other methods.  The bar chart would have bars of heights showing these percentages in the 
Recycled column.
\item[B.] (5) Which material is landfilled most, proportionally 
(that is, considering the landfilled amount as a fraction of the amount produced)?  Explain. 
\item[] {\it Solution:}  The material having the largest percentage of the amount produced
going into landfills is the Plastics category, since the $75.5\%$ for Plastics is the largest
number in the Landfilled column in the last table.   
\item[C.]  (10)  What percentage of the total weight of these classes of materials
is recycled?   What percentage of the total is landfilled?  
\item[] {\it Solution:}  The total amount produced is $68.61+23.26+33.25+11.48+4.44 = 141.04$ million tons.  The total
recycled is $44.4+7.9+3.17+2.99+1.29 = 59.75$ million tons.  This is 
$59.75/141.04 \times 100\% \doteq 42.4\%$ of the total produced.  Similarly, the total amount landfilled
is $19.47+12.82+25.1+7.04+2.58 = 67.01$ million tons.  This is $67.01/141.04 \times 100\% \doteq 47.5\%$
of the total.  Note that you \emph{cannot} do this by averaging the percents computed in part A.  
The amounts are different so they contribute to the total percent differently.  
\end{enumerate}
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\noindent
III.   
\begin{enumerate}
\item[A.]  (11) In 2010, $2.74 \times 10^5$ hybrid and electric vehicles were sold in
the US.  In 2015, the number was $4.98 \times 10^5$.  Construct a linear model for the 
number of hybrid vehicles sold as a function of $t = $ years after 2010 using this information.
What does your model predict about the number of hybrid and electric vehicles sold in 2018?
\item[] {\it Solution:}  The slope is 
$$m = \frac{498000 - 274000}{2015 - 2010} = 44800.$$
Using the point-slope form, the linear equation is
$${\rm Hybrids\ sold} - 274000 = 44800 \cdot ({\rm years\ after\ 2010}) = 44800 t,$$
or 
$$  {\rm Hybrids\ sold} = 44800 t + 274000.$$
Since $2018 - 2010 = 8$.  The model would predict hybrid sales of 
$$44800 \cdot 8 + 274000 = 632400$$
vehicles this year.  
\item[B.]  The following spreadsheet shows more complete data 
for the total number of hybrid (and electric) vehicles sold for each year from 2010 through 2015.
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\noindent
[Spreadsheet omitted here.]
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\noindent
Answer each of the following questions based on this information:
\begin{enumerate}
\item[1.]  (3)  Based on the scatter plot, the $R^2$ (``coefficient of determination'') is
probably:
\leftline{ between $0$ and $.25$ $\underline{\qquad\qquad\qquad}$}
\leftline{ between $.25$ and $.5$ $\underline{\qquad\qquad\qquad}$} 
\leftline{ between $.5$ and $.75$ $\underline{X}$} 
\leftline{ between $.75$ and $1$ $\underline{\qquad\qquad\qquad}$}
\noindent
Note that the least squares trendline (regression line) is not very close
to the data points in this case, so $R^2$ would be significantly less than $1$.  On
the other hand, there is a definite upward trend, so the $R^2$ value would not
be really small either.  This is the best estimate.  (Note:  I also gave 2/3 points for
either of the adjacent answers, though.) 
Using GoogleSheets, the computed value of $R^2$ is $R^2 \doteq .64$.   
\item[2.]  (3) The least squares regression line for this data set would
have a slope $m$ about:
\leftline{ $m = 60000$ $\underline{X}$}
\leftline{ $m = -60000$ $\underline{\qquad\qquad\qquad}$}
\leftline{ $m = 30000$ $\underline{\qquad\qquad\qquad}$}
\noindent
The slope is definitely positive, so the second value is ruled out.  Looking
at where the regression line crosses Year $= 2010$ and Year $ = 2011$, we can see that
the vertical axis value is increasing by about $60000$ for that unit change in the year.
\item[3.]  (3)  What are the units of the slope?   {\it Solution:}  vehicles per year.  
\end{enumerate}
\end{enumerate}
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\noindent
{\it Essay} (30)
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Why is there great interest in creating synthetic bacteria and other single-cell
organisms?  For what jobs might they be useful?  On the other hand, what 
potential dangers have people suggested in this kind of work?  Considering 
{\it Frankenstein} as a warning, is this something we should be pursuing with our
technology?  Are the potential good effects enough to outweigh the possible dangers?    
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\noindent
{\it Model response:}  Apart from general scientific curiosity and the desire to put
our current knowledge of genetics to work by designing a new functioning genome ``from
the ground up,'' there is interest in creating synthetic 
bacteria and other single-cell organisms for their potential uses for things like:
\begin{itemize}
\item pollution mitigation (i.e. using the new organisms to capture and dispose of
pollutants in air or water)
\item production of synthetic pharmaceuticals 
\item production of synthetic biofuels
\item production of polymers and other materials.
\end{itemize}
All of this would fall under the general category of biological engineering, and there
are great potential benefits if such organisms could be safely produced.  

On the other hand, the fact that the scientists who created this new bacterial
life form do not understand what about 30\% of its genetic information does
opens up all sorts of possibilities for unintended consequences.  It isn't 
possible to predict what the organism might do in new situations outside
the ones where it has been tested, for instance.  Might it grow uncontrollably?
Might it interact in unforeseen ways with other naturally occurring life forms?
Might it mutate into something that produces diseases in humans or other current
life forms?

How you come down on whether this sort of experimentation is justified is up to 
you.  There's no one ``right answer'' to whether we should be pursuing this
kind of research.  I'll be looking for whether you can articulate a position
clearly based on the evidence you have.  
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