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\centerline{MONT 104N -- Modeling the Environment}
\centerline{Solutions for Midterm Exam -- November 1, 2019}
\bigskip 
\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 flat field one acre in area to a depth of 
one foot.  Using the information below, answer questions A and B.
$$\matrix{ &1\ {\rm yard} = 3\ {\rm feet}\cr
           &1\ {\rm acre} = 43560\ {\rm square\ feet}\cr
           &1\ {\rm meter} \doteq 3.28\ {\rm feet}\cr}$$ 
\item{A.} (10)  How many cubic yards (${\rm yd}^3$) of water are in one acre-foot?
\item{} {\it Solution:}  
$$1\ {\rm acre-foot} = 43560\ {\rm ft}^3 = 43560\ {\rm ft}^3 \times {1\over 3^3}\ {\rm yd}^3/{\rm ft}^3
\doteq 1613.3\ {\rm yd}^3$$ 
\item{B.} (10)  How many cubic meters (${\rm m}^3$) of water are in one acre-foot?
\item{} {\it Solution:}
$$1\ {\rm acre-foot} = 43560\ {\rm ft}^3 =  43560\ {\rm ft}^3 \times 
{1\over (3.28)^3}\ {\rm m}^3/{\rm ft}^3
\doteq 1234.4\ {\rm m}^3$$ 
\noindent
II.  The following table gives amounts (in tons) of materials of different types
recycled in Washington State in two years, 1986 and 1998.
$$\matrix{ {\bf Category} & {\bf 1986} & {\bf 1998}\cr
           {\rm Paper} & 391994 & 821994\cr
           {\rm Metal} & 9528 & 318710\cr
           {\rm Plastic} & 349 & 9871\cr
           {\rm Glass} & 48013 & 113338\cr
           {\rm Organics + Others} & 352 & 903466\cr}$$  
\item{A.}  (15)   Construct a chart {\it for 1998} showing the percentages
of that year's total accounted for by each of these types of materials.  Any reasonable 
type of chart is OK.
\bigskip
\item{} {\it Solution:}  We first need to compute the total amount
of all types for 1998, then compute the percents of each type of recyclable. Here
is a pie chart (computed with Google Sheets).  (If you used a pie chart, 
it doesn't need to look this good, of course!)
\vglue 5.0truein
\item{B.}  (5)  What percent of the total {\it for 1986} was accounted for by 
Paper and Plastic together?   
\bigskip
\item{} {\it Solution:}  The total for 1986 was 450236 tons, while the sum of 
the Paper and Plastic categories was 392343 tons.  The percentage accounted
for by Paper and Plastic together was
$${392343 \over 450236} \times 100\% \doteq 87.1\%$$
\bigskip
\noindent
III.   
\item{A.}  (10) In 2012, there were about $5.3 \times 10^4$ {\it alternative
fuel vehicles} (powered by electricity, ethanol, natural gas, etc.) sold
in the U.S.  By 2016, that number had risen to $1.6 \times 10^5$.
Construct a linear model for $AFV = $ number of alternate fuel vehicles, 
as a function of $t = $ years since 2012. 
\bigskip 
\item{} {\it Solution:}  The slope is 
$$m = {1.6 \times 10^5 - 5.3 \times 10^4\over 2016-2012} = 26750$$
(vehicles per year).  The equation of the linear model is 
$${\rm AFV} = 26750 t + 5.3\times 10^4$$  
\item{B.}  (5) What does your model predict
about the number of AFV sold in 2017?  The actual number was $1.95 \times 10^5$
vehicles.  What is the percentage difference between the predicted number
(the ``comparison'') and the actual number (the ``reference'')?
\bigskip
\item{} {\it Solution:}  The model prediction is 
$$AFV = 26750\cdot 5 + 5.3 \times 10^4 = 186750 \doteq 1.9 \times 10^5$$
The percent difference between this and the actual number is
$${1.9 \times 10^5 - 1.95 \times 10^5 \over 1.95\times 10^5} \times 100\% \doteq -2.6\%$$
(In other words, the model prediction is about $2.6\%$ too small.  If you use the 
value $186750$ instead, then the percent difference is about $-4.4\%$.  That is OK 
too.) 
\bigskip
\noindent
IV.  Wind power has emerged as one of the faster-growing methods
of electricity 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.
\bigskip
\item{A.} (10) Using this information, construct an exponential model for $WP = $
wind power generation as a function of
$t = $ years after 2016.
\bigskip  
\item{} {\it Solution:}  The model is 
$$\eqalign{WP &= (301)\cdot \left(1 + {33.2\over 100}\right)^t\cr
              &= (301)\cdot (1.332)^t}$$
(the units are gigawatts).
\bigskip
\item{B.}  (5) According to your model, how long will it take for the generating
power of wind turbines to reach $3000$ gigawatts?
\bigskip
\item{} {\it Solution:}  
$$3000 = (301)\cdot (1.332)^t$$
when 
$$t = {\log(9.967)\over \log(1.332)} \doteq 8.02$$
That is, slightly more than 8 years after 2016, so 2024.   
\bigskip
\noindent
{\it Essay} (30)
\bigskip
There is an ongoing movement at this point in history (i.e. 2019) 
that is especially strong in Sweden, where it known as {\it flygskam}.  
What does this Swedish word mean in English?  What actions 
are proponents of {\it flygskam} urging other people not to continue
doing, and what are their reasons for doing this?  Part of the underlying reason
for the existence of this movement is concern about how the actions are 
influencing greenhouse gas levels in the atmosphere.   What proportion of
current $CO_2$ emissions can be attributed to these actions and which people
are contributing most to this? 
While the recommendations of proponents of {\it flygskam} might make
a lot of sense from the environmental point of view, there are also 
certainly aspects of modern life that would be drastically changed if those
recommendations were adopted by everyone.  What would we be losing by following 
those recommendations?  Are the environmental benefits great enough to justify the 
costs to individuals and society of adopting those recommendations?  
\bigskip
\noindent
{\it Model Response:}  The Swedish word {\it flygskam} means ``flying shaming'' or 
maybe better, ``shaming for flying.''  This refers to the fact that many environmental
activists, especially in Sweden, are vocally encouraging people to 
{\it cut down on their use of air travel}
by shaming them when they travel by air and bragging about using different 
transportation modes instead. 
(For instance, this is the reason that Greta Thunberg sailed from Europe to 
attend the opening of the United Nations session in New York in September this year
rather than flying.)  

The reason is that air travel is a significant
source of $CO_2$ (and other greenhouse gas) emissions into the upper atmosphere 
from jet engine exhaust.  For instance, it is estimated that air travel accounts
for between 2\% and 2.5\% of all anthropogenic (human-produced) $CO_2$ at 
present.    The individuals who are contributing to this the most
are the ``frequent flyers'' who use air travel to get to 
business destinations  (and academic meetings--see below), plus people
who are flying for tourism.  
If the demand for this was reduced and people used video conferencing for meetings, 
or made more use of less carbon-intensive 
public transportation methods such as passenger rail, then airline
schedules could be trimmed (fewer total flights), and total
$CO_2$ emissions would be reduced (at least somewhat).  

The biggest thing that we would lose if we seriously curtailed the 
availability of air travel would be {\it convenience} in planning trips.  
Fewer flights means fewer options and probably higher prices. 
Passenger rail is an attractive option for shorter distances on the same continent, 
such as the distances between many cities in Europe.  However 
land transportation for distances over about 400 miles is inevitably 
slower than air travel.  For instance, you a direct flight from Boston to Denver 
(about 2000 miles) takes about 5 hours (and including travel to and from 
the airports only adds another 2 hours or so).  On the other hand,
the currently-available rail option (Amtrak) takes about 40 hours 
(including a layover in Chicago and 2 nights on the train -- I
know about this because I recently booked a trip to the upcoming 
Joint Mathematics Meetings in Denver in January). 
Riding a bus would take even longer and be much less comfortable.
Google says you could probably drive the whole way on Interstate highways
in about 30 hours, but that time assumes you never have to stop, sleep, or eat, 
and you never hit any traffic--totally unrealistic!
   
Perhaps more importantly, it would be much harder to gain the direct 
face-to-face contact with other people and other cultures that is possible 
with modern air travel.  The quotation
from Mark Twain on the exam review sheet is an eloquent statement
about how travel can help drive out narrow-mindedness and foster understanding
of other peoples.  

Whether the environmental benefits are great enough to justify
this loss of convenience of travel is in part a matter of opinion.  I'm 
personally torn about this. The total $CO_2$ emissions due to 
aviation are not an especially large part of the total $CO_2$ emissions
and other sources are much more significant. I do try to use
rail options whenever I can, but part of that is just that I really
enjoy rail travel.  There are also times when air travel can be
necessary because of time constraints, and auto travel is necessary for
``out of the way'' destinations.      
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