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\begin{document}

\section*{Homework 8	} 
\textbf{Instructions:} Please answer the following questions with well thought out answers. You can use this sheet to write your answers or use your own.  STAPLE this sheet to the front of the rest of your work. It is important to be able to explain ideas clearly. In this assignment pretend that you are trying to explain to a friend your answers. You should strive for your answers to be: precise, accurate, succinct, and understandable.
\subsection*{Space Debris.} 
One problem that  NASA and space scientists from other countries must deal with is the 
accumulation of space debris in orbit around Earth. Such debris includes payloads that are no 
longer operating; spent stages of rockets, assorted parts and lost tools; debris from the breakup of 
larger objects or from collisions between objects; and countless small pieces, such as flakes of 
paint and even smaller objects. Because bodies in Earth orbit travel at approximately 17500 
miles per hour, a collision with even a tiny object can have catastrophic effects. In 1990, 
scientists estimated that a total of 4 million pounds of debris was in Earth orbit. They also 
estimated that at that time, we were adding 1.8 million pounds per year to the already serious 
problem, which in a few years would result in 9.5 million pounds of orbital debris. The 1990 
prediction also stated that the amount of debris being added was anticipated to increase to a rate 
of 2.7 million pounds per year by the year 2000.
\begin{question} How much is $4$ million pounds of anything? Give three concrete examples of something weighing $4$ millions pounds (example: 12549020 baseballs would weigh $4$ million pounds). 
\end{question} 
\begin{enumerate} 
\item First example:  
\vspace{ 1.5 cm} 
\item Second example: 
\vspace{ 1.5 cm} 
\item Third example: 
\vspace{1.5 cm} 

\end{enumerate} 
\subsection*{Modeling the problem: Linear Growth} 
The problem of determining the amount of debris in space and the anticipated rate of increase of 
such matter is not one that can be solved directly. We cannot locate, count, and weigh all objects 
in orbit. Nor can we predict with assurance when two of them will collide. Instead, we must rely 
on mathematical models to help us represent the problems and identify trends and expected 
outcomes. In these activities, you will create and compare various mathematical models to help 
you investigate some of the questions raised by the proliferation of orbital debris. These models 
are greatly simplified in their assumptions so that you can investigate them with calculators, 
spreadsheets, and graphing utilities, but they provide insight into the process of mathematical 
modeling and its importance. 
\newpage
\begin{question} 
When creating models, mathematicians favor the simplest model that will account for the 
phenomena in question. Generally, a linear model gives the simplest case. So, using the 
reported 1990 rate of increase of 1.8 million pounds per year and assuming 4 million pounds 
of existing debris at the beginning of 1990, write a linear model to predict the number of 
pounds of orbital debris at the end of any given year, t. Assume that t = 1 represents 1990.
\end{question} 
\begin{enumerate} 
\item Linear model using rate of increase of 1.8 million pounds per year: 
\end{enumerate} 
\vspace{ 2 cm} 
\begin{question}
Write a second linear model using the predicted 2.7 million pounds per year rate of increase 
and the initial 4 million pounds for 1909. 
\end{question} 
\begin{enumerate} 
\item  Linear model using rate of increase of 2.7 million pounds per year: 
\end{enumerate} 
\vspace{2 cm}
\begin{question} Evaluate each model for several years to determine the year in which the predicted 9.5 
million pounds of accumulated debris would occur. 
\end{question} 
\begin{enumerate} 
\item What year does the first model predict there will be 9.5 million pounds of space debris? 
\vspace{2 cm} 
\item What year does the second model predict there will 9.5 million pounds of space debris? 
\vspace{2 cm} 
\end{enumerate}  
\begin{question} 
Do you think that either of your linear models accurately represents that situation of 
escalating amounts of space debris as described in the original paragraph? Why or why not? 
\end{question}
\newpage 
\subsection*{Refining the Model: Quadratic Growth }
Does either rate, 1.8 million pounds per year or 2.7 million 
pounds per year, tell us how much debris is building up 
between 1990 and 2000? Which rate of increase should we 
use? Obviously the amount being added each year is 
changing during this period, but by how much each year? 
The problem is one of acceleration, not constant velocity, so 
we need to adjust our model. 
 
Again, let's make the simplest assumption: the rate at which 
we are adding debris increases at a constant rate from 1.8 
million pounds per year in 1990 to 2.7 million pounds per 
year in 2000. This change means that over the ten-year period from the end of 1990 through 
2000, the rate (velocity) of littering will increase by 0.9 million pounds (2.7 - 1.8 = 0.9), 
and we are making the assumption that this increase is achieved in equal annual increments of 
0.09 million pounds per year in each year of the decade. 

\begin{question} Create a table to show the amount of debris added each year and the total 
amount in orbit at the end of the year.
\begin{tabular} {|l |c | c|} 
\hline
Year & Amount added in year (millions of pounds) & Total in orbit at end of year (millions of pounds)\\ 
\hline && \\
1990 & 1.8 & 5.8 \\ \hline && \\
1991&&\\ \hline && \\
1992&& \\ \hline && \\
1993 & & \\ \hline && \\
1994 & & \\ \hline && \\
1995 & & \\ \hline && \\
1996 & & \\ \hline && \\
1997 & & \\ \hline  && \\
1998 & & \\ \hline && \\
1999 & & \\ \hline && \\
2000 & & \\ \hline
\end{tabular} 
\end{question} 

\begin{question} Since we assumed that the increase in the velocity of littering was achieved in equal annual 
increments, you can write a linear equation that describes the increase in the amount of debris 
being added each year (i.e, the increase in the annual velocity of littering as a function of the 
number of years since 1990. In this case, we let $a = 0$ in $1990$ because we are assuming that 
the 1990 rate of 1.8 million pounds per year is our baseline rate. Then $d = f(t)$ represents the 
rate of littering $t$ years after 1990.
\end{question} 
\begin{enumerate} 
\item Write a linear equation that describes the increase in the amount of debris 
being added each year: 
\end{enumerate} 
\vspace{1.5 cm} 
\newpage

\begin{question} The situation described in your equation, where the rate of increase of litter is itself 
increasing at a constant rate, is analogous to a vehicle that accelerates at a constant rate from 
an initial velocity, $v_0$, to a final velocity, $v_f$. Use the data generated in the table above to 
create a scatterplot of the total number of pounds of orbital debris that have accumulated 
relative to the year\footnote{In essence, graph the data from the table}. You graph should cover the period from 1990 through 2000. 
\end{question} 
\begin{center}
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[help lines] (0,0) grid (20,20);
\draw[->] [thick] (0,0) -- (0,20);
\draw[->][thick](0,0) -- (20,0);
\end{tikzpicture}
\end{center} 

\begin{question} 
Fit a line to your data and decide whether the accumulation of debris appears to be linear. 
Write your conclusion and describe the evidence on which you based your decision. 
\end{question}
\begin{enumerate} 
\item Do you think the accumulation of the debris appears to be linear? \vspace{ 2 cm}
\item What evidence helped you reach this conclusion?  \vspace{ 2cm}
\end{enumerate} 

\begin{question} Draw a line that approximates the data. Calculate the equation for that line. 
\end{question} 
\begin{enumerate} 
\item What is the equation for your ``best fit'' line? 
\end{enumerate} 
 \vspace{ 2cm}
\pagebreak

\begin{question} If you asked a computer to give you a quadratic function that approximated your data, it would give you the following function $g(x)=4+1.755x+.045x^2$. Graph this function on top of the scatter plot you created (use a different color if possible). 
\end{question} 



\begin{question} Compare how $g(x)=4+1.755x+.045x^2$ with the earlier linear models. 
\end{question} 
\vspace{1.5 cm} 

\begin{question} In each case, use your models to predict the accumulation of debris after twenty years, thirty 
years, and fifty years. Describe the behavior of the linear model versus the quadratic model 
over time. \end{question}
\begin{center}  
\begin{tabular} {|l|l|l|l|}
\hline
Year & Linear 1 & Linear 2 & Quadratic \\ \hline
1990 & & & \\ 
&&& \\ \hline
2010 & & & \\ 
& & &\\  \hline 
2020 & & & \\ 
& & & \\ \hline 
2030 & & & \\ 
& & & \\ \hline 
2040 & & & \\ 
& & &\\
 \hline 
\end{tabular} 
\end{center} 
\newpage
\subsection*{Refining the Model: Exponential Growth }

\begin{question} 
Instead of assuming that the acceleration of space garbage is constant, we could assume that amount of space garbage is constantly growing by $20 \%$. Under this assumption we would have an exponential model 
$$ f(t) = 4 e^{.2 t}$$ 
Make a table for $f(t)$. 
\end{question} 
\begin{center} 
\begin{tabular} { | l | c | } \hline
$t$ & \textcolor{white}{BLAN}$f(t)$ \textcolor{white}{SPACE} \\ \hline 
0 & \\
& \\ 
\hline
1 & \\
& \\ 
\hline
2 & \\
& \\ 
\hline
3 & \\
& \\ 
\hline
4 & \\
& \\ 
\hline
5 & \\
& \\ 
\hline
6 & \\
& \\ 
\hline
7 & \\
& \\ 
\hline
8 & \\
& \\ 
\hline
9 & \\
& \\ 
\hline
10 & \\
& \\ 
\hline

\end{tabular} 
\end{center} 
\begin{question} 
How does the exponential function $f$ compare to the data given in the problem? 
\end{question} 
\newpage

\subsection*{What Goes Up Might Come Down: Extending the Models} 
All the models you have developed thus far assume that the 
additional every year some of the debris slows down enough 
to re-enter the atmosphere where it burns up or, on rare 
occasions, returns to Earth. Assume for the moment that 10 
percent of the debris in orbit at the beginning of any year will 
be destroyed during that year. 

\begin{question} Modify you linear, quadratic, and exponential models to 
account for the situation in which additional debris is being 
added each year while 10 percent of what was already in 
orbit is being destroyed. Here we won't be able to write down an explicit formula. 
\subsubsection*{First Linear Model} 
\begin{multicols}{2}
\begin{enumerate} 
\item Modify your first linear model and write debris as a sequence $d_n$ \vspace{ 2 cm} 
\item Write down a recursive relationship between step $n$ and step $n+1$. \vspace{ 2 cm} 
\item Make a table for the first 10 years using these sequences. \vspace{ 2 cm}
\end{enumerate} 
\begin{center}
\begin{tabular} { | l | c | } \hline
$n$ & \textcolor{white}{BLAN}$d_n$ \textcolor{white}{SPACE} \\ \hline 
1 & \\
\hline
2 & \\
\hline
3 & \\
\hline
4 & \\
\hline
5 & \\
\hline
6 & \\
\hline
7 & \\
\hline
8 & \\
\hline
9 & \\
\hline
10 & \\
\hline

\end{tabular} 
\end{center}
\end{multicols}
\subsubsection*{Second Linear Model} 
\begin{multicols}{2}
\begin{enumerate} 
\item Modify your second linear model and write debris as a sequence $g_n$. \vspace{ 2 cm}
\item Write down a recursive relationship between step $n$ and step $n+1$. \vspace{ 2 cm}
\item Make a table for the first 10 years using these sequences. \vspace{ 2 cm}
\end{enumerate}

\begin{center}
\begin{tabular} { | l | c | } \hline
$n$ & \textcolor{white}{BLAN}$g_n$ \textcolor{white}{SPACE} \\ \hline 
1 & \\
\hline
2 & \\
\hline
3 & \\
\hline
4 & \\
\hline
5 & \\
\hline
6 & \\
\hline
7 & \\
\hline
8 & \\
\hline
9 & \\
\hline
10 & \\
\hline

\end{tabular} 
\end{center}
\end{multicols}
\newpage
\subsubsection*{Quadratic Model}
\begin{multicols}{2}
\begin{enumerate}
\item Modify your quadratic model and write debris as a sequence $h_n$ \vspace{ 2 cm}
\item Write down a recursive relationship between step $n$ and step $n+1$. \vspace{ 2 cm}
\item Make a table for the first 10 years using these sequences. \vspace{ 2 cm}
\end{enumerate}
\begin{center}
\begin{tabular} { | l | c | } \hline
$n$ & \textcolor{white}{BLAN}$h_n$ \textcolor{white}{SPACE} \\ \hline 
1 & \\
\hline
2 & \\
\hline
3 & \\
\hline
4 & \\
\hline
5 & \\
\hline
6 & \\
\hline
7 & \\
\hline
8 & \\
\hline
9 & \\
\hline
10 & \\
\hline

\end{tabular} 
\end{center}
\end{multicols}
\subsubsection*{Exponential Model} 
\begin{multicols}{2}
\begin{enumerate} 
\item Modify your first exponential mode (this will require seeing it as a geometric sequence) and write debris as a sequence $k_n$. \vspace{ 2 cm}
\item Write down a recursive relationship between step $n$ and step $n+1$. \vspace{ 2 cm}
\item Make a table for the first 10 years using these sequences. \vspace{ 2 cm}
\end{enumerate} 
\begin{center}
\begin{tabular} { | l | c | } \hline
$n$ & \textcolor{white}{BLAN}$k_n$ \textcolor{white}{SPACE} \\ \hline 
1 & \\
\hline
2 & \\
\hline
3 & \\
\hline
4 & \\
\hline
5 & \\
\hline
6 & \\
\hline
7 & \\
\hline
8 & \\
\hline
9 & \\
\hline
10 & \\
\hline

\end{tabular} 
\end{center}
\vspace{1cm}
\end{multicols}
\begin{question}
In which case - linear, quadratic, or exponential - does the assumption of a $10 \%$ reentry rate have the greatest effect? 
\end{question}

\end{question} 
\newpage
\subsection*{Putting Your Models to Work}
The power of mathematical models comes from their ability to enable us to ask "What if?" 
questions. An earlier question is an example: What if we could increase the rate at which orbital 
debris is destroyed? Other questions might include these: What if we decrease the rate at which 
we are adding debris and find a way to increase the rate at which existing debris is destroyed? 
Such questions lead to open-ended investigations using mathematical models.  

\question{ Working with a partner or a small group, generate two or three specific questions that you 
would like to investigate.}
\begin{enumerate} 
\item What if .... 
\item What if ... 
\item What if ... 
\end{enumerate} 

\begin{question} Describe a plan for investigating your questions using spreadsheets, graphing utilities, or other appropriate technology. 
\end{question} 



\end{document} 