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\usepackage{quiz}
\usepackage{alltt}
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\usepackage{hyperref}
\usepackage{graphicx}
\begin{document}
\section*{Sample Quiz 4}
{\bf Sample Quiz4 Problem 1}.
The velocity of a crossbow arrow fired upward from the ground is given at different times in the following table.


\begin{tabular}[b]{|l|c|l|}
\hline
Time $t$ in seconds & Velocity $v(t)$ in ft/sec & Location\\
\hline
0.000 & 50 & Ground \\
1.413 & 0 & Maximum \\
2.980 & -45 & Near Ground Impact\\
\hline
\end{tabular}
\hfill
\includegraphics[scale=0.3]{crossbowArrow}


\begin{itemize}
\item[{\bf (a)}] The velocity can be approximated by a quadratic polynomial
$$v(t) = at^2 + bt + c$$
which reproduces the table data.
Find three equations for the coefficients $a , b , c$. Then solve for them to obtain $a \approx 2.238$, $b\approx -38.55$, $c=50$.

\item[{\bf (b)}] Assume a drag model $v'=-32-\rho v$. Substitute the polynomial answer of (a) into this differential equation, then substitute $t=0$ and solve for $\rho \approx 0.131$.

\item[{\bf (c)}] Solve the model $w'=-32-\rho w$, $w(0)=50$ with $\rho=0.131$.

\item[{\bf (d)}] Compare $w(t)$ and $v(t)$ in a plot. Comment on the plot and what it means.
\end{itemize}

{\bf References}. Edwards-Penney sections 2.3, 3.1, 3.2.
Course documents on
\href{http://www.math.utah.edu/~gustafso/s2015/2280/linearequDRAFT.pdf}{Linear algebraic equations}
and
\href{http://www.math.utah.edu/~gustafso/s2015/2280/newtonModelsDE2008.pdf}{Newton kinematics}.


{\bf Sample Quiz4 Problem 2}.
Consider the system of differential equations
$$
\begin{array}{lcrcrcrcr}
x_1' &=& -\frac16 x_1& & & +& \frac16 x_3, \\[2ex]
x_2' &=& \frac16 x_1 & -& \frac13 x_2, & & \\[2ex]
x_3' &=& & &  \frac13 x_2  &-& \frac16 x_3,
\end{array}
$$
for the amounts $x_1, x_2, x_3$ of salt in recirculating brine tanks, as in the figure:
\begin{center}
\includegraphics[scale=0.75]{brineTanksRecycled.png}
 \\
 {\bf Recirculating Brine Tanks A, B, C} \\
 The volumes are $60,30,60$ for $A,B,C$, respectively.
\end{center}
The steady-state salt amounts in the three tanks are found by formally setting $x_1'=x_2'=x_3'=0$
and then solving for the symbols $x_1,x_2,x_3$.
Solve the corresponding linear system of algebraic equations to obtain the answer $x_1=x_3=2c$, $x_2=c$, which means the total amount of salt is uniformly distributed in the tanks in ratio $2:1:2$.


{\bf References}. Edwards-Penney sections 3.1, 3.2, 7.3 Figure 5.
Course documents on
\href{http://www.math.utah.edu/~gustafso/s2015/2280/linearequDRAFT.pdf}{Linear algebraic equations}
and
\href{http://www.math.utah.edu/~gustafso/s2015/2280/systemsBrineTank.pdf}{Systems and Brine Tanks}.

% A:=Matrix([[-1/6,0,1/6],[1/6,-1/3,0],[0,1/3,-1/6]]);
% linalg[linsolve](A,<0,0,0>);
%  ans=Vector[row](3, {(1) = 2*_t[1], (2) = _t[1], (3) = 2*_t[1]})

%{\bf Sample Quiz4 Problem 3}. Resistive electrical network with direct current sources.
%
%\begin{itemize}
%\item[{\bf 1}.]
%\item[{\bf 2}.]
%\item[{\bf 3}.]
%\end{itemize}
%
%{\bf References}. Edwards-Penney Sections 3.1, 3.2, 3.3.
%\href{http://www.math.utah.edu/~gustafso/s2015/2280/linearequDRAFT.pdf}{Linear algebraic equations}

\end{document}
