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\begin{document}
\section*{Quiz 4}
{\bf Quiz4 Problem 1}.
The velocity of a crossbow bolt launched upward from the ground was determined from a video and a speed gun to complete the following table.

\begin{tabular}[b]{|l|c|l|}
\hline
Time $t$ in seconds & Velocity $v(t)$ in ft/sec & Location\\
\hline
0.000 & 60 & Ground \\
1.7 & 0 & Maximum \\
3.5 & -52 & Near Ground Impact\\
\hline
\end{tabular}
\hfill
\includegraphics[scale=0.2]{crossbowBolt}

\begin{itemize}
\item[{\bf (a)}] The bolt 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 the coefficients.

\item[{\bf (b)}] Assume a linear drag model $v'=-32-\rho v$. Substitute the polynomial answer of (a) into this differential equation, then substitute $t=0$ and solve for $\rho \approx 0.11$.

\item[{\bf (c)}] Solve the model $w'=-32-\rho w$, $w(0)=60$ with $\rho = 0.11$.

\item[{\bf (d)}] The error between $v(t)$ and $w(t)$ can be measured. Is the drag coefficient value $\rho=0.11$ reasonable?
\end{itemize}

{\bf References}. Edwards-Penney sections 2.3, 3.1, 3.2.
Course documents on
\href{http://www.math.utah.edu/~gustafso/s2014/2250/linearequDRAFT.pdf}{Linear algebraic equations}
and
\href{http://www.math.utah.edu/~gustafso/s2014/2250/newtonModelsDE2008.pdf}{Newton kinematics}.


{\bf Quiz4 Problem 2}.
Consider the system of differential equations
$$
\begin{array}{lcrcrcrcr}
x_1' &=& -\frac15 x_1& & & +& \frac17 x_3, \\[2ex]
x_2' &=& \frac15 x_1 & -& \frac13 x_2, & & \\[2ex]
x_3' &=& & &  \frac13 x_2  &-& \frac17 x_3,
\end{array}
$$
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 $50,30,70$ 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 $x_1,x_2,x_3$.
\begin{itemize}
\item[{\bf (a)}]
Solve the corresponding linear system of algebraic equations for answers $x_1,x_2,x_3$.
 \item[{\bf (b)}]
 The total amount of salt is uniformly distributed in the tanks in ratio $5:3:7$. Explain this mathematically from the answer in (a).
 % A:=Matrix([[-1/5,0,1/7],[1/5,-1/3,0],[0,1/3,-1/7]]);linalg[linsolve](A,<0,0,0>);
\end{itemize}

{\bf References}. Edwards-Penney sections 3.1, 3.2, 7.3 Figure 5.
Course documents on
\href{http://www.math.utah.edu/~gustafso/s2014/2250/linearequDRAFT.pdf}{Linear algebraic equations}
and
\href{http://www.math.utah.edu/~gustafso/s2014/2250/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/s2014/2250/linearequDRAFT.pdf}{Linear algebraic equations}

\end{document}
