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{\large\bf Quiz 1}. Please prepare your own report on 8x11 paper,
handwritten. Work alone or in groups.

\bigskip

{\bf Quiz1 Problem 1}. An answer check for the differential equation and
initial condition
\begin{equation} \label{quiz1.eq.ivp}
\frac{dy}{dx} = k(73-y(x)), \quad y(0)=28\end{equation}
requires substitution of the candidate solution $y(x)=73 - 45 \, e^{-kx}$ into the left side ($\LHS$) and right side ($\RHS$), then compare the
expressions for equality for all symbols. The process of testing $\LHS=\RHS$ applies to both the differential equation and the initial condition, making the answer check have {\bf two} presentation panels. Complete the following:
\begin{itemize}
\item[{\bf 1}.]
Show the two panels in an answer check for initial value problem (\ref{quiz1.eq.ivp}).

\item[{\bf 2}.] Relate (\ref{quiz1.eq.ivp}) to a Newton cooling model for warming a $28$ F ice cream bar to room temperature $73$ F.

\item[{\bf 3}.] Let $x$ be the time in minutes. Find the Newton cooling constant $k$, given the additional information that the ice cream bar reaches $34$ F in $5$ minutes.
\end{itemize}
{\bf References}. Edwards-Penney sections 1.1, 1.4, 1.5. Newton cooling
in Serway and Vuille, {\em College Physics 9/E}, Brooks-Cole (2011),
ISBN-10: 0840062060. Newton cooling differential equation
$\frac{du}{dt}=-h(u(t)-u_1)$, 
Math 2280 slide \href{http://www.math.utah.edu/~gustafso/s2017/2280/2250ThreeExamples.pdf}{Three Examples}. 
Math 2280 slide on \href{http://www.math.utah.edu/~gustafso/s2017/2280/FTC-Method-of-Quadrature.pdf}{Answer checks}.

\vspace{5pc}

{\bf Quiz1 Problem 2}. A 2-ft high conical water urn drains from an orifice $6$ inches above the base. The tank drains according to the Torricelli model
\begin{equation} \label{quiz1.eq.tank-drain}
|y(x)|^2\frac{dy}{dx} = -0.021\sqrt{|y(x)|}, \quad y(0)=y_0.\end{equation}
Symbol $y(x)\ge 0$ is the tank water height in feet above the orifice at time $x$ seconds, while $y_0 \ge 0$ is the water height at time $x=0$.

Establish these facts about the physical problem.

\begin{itemize}

\item[{\bf 1}.] If $y_0>0$, then the solution $y(x)$ is uniquely determined and computable by numerical software. Justify using Picard's existence-uniqueness theorem.
 \item[{\bf 2}.] Solve equation (\ref{quiz1.eq.tank-drain}) using separation of variables when $y_0$ is $18$ inches, then numerically find the drain time. Check your answer with technology.
\end{itemize}
% http://vimeo.com/26340791
% Drain a cylindrical tank dh/dt = - sqrt(2g)(r^2/R^2)sqrt(h)
% Chem engineering example
% sqrt(2g)*(r^2/R^2) = sqrt(2*32.2)*((0.5/2)/(10/2))^2 = 0.02006240265
% de:=diff(y(x),x)=-0.021*sqrt(y(x))/y(x)^2; ic:=y(0)=18/12;
% p:=dsolve([de,ic],y(x));
% solve(subs(y(x)=0,p),x);

{\bf References}. Edwards-Penney, Picard's theorem 1 section 1.3 and
Torricelli's Law section 1.4. Tank draining {\tt Mathematica} demo at
\href{http://demonstrations.wolfram.com/TimeToDrainATankUsingTorricellisLaw/}{Wolfram Research}.  
Carl Schaschke, {\em Fluid Mechanics: Worked Examples for Engineers}, The Institution of Chemical Engineers (2005), ISBN-10: 0852954980, Chapter 6. Math 2280 slide on \href{http://www.math.utah.edu/~gustafso/s2017/2280/Picard+DirectionFields.pdf}{Picard and Peano Theorems}. Manuscript on applications of first order equations
\href{http://www.math.utah.edu/~gustafso/2250SciEngApplications.pdf}{http://www.math.utah.edu/~gustafso/2250SciEngApplications.pdf}, Example 35.

% de:=diff(y(x),x)=-k*y(x)+k*73;dsolve([de,y(0)=28],y(x));
% solve(73-45*exp(-k*5)=34,k);evalf(%);

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