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\title{\vspace*{-1.5in}\textbf{Differential Equations: Applications (Ch 7)}}
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%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\chead{April 17, 2018}
\rhead{Diff. Eq. Applications}
\vspace{-.8in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item If the rate at which $y(t)$ changes is proportional to the value of $y$ at a time $t$, we have the differential equation \[\frac{dy}{dt} = k y.\] Some examples where this occurs include:\vskip.5cm
\item \textbf{Newton's Law of Cooling} states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings (provided that this difference is not too large). This is summarized in the differential equation \[\frac{dT}{dt} = k(T-T_s),\] where $T = T(t)$ is the temperature of the object at time $t$, and $T_s$ is the temperature of the surroundings.
\end{itemize}


\noindent
\textbf{Exercises:}\vskip12pt
\begin{enumerate}
%\item In an elementary chemical reaction, molecules of two reactants A and B form a molecule of the product C: $\rm A + B \to C$. (Examples are $\rm 2NO + 2H_2 \to N_2$ or $\rm H_2 + I_2 \to 2HI$.) The law of mass action states that the rate of the reaction is proportional to the product of the concentrations of A and B:\[\frac{d[\text{C}]}{dt} = k[\text{A}][\text{B}],\]
%where [A], [B], and [C] represent the concentrations of molecules of substances A, B, and C, respectively. Suppose $k=2$ moles/L/sec and the initial concentrations of the substances are $[\text{A}]_0 = 3$ moles/L and $[\text{B}]_0 = 4$ moles/L, and $[\text{C}]_0 = 0$ moles/L.
%\begin{enumerate}[(a)]
%\item  we write $x = [\text{C}]$ (Over time, the concentration of molecules of substance C will change.)
%\end{enumerate}

\item Strontium-90 has a half-life of 28 days.
\begin{enumerate}[(a)]
\item Write down a differential equation to model this situation. Let $m = m(t)$ be the mass of strontium after $t$ days.\vskip2cm 

\item A sample has a mass of 50 mg initially. Find a formula for the mass remaining after $t$ days. \vskip1.5in
\item Find the mass remaining after 40 days.\vskip2cm

\item How long does it take the sample to decay to a mass of 2mg?\newpage

\end{enumerate}
\item When \textit{E. coli} grows in a nutrient-broth meduim each cell divides into two cells every 20 minutes. Suppose the initial population of a culture of E. coli is 60 cells.
\begin{enumerate}[(a)]
\item Find an expression for the number of cells after $t$ hours.\vskip3in
\item What is the relative growth rate (i.e. $k$)?\vskip2cm
\item Find the number of cells after 8 hours.\vskip2cm
\item Find the rate of growth after $8$ hours.\vskip2cm
\item When will the population reach 20,000 cells? 
\end{enumerate}
\newpage
\item A freshly brewed cup of coffee has temperature $95^\circ\rm C$ in a $20^\circ \rm C$ room. When its temperature is $70^\circ \rm C$, it is cooling at a rate of $1^\circ \rm C$ per minute. When does this occur?

\newpage
\item Experiments show that the reaction $\rm H_2 + Br_2 \to 2HBr$ satisfies the rate law \[\frac{d[\text{HBr}]}{dt} = k[\text{H}_2][\text{Br}_2]^{1/2},\] where [HBr], $\rm[H_2]$, and $\rm[Br_2]$ represent the concentrations of molecules of Hydrogen Bromide, Hydrogen, and Bromine, respectively. Suppose that the initial concentrations of the reactants are $[\text{H}_2]_0 = a$ moles/L and $[\text{Br}_2]_0 = b$ moles/L.
\begin{enumerate}
\item Write a differential equation that relates the rate of change of the concentration [HBr] to the concentrations $\rm[H_2]$ and $\rm[Br_2]$. (\textbf{Hint:} Let $x = x(t)$ be the concentration of [HBr] at time $t$.)\vskip4cm
\item Find $x$ as a function of $t$ in the case where $a=b$. Use the fact that $x(0) = 0$ (Why does this make sense?).
\end{enumerate}\newpage


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