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\flushleft\underline{Some Problems --- 22 January 2018} \\
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\item The development of an amusement park on the outskirts of a city will increase the city's population at the rate of
$$4500 \sqrt{t}+1000 \hspace{5pt} \dfrac{\mathrm{people}}{\mathrm{year}}$$
$t$ years from the date of construction. The population before construction is $30,000$. Determine the projected population $9$ years after construction of the park.
\item Nineteenth-century physician Jean Louis Marie Poiseuille discovered that the rate of change of the velocity of blood $r$ cm from the central axis of an artery (in $\dfrac{\frac{\mathrm{cm}}{\mathrm{sec}}}{\mathrm{cm}}$) is given by $a(r)=-kr$, where $k$ is a constant. If the radius of an artery is $R$ cm, find an expression for the velocity of blood as a function of $r$. Hint: $v'(r)=a(r)$ and $v(R)=0$ (Why?)
\item The rate of change of coal exports from 2010 to 2012 was given by
$$f(t)=31.863 t^{-0.61} \hspace{5pt} \dfrac{\mathrm{million \hspace{2pt} short \hspace{2pt} tons}}{\mathrm{year}}$$
for $0 \leq t \leq 2$, where $t$ is measured in years with $t=0$ corresponding to $2010$. Find an expression for U.S. coal exports in year $t$. Assuming the trend continued through 2013, what were the U.S. coal exports that year?
\item The population of a certain city is projected to grow at the rate of
$$r(t) = 400 \left( 1 + \dfrac{2t}{24+t^2} \right) \hspace{5pt} \dfrac{\mathrm{people}}{\mathrm{year}}$$
for $0 \leq t \leq 5$. The current population is $60,000$. What will be the population 5 years from now?
\item A drug is carried into an organ of volume $V \mathrm{cm}^3$ by a liquid that enters the organ at a rate if $a \dfrac{\mathrm{cm}^3}{\mathrm{sec}}$ and leaves it at the rate of $b \dfrac{\mathrm{cm}^3}{\mathrm{sec}}$. The concentration of the drug in the liquid entering the organ is $c \dfrac{\mathrm{g}}{\mathrm{cm}^3}$. If the concentration of the drug in the organ at time $t$ (in seconds) is increasing at the rate of
$$x'(t) = \dfrac{1}{V}(ac-bx_0)e^{-\frac{bt}{V}} \hspace{5pt} \dfrac{\frac{\mathrm{g}}{\mathrm{cm}^3}}{\mathrm{sec}}$$
and the concentration of the drug in the organ initially is $x_0 \dfrac{\mathrm{g}}{\mathrm{cm}^3}$, show that the concentration of the drug in the organ at time $t$ is given by
$$x(t) = \dfrac{ac}{b} + \left( x_0 - \dfrac{ac}{b} \right) e^{-\frac{bt}{V}}.$$
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