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%  Rates of Change                                                 %
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\problemgrouptitle{Rates of Change}

\begin{problemblock}{The following problems consider the meaning of the derivative as a rate of change.}

\item The concentration of carbon dioxide in the Earth's atmosphere has been observed at Mauna Loa Observatory in Hawaii to be steadily increasing (neglecting seasonal oscillations) since~1958.  A best fit curve to the data measurements taken from 1982 to 2009 yields the following function for the concentration in parts per million (ppm) as a function of the year $t$\,:
% The following is a quadratic fit to data from:
% Monthly atmospheric CO2 concentrations (ppm) derived from flask air samples
% Mauna Loa, Hawaii 156W 20N
% Source: R. F. Keeling, S. C. Piper, A. F. Bollenbacher and S. J. Walker
% Scripps CO2 Program ( http://scrippsco2.ucsd.edu )
% Scripps Institution of Oceanography (SIO)
% University of California
% La Jolla, California USA 92093-0244
\[C(t)=0.0143(t-1982)^2+1.28(t-1982)+341\]
\begin{enumerate}
\item What was the level of CO$_2$ in the air in the year 2000?
\item At what rate was the CO$_2$ level changing with respect to time in the year 2000?
\item By what percentage did the CO$_2$ level change between 2000 and 2005?
\end{enumerate}
\answer{ 
\begin{enumerate}
\item $C(2000)=368.6732\approx369$ ppm
\item $C'(2000)=1.7948\approx1.79$ ppm/year
\item $\displaystyle\frac{C(2005)-C(2000)}{C(2000)}\times 100=2.53\%$.
\end{enumerate}
}

\item A conical tank has a height of 5~metres and radius at the top of 2~metres.
\begin{enumerate}
\item Show that the volume of liquid in the tank when it is filled to a depth $y$ is given by
\[V=\frac{4\pi}{75}y^3\]
\item What is the rate of change of volume with respect to depth when the tank is filled to 4~metres?
\end{enumerate}
\answer{\begin{enumerate}
\item The volume of a cone is $V=\frac{1}{3}\pi r^2 h$.  Use similar triangles to show that the radius of the surface of the liquid is $r=\frac{2}{5}y$\,.
\item $\displaystyle\left.\frac{dV}{dy}\right|_{y=4\,\mbox{\tiny m}}=\frac{64\pi}{25}\approx8.04\ \frac{\mbox{m}^3}{\mbox{m}}$
\end{enumerate}
}

\end{problemblock}
