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\title{\vspace*{-1.5in}\textbf{Math 2300-007: Quiz 6}}
\date{}
\author{}


\begin{document}
\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-013: Calculus II}
\rhead{Feb. 23, 2018}
\chead{Quiz 6}
\vspace{-.5in}
\noindent
{Name: \underline{\phantom{Aalsdkjflaskdjflaskjflksdjflkdsjf}}} \hfill{Score: \underline{\phantom{lksdjflkdsjf}}}\\[12pt]

\noindent
\begin{enumerate}
\item (1 point) Write down a formula for work in terms of force and distance.\vskip.5in

\item (1 point) In a sentence or two, describe your plan for solving the problem below.\vskip.8in

\item (5 points) Water is pumped from the top of a conical tank of height $4$ meters and base radius $2$ meters depicted below. How much work is required to empty the tank of water if the initial height of the water is $3$ meters? You may assume that the mass of water on Earth is $1000$ kilograms per cubic meter and that $g = 9.8 \frac{\text{m}}{\text{sec}^2}$ is the acceleration due to gravity on Earth. (\textbf{Set up, but do not evaluate the integral.})

\begin{flushleft}
\begin{tikzpicture}[x=2cm, y=2cm]
\draw (0,2) ellipse (1 and .05);
\fill[pattern=north west lines, pattern color = blue]
(-.75,1.5)--(0,0)--(.75,1.5)--cycle;
\draw[fill = white] (0,1.5) ellipse (.75 and .06);
\draw[fill = gray, opacity = .3] (0,1.5) ellipse (.75 and .06);
\draw (-1,2)--(0,0)--(1,2);
\draw[|-|] (1,0)--(1,.75) node[right]{$3$m}--(1,1.5);
\draw[|-|] (-1.2,0)--(-1.2,1) node[left]{$4$m}--(-1.2,2);
\draw[|-|] (0,2.25)--(-.5,2.25) node[above]{$2$m}--(-1,2.25);
\end{tikzpicture}
\end{flushleft}
\newpage
\item The lamina depicted below has centroid $(5,6)$ and mass density $\rho = 1$ kilogram per square meter.  Suppose $M_x$ is the moment of the system about the $x$-axis and $M_y$ is the moment of the system about the $y$-axis. 
\begin{center}
\begin{tikzpicture}
\begin{axis}[axis lines = center, xlabel= {$x$ (meters)}, ylabel = {$y$ (meters)}, xlabel near ticks, ylabel near ticks]
\addplot[samples=100, domain = 0:12, line width = 1pt]({12*(x/12)^4},{x});
\addplot[samples=100, domain = 0:12, line width = 1pt]{12*(x/12)^2};
\addplot[mark=*] coordinates {(5,6)};
\end{axis}
\end{tikzpicture}
\end{center}
Use the figure provided to answer the following questions. For each question, choose the best answer. 
\begin{enumerate}[(a)]
\item(1 point) If the density is changed from $\rho = 1$ $\text{kg}/\text{m}^2$ to $\rho = 2$ $\text{kg}/\text{m}^2$, then:

\begin{enumerate}[(i)]
\item $M_x$ increases, $M_y$ increases, and the centroid stays at $(5,6)$;
\item $M_x$ increases, $M_y$ decreases, and the centroid stays at $(5,6)$;
\item $M_x$ increases, $M_y$ increases, and the centroid moves to the right of $(5,6)$;
\item $M_x$ decreases, $M_y$ increases, and the centroid moves to the left of $(5,6)$;
\item $M_x$ decreases, $M_y$ decreases, and the centroid stays at $(5,6)$.
\item $M_x$ and $M_y$ do not change, and the centroid stays at $(5,6)$.
\end{enumerate}\bigskip

\item(1 point) If a point mass of $3$ kg is added to the system at the point $(7,6)$, then:
\begin{enumerate}[(i)]
\item $M_x$ increases, $M_y$ does not change, and the centroid stays at $(5,6)$;
\item $M_x$ increases, $M_y$ decreases, and the centroid stays at $(5,6)$;
\item $M_x$ increases, $M_y$ increases and the centroid moves to the right of $(5,6)$;
\item $M_x$ decreases, $M_y$ does not change and the centroid moves to the left of $(5,6)$;
%\item $M_x$ does not change, $M_y$ increases and the centroid moves to the left of $(5,6)$.
\item $M_x$ and $M_y$ do not change, and the centroid stays at $(5,6)$.
\end{enumerate}
\end{enumerate}

\end{enumerate}
\end{document}
