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\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\CONE TO CUBE FLOW}






\markboth{Water Flow from Cone to Cube and Out}{Water Flow from Cone to Cube and Out}






\author{Sania Qureshi\\
Basic Sciences and Related Studies\\ 
Mehran University of Engineering and Technology\\
Jamshoro, Sindh PAKISTAN
\teacher{\\ sania.qureshi@faculty.muet.edu.pk}}
\makeStitlePDFLaTex



\begin{abstract}
We consider a configuration of two containers. An inverted right circular cone with a hole in point at the bottom  is suspended above an open-topped cube which also has a hole in the center of the bottom. The cone is filled with water and we wish to model the water flow from cone to cube and out the bottom of the cube. This uses two applications of Torricelli's Law.
\end{abstract}


\section*{STATEMENT}


Consider the following configuration as seen in Figure~\ref{fig:1-29-ConeCube}. We have an inverted (point down) right circular conical tank, with a top (base) radius $R$ meters and a total height of $H$ meters, filled with water to a height of $h$ meters. There is a circular hole of radius $r_C$ meters at the very bottom of the cone from where water flows out of the cone. This water falls into an open-topped  cubicle tank sitting directly below the cone. Finally, this cubicle tank  also has a circular hole at the center of its base with radius $r_B$ meters and water flows out of the bottom cubicle tank. 


\begin{figure}[h]
\centering
\includegraphics[width=6cm]{1-29-ConeCube.eps}


\parbox{9.7 true cm}{\caption{\label{fig:1-29-ConeCube}  Shown is a cone   of base radius $R$ and height $H$ sitting above a cube of side $s$ with the water running out of the cone into the cube, whereupon the water runs out of the cube.}}

\end{figure}

Can we create a mathematical model to describe this situation? 

\begin{enumerate}
\item How long does it take the conical container to empty? 
\item How long does it take both containers to  empty? 
\item Further, what if the circular holes in the containers offer friction or resistance? 
\item Offer comments on the reality of your solutions.
\end{enumerate}

Here are a reasonable set of working dimensions, parameters, and constants:

\begin{itemize}
\item top (base) radius of the cone, $R = 0.3$ m;
\item height of the  cone, $H = 0.8$ m;
\item radius of hole at bottom tip of the cone, $r_C  = 0.02$ m;
\item length of side of the cube, $s = 0.5$ m;
\item radius of hole at bottom of the cube, $r_B  = 0.03$ m;
\item acceleration due to gravity, $g = 9.8$ m$^2$/s;
\item (for the cone)  the empirically determined resistance percentage, $\alpha$, of the flow  rate of the liquid which actually goes through the hole in the  bottom of the cone is the {\it contraction coefficient\/}, here $\alpha = 0.70$; 
\item (for the cube) the empirically determined resistance percentage, $\beta$, or the percent of the flow  rate of the liquid which actually goes through the hole in the bottom of the cube is the {\it contraction coefficient\/}, here $\beta = 0.60$.
\end{itemize}


\teacher{\newpage
\section*{NOTES FOR TEACHERS}

This activity is a direct application of Torricelli's Law as offered in \cite{Winkel2015}; two applications, in fact. One will be for the cone and one will be for cube. We offer a complete analysis in an accompanying notebook 1-029-T-Mma-ConeToCubeFlow-Teacher-Version.nb and a pdf version of this file for those who do not use Mathematica.

Key to the modeling involves determining the time at which the cone empties, for from that time forth the cube is no longer accumulating  water, but rather is only losing water. We show this in Figure~\ref{fig:1-29-Plots}.



\begin{figure}[h]
\centering
\includegraphics[width=12cm]{1-29-Plots.eps}


\parbox{10.2 true cm}{\caption{\label{fig:1-29-Plots}  Plots of volume of water in the cone, in the cube, and in both, showing the time (around $t = 26$ s) at which the cone empties totally.}}

\end{figure}


\begin{thebibliography}{99}




\bibitem{Winkel2015} Winkel, B. 2015. 1-015-T-Torricelli. 





\end{thebibliography}
}





\end{document}
