%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{makeidx}
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="23">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="How to Measure the Dam Force">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 23:54:01 1997}
%TCIDATA{CSTFile=Math.cst}
%TCIDATA{PageSetup=72,72,36,36,0}
%TCIDATA{<LINK REL="contents" HREF="activity.tex">}
%TCIDATA{AllPages=
%F=36,\PARA{038<p type="texpara" tag="Body Text" >\hfill \thepage}
%}


\input{tcilatex}
\begin{document}


\section{How to Measure the Dam Force}

\begin{quotation}
Water pressure depends on depth. Finding the total force on a dam requires
calculating the pressure at various levels and adding together the resulting
forces.
\end{quotation}

A vertical concrete dam in the shape of a parabola is all that separates the
town of Maple Grove from millions of cubic feet of water. Calculate the
hydrostatic force against the dam, assuming the parabola has the equation $y=%
\frac{1}{50}x^{2}-200$.

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$\frac{1}{50}x^{2}-200$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{-1};linecolor
"blue";linestyle 1;linethickness 3;pointstyle "point";function
\TEXUX{-2};linecolor "cyan";linestyle 1;linethickness 3;pointstyle
"point";xmin "-100";xmax "100";xviewmin "-104";xviewmax "104.1";yviewmin
"-204";yviewmax "4.08";rangeset"X";phi 45;theta 45;plottype 4;numpoints
49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

\subsection{Mary's Solution}

\FRAME{itbpF}{0.646in}{0.4315in}{0in}{}{}{solution.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"PICT";valid-file "F";width 0.646in;height 0.4315in;depth 0in;original-width
216.8125pt;original-height 144.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Solution.wmf';file-properties "XNPEU";}}

To find the hydrostatic force against the dam, it is convenient to integrate
with respect to depth. Solving the equation $y=\frac{1}{50}x^{2}-200$ $%
\allowbreak $ for $x$ in terms of $y,$ we get two potential solutions 
\begin{eqnarray*}
x &=&-5\sqrt{\left( 2y+400\right) } \\
x &=&5\sqrt{\left( 2y+400\right) }
\end{eqnarray*}

{\small 
\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
0in;original-width 29.375pt;original-height 30.875pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"XNPEU";}}]{Margin Hint}{{\small With the insertion point in the equation $y=%
\frac{1}{50}x^{2}-200$, choose Solve + Exact (Variable to Solve for: $x$).}}}

We choose the positive solution, and note that at depth $y$ the total width
of the dam is given by 
\[
w(y)=10\sqrt{2y+400} 
\]
The water pressure at depth $y$ ($-200\leq y\leq 0$) is given by 
\[
p(y)=-62.5y 
\]
The hydrostatic force against the dam is given by the integral 
\[
\int_{-200}^{0}p(y)w(y)dy=\allowbreak 1.\,3333\times 10^{8} 
\]
or roughly 133,000,000 lb$.$

{\small 
\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
0in;original-width 29.375pt;original-height 30.875pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"XNPEU";}}]{Margin Hint}{{\small \ With the insertion point in the equation $%
w(y)=10\sqrt{2y+400}$, choose Define + New Definition. Repeat for the
equation $p(y)=-62.5y$. With the insertion point in the integral $%
\int_{-200}^{0}p(y)w(y)dy$, choose Evaluate.}}}\medskip

\subsection{Activity}

\FRAME{itbpF}{0.5881in}{0.3935in}{0in}{}{}{activity.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"PICT";valid-file "F";width 0.5881in;height 0.3935in;depth
0in;original-width 138.3125pt;original-height 92.3125pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Activity.wmf';file-properties
"XNPEU";}}

Livestock ponds are often limited to a depth of $10\unit{ft}$. A valve at
the bottom of such a pond is circular with diameter $1\unit{ft}$, and has a
vertical orientation. Imagine the valve subdivided into horizontal strips,
as in the following figure: \FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height
2.0003in;depth 0pt;display "PICT";function \TEXUX{$\
\sqrt{25-x^{2}}$};linecolor "black";linestyle 1;linethickness 1;pointstyle
"point";function \TEXUX{$-\sqrt{25-x^{2}}$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\left( -4,3,4,3\right)
$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$\left( -4,-3,4,-3\right) $};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\left(
-\sqrt{25-1},1,\sqrt{25-1},1\right) $};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\left(
-\sqrt{25-1},-1,\sqrt{25-1},-1\right) $};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "-5";xmax "5";xviewmin
"-5";xviewmax "5";yviewmin "-5.2";yviewmax "5.204";phi 45;theta 45;plottype
4;constrained TRUE;numpoints 49;axesstyle "none";xis \TEXUX{x};var1name
\TEXUX{$x$};}}If the bottom of the circle rests on the bottom of the pond ($%
10\unit{ft}$ below the surface), make a table that shows the approximate
force applied to each of the five horizontal strips, assuming pressure is $%
62.5y\unit{lb}$ where $y$ is the depth in feet. Describe the methods used to
create the approximations. Calculate the approximate total force against the
circular valve. Set up an integral that represents the force against the
circular valve. Compare the evaluated integral with your estimate. Explain
any discrepancy.

\end{document}
