%% 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="Making a Big Box">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 22:23:20 1997}
%TCIDATA{CSTFile=Math.cst}
%TCIDATA{PageSetup=72,72,72,72,0}
%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{Making a Big Box}

\begin{quotation}
Calculus provides useful tools for solving problems that ask for the
``best'' way to perform a task. Such problems may be stated\strut
\end{quotation}

\begin{itemize}
\item  in economic terms: What is the cheapest or most profitable?\medskip 

\item  in physical terms: What is the easiest?\medskip 

\item  in geometric terms: What is the largest or smallest? Longest or
shortest?\medskip 

\item  in terms of time: What is the fastest?
\end{itemize}

\strut

\begin{quotation}
Usually several steps are required to solve such a problem.
\end{quotation}

\begin{enumerate}
\item[1.]  Read and understand the problem. What is known? What is the
unknown?

\item[2.]  Draw a diagram and label the parts with appropriate symbols.

\item[3.]  Write an equation that expresses the unknown quantity in terms of
the known quantities.

\item[4.]  Plot a graph to locate approximately the extreme values.

\item[5.]  Use calculus to locate precisely the extreme values.
\end{enumerate}

\strut

\hspace{-1in}A box with an open top is to be constructed from a rectangular
piece of cardboard of dimensions $8$ inches by $10$ inches by cutting out a
square from each of the four corners and bending up the sides. Draw a
diagram of the box and label the parts. Determine a formula for the volume
of the box in terms of the dimension of an edge of a corner that is cut out.
Explain why the interval $\left[ 0,4\right] $ is a reasonable domain for the
volume function. Plot a graph of the volume versus corner dimension. Find
the largest volume that such a box can have.

\subsection{Carla's Solution}

Let $x$ denote the length of the edge of a corner that is cut out. Fold the
cardboard along the dotted lines and fold to make an open-top box, as
indicated to the right.

\begin{center}
\FRAME{itbpF}{2.0003in}{2.0003in}{0in}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 2.0003in;height 2.0003in;depth 0in;display
"PICT";function
\TEXUX{$[-5,-4,5,-4,5,4,-5,4,-5,-4]$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";function
\TEXUX{$[-5,-2,5,-2]$};linecolor "black";linestyle 3;linethickness
1;pointstyle "point";function \TEXUX{$[-5,2,5,2]$};linecolor
"black";linestyle 3;linethickness 1;pointstyle "point";function
\TEXUX{$[-3,-4,-3,4]$};linecolor "black";linestyle 3;linethickness
1;pointstyle "point";function \TEXUX{$[3,-4,3,4]$};linecolor
"black";linestyle 3;linethickness 1;pointstyle "point";xmin "-5";xmax
"5";xviewmin "-5.2";xviewmax "5.204";yviewmin "-4.16";yviewmax
"4.163";rangeset"X";phi 45;theta 45;plottype 4;constrained TRUE;numpoints
49;axesstyle "none";xis \TEXUX{x};var1name \TEXUX{$x$};}}\quad \FRAME{itbpF}{%
2.0003in}{2.0003in}{0.0104in}{}{}{}{\special{language "Scientific Word";type
"MAPLEPLOT";width 2.0003in;height 2.0003in;depth 0.0104in;display
"PICT";function
\TEXUX{$[0,0,4,4,4,6,2,8,-2,4,-2,2,0,0,0,2]$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";function
\TEXUX{$[-2,4,0,2,4,6]$};linecolor "black";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$[3,5,2,6,2,8,2,6,-1,3]$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "-2";xmax
"4";xviewmin "-2.12";xviewmax "4.122";yviewmin "-0.16";yviewmax
"8.163";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle
"none";xis \TEXUX{x};var1name \TEXUX{$x$};}}
\end{center}

{\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 list $%
\left( -5,-4,5,-4,5,4,-5,4,-5,-4\right) $, choose Plot 2D + Rectangular.
Select with the mouse and drag each of the following to the plot 
\begin{eqnarray*}
\left( -5,-2,5,-2\right) \\
\left( -5,2,5,2\right) \\
\left( -3,-4,-3,4\right) \\
\left( 3,-4,3,4\right)
\end{eqnarray*}
Choose Edit + Properties, Plot Components page. Set\ Line Style to Dots for
Item Numbers 2, 3, 4, 5. Choose Axes page, set Axes Type to None, and check
Equal Scaling Along Each Axis. Choose OK. With the insertion point in the
list $\left( 0,0,4,4,4,6,2,8,-2,4,-2,2,0,0,0,2\right) $, choose Plot 2D +
Rectangular. Select with the mouse and drag each of the following to the
plot: $\left( -2,4,0,2,4,6\right) $, $\left( 3,5,2,6,2,8,2,6,-1,3\right) $.
Choose Edit + Properties, Axes page; set Axes Type to None, and check Equal
Scaling Along Each Axis. Choose OK.}}}

The open-top box will have dimensions $x$ by $8-2x$ by $10-2x$, and hence
the volume is given by $V=x(8-2x)(10-2x)=80x-36x^{2}+4x^{3}$. The domain of
the volume function is $\left[ 0,4\right] $ because for $x>4$ there is
nothing left to fold up. The following is a graph of the volume function: 
\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$80x-36x^{2}+4x^{3}$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "0.01458";xmax "4.04085";xviewmin
"-0.06595";xviewmax "4.123";yviewmin "-1.696";yviewmax
"53.57";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle
"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

{\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 expression $%
x(8-2x)(10-2x)$, choose Expand. With the insertion point in the expression $%
80x-36x^{2}+4x^{3}$, choose Plot 2D + Rectangular. Change Domain Interval to 
$0<x<4$.}}}

There appears to be a critical number at approximately $1.5.$ Solving 
\[
\dfrac{d}{dx}\left[ 80x-36x^{2}+4x^{3}\right] =0 
\]
we get the two potential solutions $x=1.\,4725$ and $x=4.\,5275$. The second
potential solution must be thrown out because it is not in the domain. The
maximum volume is given by 
\[
\left[ 80x-36x^{2}+4x^{3}\right] _{x=1.\,4725}=\allowbreak 52.\,514\text{
cubic inches} 
\]

{\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 $%
\dfrac{d}{dx}\left[ 80x-36x^{2}+4x^{3}\right] =0$, choose Solve + Numeric.
With the insertion point in the expression $\left[ 80x-36x^{2}+4x^{3}\right]
_{x=1.\,4725}$, choose Evaluate.\strut }}}

\subsection{Activities}

\hspace{-1in}\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 394.4375pt;original-height 263.25pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Activity.wmf';file-properties
"XNPEU";}}\hspace{0.36in}

\begin{enumerate}
\item  A local building code requires that a rectangular-shaped
manufacturing plant be bordered by 100 feet of landscaping on the front and
rear and by 80 feet of landscaping on each side. If 800,000 square feet of
land are necessary in order to have room for parking lots, buildings, and so
forth, what is the minimum number of square feet that must be purchased for
the manufacturing site? Label the inner rectangle with sides of length $x$
and $y$, and use the stated conditions to write $y$ in terms of $x$. Write a
formula for the area of the large rectangle in terms of $x$. What is an
appropriate domain? Plot a graph of the area of the large rectangle versus $%
x $. Use calculus to find precise estimates for $x$ and $y$ that minimize
the area of the large rectangle.\FRAME{dtbpFU}{3in}{2.0003in}{0pt}{\Qcb{%
Manufacturing plant site}}{}{}{\special{language "Scientific Word";type
"MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$\left[
0,0,1000,0,1000,800,0,800,0,0\right] $};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\left[
-100,-80,1100,-80,1100,880,-100,880,-100,-80\right] $};linecolor
"black";linestyle 3;linethickness 1;pointstyle "point";xmin "-100";xmax
"1100";xviewmin "-124";xviewmax "1124";yviewmin "-99.2";yviewmax
"899.6";rangeset"X";phi 45;theta 45;plottype 4;constrained TRUE;numpoints
49;axesstyle "none";xis \TEXUX{x};var1name \TEXUX{$x$};}}\bigskip

\item  A rain gutter is to be constructed from a metal sheet of width 25 cm
by bending 10 cm on each side so that one side is vertical for attaching to
a vertical wall, the opposite side is bent at an angle $\theta $, and the
middle 5 cm is horizontal. How should $\theta $ be chosen so that the gutter
will carry the maximum amount of water? Plot the cross-sectional area versus
angle $\theta $. What is a reasonable domain for the angles $\theta $?
\end{enumerate}

\begin{center}
\FRAME{itbpFU}{2.0081in}{1.3353in}{0in}{\Qcb{Cross section of gutter}}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 2.0081in;height
1.3353in;depth 0in;display "PICT";function
\TEXUX{$(0,10,0,0,5,0,10,8.67)$};linecolor "black";linestyle 1;linethickness
2;pointstyle "point";function \TEXUX{$\left( 0,-5,0,15\right) $};linecolor
"gray";linestyle 3;linethickness 1;pointstyle "point";function
\TEXUX{$\left( 0,8.67,10,8.67\right) $};linecolor "cyan";linestyle
3;linethickness 1;pointstyle "point";xmin "0";xmax "10";xviewmin
"-0.2";xviewmax "10.2";yviewmin "-5.4";yviewmax "15.41";rangeset"X";phi
45;theta 45;plottype 4;constrained TRUE;numpoints 49;axesstyle "none";xis
\TEXUX{x};var1name \TEXUX{$x$};}}\FRAME{itbpFU}{2.0081in}{1.3353in}{%
0in}{\Qcb{Cross section of gutter}}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 2.0081in;height 1.3353in;depth 0in;display
"PICT";function \TEXUX{$(0,10,0,0,5,0,13,6)$};linecolor
"black";linestyle 1;linethickness 2;pointstyle "point";function
\TEXUX{$\left( 0,-5,0,15\right) $};linecolor "gray";linestyle
3;linethickness 1;pointstyle "point";function \TEXUX{$\left( 0,6,13,6\right)
$};linecolor "cyan";linestyle 3;linethickness 1;pointstyle "point";xmin
"0";xmax "13";xviewmin "-0.26";xviewmax "13.27";yviewmin "-5.4";yviewmax
"15.41";rangeset"X";phi 45;theta 45;plottype 4;constrained TRUE;numpoints
49;axesstyle "none";xis \TEXUX{x};var1name \TEXUX{$x$};}}
\end{center}

\end{document}
