%
%	gbb60166@gmail.com	https://unilab.gbb60166.jp/prekou/prekou.htm
%
% aspectratio= は 1610, 169, 149, 54, 32 の中から選べる（省略時は 43）
% C:\texlive\2024\texmf-dist\tex\latex\beamer\beamer.cls
%\documentclass[20pt,dvipdfmx,usepdftitle=false,hyperref={pdfpagemode=FullScreen},aspectratio=169]{beamer}
%\documentclass[20pt,dvipdfmx,usepdftitle=false,hyperref={pdfstartview={XYZ null null 8.00}},aspectratio=169]{beamer}
\documentclass[20pt,dvipdfmx,usepdftitle=false,aspectratio=169]{beamer}
% pdfの栞の字化けを防ぐ
%\AtBeginDvi{\special{pdf:tounicode EUC-UCS2}}
% テーマ
\usetheme{Copenhagen}
% navi. symbolsは目立たないが，dvipdfmxを使うと機能しないので非表示に
\setbeamertemplate{navigation symbols}{} 
\usepackage{bxdpx-beamer,pxjahyper,minijs}
%\usepackage{graphicx}
%\usepackage{tkokugo,furikana,tsayusen,shiika,sfkanbun,jdkintou,plext}
\usepackage{setspace}
\usepackage{type1cm}
\usepackage[deluxe]{otf}
\usepackage{tikzsymbols}
%\usepackage[dvipdfmx]{graphicx}%
\def\pgfsysdriver{pgfsys-dvipdfmx.def}%（graphicxパッケージを使用しない場合はこの行を有効に）
%\def\pgfsysdriver{pgfsys-dvips.def}%デフォルト
%\usepackage[e]{esvect}
\usepackage{tikz}%（これで、pgfとpgfforが読み込まれます。）
%\usepackage{tikz-3dplot}
\usetikzlibrary{arrows.meta,bending}
%\usetikzlibrary{calc,patterns,angles,intersections,backgrounds}
%\usepackage{animate}
%\usepackage{pgfplots}
%\pgfplotsset{compat=newest}
%\pgfplotsset{compat=newest, every axis/.append style={line width=1pt}}
%\PassOptionsToPackage{dvipdfmx}{graphicx}
\nofiles

\hypersetup{% Jul 18 2023
	pdftitle={プレ高数学科（LaTeX Beamer で汎用PDFプレゼンを作る）},
	pdfauthor={gbb60166@gmail.com},
	pdfsubject={https://unilab.gbb60166.jp/prekou/prekou.htm}
	}

% フォントはお好みで
%\usepackage{txfonts}

\mathversion{bold}
\renewcommand{\familydefault}{\sfdefault}
\renewcommand{\seriesdefault}{bx}
\renewcommand{\kanjifamilydefault}{\gtdefault}
\renewcommand{\kanjiseriesdefault}{bx}
\setbeamerfont{title}{size=\normalsize,series=\bfseries}
\setbeamerfont{frametitle}{size=\normalsize,series=\bfseries}
\setbeamertemplate{frametitle}[default][center]
\usefonttheme{professionalfonts}

% 参考にしたURL
% http://windom.phys.hirosaki-u.ac.jp/fswiki/wiki.cgi?page=LaTeX+Beamer%A4%C7%A5%D7%A5%EC%A5%BC%A5%F3%A5%C6%A1%BC%A5%B7%A5%E7%A5%F3

\newcommand{\Slash}[1]{\ooalign{\hfil\kern-3pt／\hfil\crcr$#1$}}
\everymath{\displaystyle}

\newcommand{\maruwaku}[1]{\begin{tikzpicture}[scale=0.7, baseline={([yshift=-22pt] current bounding box.north)}]
\filldraw[color=CUDBlue, line width=1pt, rounded corners=2pt] (-0.1,0)--(2.1,0)--(2.1,1.1)--(-0.1,1.1)--cycle;
\draw(1,0.5) node[white]{#1};
\end{tikzpicture}
}

\setbeamersize{text margin left=5mm,text margin right=5mm}
\fboxrule=0.8pt

\makeatletter
\newcommand{\hooklen}[2]{\settowidth{\@tempdima}{\(#1\)}
%\advance\@tempdima by.3ex
% ↑ 数式モードで式の前後に入るスペースを制御したかったが、
%    難しいのでやめた。段々難解なコードになっているのでやめた方がよい？
\hbox to\@tempdima{\hfil \(#2\)\hfil}}
%■等号付き不等号 by 美文書作成入門
\newcommand{\myleqq}{\mathrel{\mathpalette\gl@align<}}
\newcommand{\mygeqq}{\mathrel{\mathpalette\gl@align>}}
\newcommand{\gl@align}[2]{\lower.6ex\vbox{\baselineskip\z@skip\lineskip\z@
	\ialign{$\m@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}}
\makeatother

% カラーユニバーサルデザインを調べたつもりだがあまり自信がありません
% http://www.fukushihoken.metro.tokyo.jp/kiban/machizukuri/kanren/color.files/colorudguideline.pdf
% http://jfly.iam.u-tokyo.ac.jp/colorset/

% ■ アクセントカラー小面積を目立たせる高彩度色
\definecolor{CUDRed}{RGB}{255,75,0}
\definecolor{CUDGreen}{RGB}{3,175,122}
\definecolor{CUDBlue}{RGB}{0,90,255}
\definecolor{CUDCyan}{RGB}{77,196,255}
\definecolor{CUDMagenta}{RGB}{153,0,153}
\definecolor{CUDYellow}{RGB}{255,241,0}
\definecolor{CUDBrown}{RGB}{128,64,0}

\definecolor{CUDOrange}{RGB}{246,170,0}

% ■ ベースカラー広い面積の塗り分けに用いる低・中彩度色
\definecolor{CUDPink}{RGB}{255,202,191}
\definecolor{CUDBrightGreen}{RGB}{119,217,168}
\definecolor{CUDLime}{RGB}{216,242,85}
\definecolor{CUDCream}{RGB}{255,255,128}
\definecolor{CUDBrightCyan}{RGB}{191,228,255}
%\definecolor{CUD}{RGB}{}%

\setbeamercolor{CUDBrightGreen}{fg=black,bg=CUDBrightGreen!50}
\setbeamercolor{CUDBrightCyan}{fg=black,bg=CUDBrightCyan}
\setbeamercolor{CUDCream}{fg=black,bg=CUDCream!75}

%\setbeamercolor{frametitle}{use=structure,fg=white,bg=structure.fg}
%\setbeamercolor{palette primary}{use=structure,fg=white,bg=structure.fg}
%\setbeamercolor{frametitle}{bg=CUDOrange}
%\setbeamercolor{palette primary}{bg=CUDOrange}
%\setbeamercolor{frametitle}{bg=CUDCyan!25,fg=black}
%\setbeamercolor{palette primary}{bg=CUDCyan!25,fg=black}
%\setbeamercolor{frametitle}{bg=CUDCream!75,fg=black!80}
%\setbeamercolor{palette primary}{bg=CUDCream!75,fg=black!80}

\newcommand{\myfrac}[2]{\mathchoice%■ 修正 Oct 8,2023
	{%\fontsize{20}{20}\selectfont%	\displaystyle
		\frac{\raisebox{-.6ex}{\(#1\)}}{\raisebox{.4ex}{\(#2\)}}}%
	{\textstyle{\frac{#1}{#2}}}%	\textstyle
	{%\fontsize{13}{13}\selectfont%	\scriptstyle
		\frac{\raisebox{0ex}{\(#1\)}}{\raisebox{-.4ex}{\(#2\)}}}%
	{%\fontsize{11}{11}\selectfont%	\scriptscriptstyle
		\frac{\raisebox{0ex}{\(#1\)}}{\raisebox{-.4ex}{\(#2\)}}}%
	}


\begin{document}

\title{プレ高数学科}\author{gbb60166}

%■ 本文と数式モードの間に挿入される空白スペース
%\mathsurround=-1pt
%■ \( \log\log \)でlogとlogの間に挿入される細スペース（省略形は \,）
%\thinmuskip=3mu
%■ \( a+b \)で+の前後に挿入される中スペース（省略形は \>）
\medmuskip=1mu plus 1mu minus 1mu
%■ \( x=5 \)で=の前後に挿入される太スペース（省略形は \;）
%\thickmuskip=5mu plus 5mu
% mu は math units の省略で長さの単位

\def\ittan{\begin{tikzpicture}[scale=0.7,
	baseline={([yshift=-21pt] current bounding box.north)}]
\filldraw[very thick,draw=CUDBlue,fill=CUDYellow,rounded corners=1pt]
	(0,0)--++(0.8,0)--++(0,0.8)--++(-0.8,0)--cycle;
\draw(0.4,0.4) node[CUDBlue]
	{\fontsize{7pt}{7pt}\selectfont
		\shortstack{一旦\\[-2pt]停止}};
\end{tikzpicture}
	}

\def\ensui{%
\coordinate (O) at (0,0);
\coordinate (A) at (-1.2,0);
\coordinate (AA) at (-1.18,0.11);
\coordinate (B) at (0,2.3);
\coordinate (C) at (1.2,0);
\coordinate (CC) at (1.15,0.15);

\draw(A) arc [start angle=-177, end angle=-190,x radius=1.6, y radius=0.5];
\draw(C) arc [start angle=0, end angle=15,x radius=1.6, y radius=0.5];
\draw(AA)--(B)--(CC);
\draw(A) arc [start angle=-180, end angle=0,x radius=1.2, y radius=0.5];
\draw[dashed](C) arc[start angle=0,end angle=160,x radius=1.2, y radius=0.5];
}

\def\sunpo{%
\draw[dashed](A)--(O)--(B);
\draw(-0.2,0)--(-0.2,0.2)--(0,0.2);

\draw(-0.6,0) node[inner sep=4pt,fill=white]{$\scriptstyle 2$};
\draw(-0.8,1.2) node[inner sep=1pt,fill=white]{$\scriptstyle 6$};
}


%■■■■■■■■■■■■■　テスト領域　■■■■■■■■■■■■■■


%\end{document}
%■■■■■■■■■■■■■　完成品　■■■■■■■■■■■■■■

% 昔書いたコードで、今見直すと気になる箇所が多いので
% 少し手直ししたが、時間がなく完全には仕上がっていない

\begin{frame}[t]
\frametitle{表面積と体積を求めなさい}
\begin{tikzpicture}[scale=2, very thick,>=stealth]
\ensui
\visible<3->{\draw[CUDRed,line width=0.3ex](A)
	arc[start angle=-180,end angle=180,x radius=1.2,y radius=0.5];
	\draw[CUDBlue,line width=0.2ex](A)++(0.024,0.002)
		arc[start angle=-180,end angle=180,
			x radius=1.175,y radius=0.475];
	}
\draw[CUDGreen,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},
	line width=2pt](A)--node[fill=white,inner sep=2pt]{$2$}(O);
\draw[CUDGreen,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},
	line width=2.2pt](AA)--node[circle,fill=white,inner sep=2pt]{$6$}(B);
\fill (O) circle (1pt);
\end{tikzpicture}
\hskip3zw\pause
\begin{tikzpicture}[scale=0.5, very thick,>=stealth]
\draw(-60:6) arc (-60:60:6)--(0,0)--cycle;
\draw[CUDGreen,{Stealth[length=1ex]}-{Stealth[length=1ex]},ultra thick](8,0)--
	node[yshift=1ex]{\footnotesize $2$}(6,0);
\draw(8,0) circle (2);
\fill(8,0) circle (3pt);
\draw[CUDGreen,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},ultra thick]
	(0,0)--	node[pos=0.4,yshift=2ex]{\footnotesize $6$}(60:6);
\visible<3->{\draw[CUDRed,ultra thick](-60:6) arc (-60:60:6);
	\draw[CUDBlue,ultra thick](8,0) circle (2);
	\draw(10,5) node{長さは同じ};
	}
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{まず中心角度を求めるぞ！}
\begin{minipage}{0.6\textwidth}
\begin{tikzpicture}[scale=0.5,>=stealth,very thick]
\draw(60:6)--(0,0)--(-60:6);
\draw(8,0)--(6,0);
\draw[CUDBlue,ultra thick](8,0) circle (2);
\filldraw(8,0) circle (3pt);
\begin{scope}
\footnotesize
\draw(1,3) node{$6$};
\draw(7,0.5) node{$2$};
\visible<4->{\draw(3,0) node[CUDGreen]{$120^\circ$};
	\draw[CUDGreen,
		{Stealth[length=1.6ex,flex=1]}-{Stealth[length=1.6ex,flex=1]},
		ultra thick](-60:1.5) arc (-60:60:1.5);
	}
\end{scope}
\draw[dashed,CUDRed](0,0) circle (6);
%\draw[CUDRed,ultra thick](-60:6) arc(-60:60:6);
\draw(-4,0) node[CUDRed]{長さ$12\pi$};
\draw(9,3) node[CUDBlue]{長さ$4\pi$};
\draw(3,-3.3) node[fill=CUDCream]{\small 円周の長さ\(=2\pi\times\text{半径}\)};
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\begin{eqnarray*}
\pause
& & \myfrac{\textcolor{CUDBlue}{4\pi}}{\textcolor{CUDRed}{12\pi}}
		\times360^\circ\\[1ex] \pause
&=& \hooklen{12\pi}{\myfrac{\,1\,}{3}}\times360^\circ\\[1ex] \pause
&=& \quad \textcolor{CUDGreen}{120^\circ}\ \ittan
\end{eqnarray*}
\end{minipage}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{よって表面積は}
\begin{minipage}{0.35\textwidth}
\begin{tikzpicture}[scale=0.5, very thick,>=stealth]
\filldraw[fill=CUDPink,draw=CUDRed](-60:6) arc(-60:60:6)--(0,0)--cycle;
\filldraw[fill=CUDBrightCyan,draw=CUDBlue](8,0) circle (2);

\draw(8,0)--(6,0);
\filldraw(8,0) circle (3pt);
\footnotesize
\draw(1,3) node{$6$};
\draw(7,0.5) node{$2$};
\draw(3,0) node[CUDGreen]{$120^\circ$};
\draw[CUDGreen,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex,flex=1]},ultra thick](-60:1.5) arc (-60:60:1.5);
\draw(5,-6) node[fill=CUDCream]
	{\small\(\text{円の面積}=\pi\times\text{半径}^2\)};
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\vspace*{-6ex}
\begin{eqnarray*}
\pause
& & \colorbox{CUDPink}{\(\pi\times 6^2\times
	\myfrac{\textcolor{CUDGreen}{120^\circ}}{360^\circ}\)}
	+ \colorbox{CUDBrightCyan}{\(\pi\times2^2\)}\\[1ex] \pause
&=& \hooklen{\colorbox{CUDPink}{\(\pi\times 6^2\times
		\myfrac{120^\circ}{360^\circ}\)}}{12\pi}
	+ \hooklen{\colorbox{CUDBlue}{\(\pi\times2^2\)}}{4\pi}
		\\[1ex] \pause
&=& \hskip4em 16\pi \quad\text{\□答}
\end{eqnarray*}
\end{minipage}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{おうぎ形の面積を求める、別の公式}
\begin{tikzpicture}[scale=1, ultra thick,>=stealth,
	gbb/.style={fill=white,inner sep=1pt},
	baseline={([yshift=-24pt] current bounding box.north)}]
\fill[CUDCream](-150:5) arc (-150:-30:5)--(0,0)--cycle;
\draw[line cap=round](-30:5)--(0,0);

\draw[CUDBlue,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},line width=2.4pt] (-150:5)--
	 node[CUDBlue,gbb]{半径$r$}(0,0);

%\draw[decorate,decoration={text along path,text={texte}}]は
%2バイト文字は使用不可のようだ。\usetikzlibrary{decorations.text}が必要です
\draw[CUDRed,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},
	line width=2.4pt](-150:5) arc(-150:-30:5);
%		node[midway,fill=white,inner sep=2pt]{弧の長さ$\ell$};
%\draw[CUDRed](-106:5) node[rotate=-16,gbb]{弧};
%\draw[CUDRed](-98:5) node[rotate=-8,gbb]{の};
%\draw[CUDRed](-90:5) node[rotate=0,gbb]{長};
%\draw[CUDRed](-82:5) node[rotate=8,gbb]{さ};
\foreach \i/\l in {0/弧,1/の,2/長,3/さ}
	\draw[CUDRed]({-106+8*\i}:5) node[rotate={-16+8*\i},gbb]{\l};
\draw[CUDRed](-75:5) node[rotate=15,fill=white,inner sep=3pt]{\(\ell\)};
\draw(-90:3) node{\( \text{面積}=\myfrac{\,1\,}{2}\,
	\textcolor{CUDRed}{\ell}\,\textcolor{CUDBlue}{r}\)};

\draw(6,-0.5) node{面積を求めるだけなら};
\draw(7,-2.5) node{を使うと楽！};
\draw(7,-4.5) node[text width=8em,align=center]
	{\begin{spacing}{1}
	\scriptsize
	でも中心角度を求めるならさっきやった方法も必要かも
	\end{spacing}
	};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{さっきの公式を使うと表面積は}
\vspace*{-1ex}
\begin{tikzpicture}[scale=0.5, very thick, >=stealth,
	baseline={([yshift=-36pt] current bounding box.north)}]
%\draw[help lines](0,0) grid (1,1);
\useasboundingbox(-0.2,-6) grid (11,6);
\footnotesize
\fill[fill=CUDPink](60:6)--(0,0)--(-60:6) arc (-60:60:6);
\filldraw[fill=CUDBrightCyan,draw=CUDBlue] (8,0) circle (2);
\draw[CUDRed] (60:6)--(0,0)--(-60:6);
\draw (8,0)--(6,0);
\filldraw (8,0) circle (3pt);
\draw(7,0.5) node{$2$};
\draw[CUDGreen,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},line width=2pt](-60:6) arc (-60:60:6)
	node[pos=0.8,CUDGreen,right,xshift=-2ex,fill=white,inner sep=2pt]
		{$4\pi${\scriptsize（底面の円周と同じ長さになるので）}};

\draw[CUDRed,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},line width=2pt](0,0)--
	node[pos=0.4,yshift=2.6ex,fill=white,inner sep=1pt]{$6$}(60:6);
\draw(-45:2)++(8,0) node[CUDBlue, fill=white,inner sep=1pt]
	{{\scriptsize 円周} $4\pi$};
\end{tikzpicture}
\begin{minipage}[t]{0.5\textwidth}
\begin{eqnarray*}
\pause
& & \colorbox{CUDPink}{\(\myfrac{\,1\,}{2}\times \textcolor{CUDGreen}{4\pi}
	\times\textcolor{CUDRed}{6}\)}
	+\colorbox{CUDBrightCyan}{\(\textcolor{CUDBlue}{\pi\times2^2}\)}
							\\[1ex] \pause
&=& \hooklen{\colorbox{CUDPink}{\(\myfrac{\,1\,}{2}\times4\pi\times6\)}}
		{12\pi}
	+ \hooklen{\pi\times2^2}{4\pi}\\[1ex] \pause
&=& \hskip3em 16\pi \quad\text{\□答}
\end{eqnarray*}
\end{minipage}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{実は一発で円錐の表面積を求める公式もある}
\begin{tikzpicture}[scale=2, very thick, >=stealth,
	baseline={([yshift=-24pt] current bounding box.north)}]
\ensui

\draw[CUDRed,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},ultra thick](AA)--
	node[CUDRed,inner sep=6pt,fill=white]{母線$L$} (B);
\draw[CUDBlue,{Stealth[length=1.6ex]}-{Stealth[length=1.6ex]},ultra thick](A)--
	node[inner sep=0pt,yshift=-1.2ex]
		{\footnotesize 半径$r$}(O);

\draw(3.4,1.2) node[text width=10em]{%
	\begin{beamercolorbox}[wd=\textwidth,ht=5ex,dp=3ex,
		sep=2pt,shadow=true,rounded=true, center]{CUDCream}
			円錐の表面積（公式）\\[1ex]
	\( =\pi \textcolor{CUDBlue}{r}\,(\textcolor{CUDRed}{L}
		+ \textcolor{CUDBlue}{r}) \)
	\end{beamercolorbox}
	};
\draw(3.6,-0.1) node[text width=10.5em]{
	\begin{spacing}{1}
	\footnotesize 
	教科書に載っている公式ではないので、試験で
	使っても正当な解答と認められないかも…
	\end{spacing}};
\fill (O) circle (1pt);
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{次に体積は？}
\begin{tikzpicture}[scale=2, very thick, >=stealth,
	baseline={([yshift=-24pt] current bounding box.north)}]
\ensui
\sunpo

\alt<6->{\draw(0,1.2) node[inner sep=2pt,fill=CUDCream]
			{$\scriptstyle 4\sqrt{2}$};}
	{\draw(0,1.2) node[inner sep=4pt,fill=white]{$\scriptstyle h$};}
\end{tikzpicture}
\begin{minipage}[t]{0.5\textwidth}
三平方の定理より
\begin{eqnarray*}
\pause
h^2+2^2	&=& 6^2\\ \pause
h^2 \phantom{{}+2^2} &=& 36-4\\ \pause
h^2 	&=& \hooklen{36-4}{32}\\ \pause
\text{\scriptsize\(h>0\)より}\quad h &=& \sqrt{32}\\ \pause
h	&=& 4\sqrt{2}\ \ittan
\end{eqnarray*}
\end{minipage}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{よって体積は}
\begin{tikzpicture}[scale=2, very thick, >=stealth,
	baseline={([yshift=-24pt] current bounding box.north)}]
\visible<2->{\fill[CUDBrightCyan](A) arc[start angle=-180,end angle=180,
			x radius=1.2, y radius=0.5];
	}
\ensui
\sunpo
\draw(0,1.2) node[inner sep=2pt,fill=CUDCream]{$\scriptstyle 4\sqrt{2}$};
\end{tikzpicture}
\begin{minipage}[t]{0.5\textwidth}
\vspace*{-3ex}
\begin{eqnarray*}
& & \myfrac{\,1\,}{3}\times\text{底面積}\times\text{高さ}\\[1ex]\pause
&=& \myfrac{\,1\,}{3}\times
	\hooklen{\text{底面積}}{\colorbox{CUDBrightCyan}{\(4\pi\)}}\times
	\hooklen{\text{高さ}}{\colorbox{CUDCream}{\(4\sqrt{2}\)}}\\[1ex]\pause
&=& \myfrac{\ 16\sqrt{2}\pi\ }{3}\quad\text{\□答}
\end{eqnarray*}
\end{minipage}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\end{document}


