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z%c;e!M+;lbIfv}7N@Cnz1|ca;e~SJ9ZFl*|T7MI_%+OR@iD}yG8At^!FSZZ#Anm1Z zy0-S#dz9Kq)19in^hU-1+!??XudVuWU1A`@)^URf`WkB-!B{&IWi8I}vKR<#@kcNy zNv!6%7i)xZ2&-lo;JeRF?m!#9ZnD4x}TN(*Yk;1E3l LMMZO)O9KA^d0?am diff --git a/latex/main.tex b/latex/main.tex index 9d82fe6..e3790ce 100644 --- a/latex/main.tex +++ b/latex/main.tex @@ -1,4 +1,6 @@ -\documentclass{article} +\documentclass[11pt]{article} +\usepackage[a4paper, margin=2.5cm]{geometry} +\usepackage{setspace} \usepackage{kotex} \usepackage[toc, page]{appendix} \usepackage{hyperref} @@ -14,20 +16,28 @@ \usepackage[style=apa,backend=biber]{biblatex} \addbibresource{references.bib} +\usepackage{booktabs} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \setlength{\parindent}{0pt} \setlength{\parskip}{0.6em} -\linespread{1.08} - -\title{κ³ μ •λœ 겉넓이λ₯Ό κ°–λŠ” κΈ°λ‘₯의 μ΅œλŒ€ 뢀피에 λŒ€ν•œ 탐ꡬ} -\author{μ΄μŠΉμ€€} -\date{\today} +\doublespacing \begin{document} -\maketitle +\begin{titlepage} + \centering + \vspace*{\fill} + {\Large κ³ μ •λœ 겉넓이λ₯Ό κ°–λŠ” κΈ°λ‘₯의 μ΅œλŒ€ 뢀피에 λŒ€ν•œ 탐ꡬ} + + \vspace{2em} + + {\large 20νŽ˜μ΄μ§€} + \vspace*{\fill} +\end{titlepage} + +\setcounter{page}{1} \newpage @@ -50,20 +60,13 @@ \section{연ꡬ 섀계} -λ³Έ μ €μžκ°€ μ›ν•˜λŠ” μ»¨ν…Œμ΄λ„ˆλŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯이닀. - -ν•΄λ‹Ή μ»¨ν…Œμ΄λ„ˆλ₯Ό λ§Œλ“€λ•Œ μ‚¬μš©ν•  수 μžˆλŠ” ν•„λΌλ©˜νŠΈμ˜ 양을 μ œν•œν•œλ‹€. - -μ΄λ•Œ 속이 빈 μ»¨ν…Œμ΄λ„ˆλ₯Ό 좜λ ₯ν• λ•Œ ν•„μš”ν•œ ν•„λΌλ©˜νŠΈμ˜ 양은 μ»¨ν…Œμ΄λ„ˆμ˜ 겉넓이와 λΉ„λ‘€ν•œλ‹€. -벽의 λ‘κ»˜λŠ” λ¬΄μ‹œν•œλ‹€. - -κ·Έλ ‡λ‹€λ©΄ κ³ μ •λœ ν•„λΌλ©˜νŠΈ μ–‘μœΌλ‘œ μ œμž‘ν•  수 μžˆλŠ” μ΅œλŒ€ λΆ€ν”Όμ˜ μ»¨ν…Œμ΄λ„ˆλ₯Ό κ΅¬ν•˜λŠ” 것은, -κ³ μ •λœ 겉넓이λ₯Ό κ°€μ§„ μ΅œλŒ€ λΆ€ν”Όμ˜ μ»¨ν…Œμ΄λ„ˆλ₯Ό κ΅¬ν•˜λŠ” 것과 κ°™λ‹€. - -μš°μ„  밑면이 μ§μ‚¬κ°ν˜•μΈ κΈ°λ‘₯인 μ§μœ‘λ©΄μ²΄μ— λŒ€ν•΄μ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό 탐ꡬ해본 λ’€, -밑면이 μ •λ‹€κ°ν˜•μΈ κΈ°λ‘₯에 λŒ€ν•΄μ„œλ„ νƒκ΅¬ν•˜μ—¬ μ–΄λ–€ 밑면이 μ–΄λ–€ λ„ν˜•μΌλ•Œ λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ˜λŠ”μ§€ νƒκ΅¬ν•œλ‹€. +λ³Έ μ €μžκ°€ μ›ν•˜λŠ” μ»¨ν…Œμ΄λ„ˆλŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯이며, ν•΄λ‹Ή μ»¨ν…Œμ΄λ„ˆλ₯Ό λ§Œλ“€λ•Œ μ‚¬μš©ν•  수 μžˆλŠ” ν•„λΌλ©˜νŠΈμ˜ 양을 μ œν•œν•œλ‹€. +μ΄λ•Œ 속이 빈 μ»¨ν…Œμ΄λ„ˆλ₯Ό 좜λ ₯ν• λ•Œ ν•„μš”ν•œ ν•„λΌλ©˜νŠΈμ˜ 양은 μ»¨ν…Œμ΄λ„ˆμ˜ 겉넓이와 λΉ„λ‘€ν•˜λ©°, 벽의 λ‘κ»˜λŠ” λ¬΄μ‹œν•œλ‹€. +κ·Έλ ‡λ‹€λ©΄ κ³ μ •λœ ν•„λΌλ©˜νŠΈ μ–‘μœΌλ‘œ μ œμž‘ν•  수 μžˆλŠ” μ΅œλŒ€ λΆ€ν”Όμ˜ μ»¨ν…Œμ΄λ„ˆλ₯Ό κ΅¬ν•˜λŠ” 것은, κ³ μ •λœ 겉넓이λ₯Ό κ°€μ§„ μ΅œλŒ€ λΆ€ν”Όμ˜ μ»¨ν…Œμ΄λ„ˆλ₯Ό κ΅¬ν•˜λŠ” 것과 κ°™λ‹€. +μš°μ„  밑면이 μ§μ‚¬κ°ν˜•μΈ κΈ°λ‘₯인 μ§μœ‘λ©΄μ²΄μ— λŒ€ν•΄μ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό 탐ꡬ해본 λ’€, 밑면이 μ •λ‹€κ°ν˜•μΈ κΈ°λ‘₯에 λŒ€ν•΄μ„œλ„ νƒκ΅¬ν•˜μ—¬ μ–΄λ–€ 밑면이 μ–΄λ–€ λ„ν˜•μΌλ•Œ λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ˜λŠ”μ§€ νƒκ΅¬ν•œλ‹€. κ²‰λ„“μ΄λŠ” $k$λΌλŠ” μƒμˆ˜λ‘œ κ³ μ •ν•œλ‹€. +μ²˜μŒμ—λŠ” λ³€μ˜ μˆ˜κ°€ λ§Žμ€ μ •λ‹€κ°ν˜•μΌμˆ˜λ‘ λΆ€ν”Όκ°€ 더 클 것이라 μ§κ΄€μ μœΌλ‘œ μ˜ˆμƒν•˜μ˜€λ‹€. 이 νƒκ΅¬μ—μ„œλŠ” κ·Έ μ˜ˆμƒμ΄ μˆ˜ν•™μ μœΌλ‘œ νƒ€λ‹Ήν•œμ§€λ₯Ό κ²€μ¦ν•˜κ³ , λ‚˜μ•„κ°€ μ–΄λ–€ λ„ν˜•μ΄ μ§„μ •ν•œ μ΅œμ μΈμ§€λ₯Ό μ—„λ°€ν•˜κ²Œ λ°νžŒλ‹€. \section{직윑면체의 μ΅œμ ν™”} @@ -81,11 +84,11 @@ λ³Έ μ—°κ΅¬μ—μ„œλŠ” μ‚¬μš©ν•  수 μžˆλŠ” ν•„λΌλ©˜νŠΈ 양이 κ³ μ •λ˜μ–΄ μžˆμœΌλ―€λ‘œ(λ‘κ»˜ λ¬΄μ‹œ), μ»¨ν…Œμ΄λ„ˆμ˜ 겉넓이λ₯Ό μƒμˆ˜ $k$둜 κ³ μ •ν•œλ‹€. -μœ—λ©΄μ΄ μ—†μœΌλ―€λ‘œ κ²‰λ„“μ΄λŠ” λ°”λ‹₯ $ab$와 λ„€ μ˜†λ©΄ $2bc+2ca$의 합이닀: +μœ—λ©΄μ΄ μ—†μœΌλ―€λ‘œ 겉넓이 $S$λŠ” λ°”λ‹₯ $ab$와 λ„€ μ˜†λ©΄ $2bc+2ca$의 합이닀: -\begin{equation} +\begin{equation*} S(a, b, c) = ab + 2bc + 2ca -\end{equation} +\end{equation*} λ”°λΌμ„œ μ œμ•½μ‘°κ±΄μ€ \begin{equation} @@ -94,19 +97,19 @@ ab + 2c(a + b) = k \label{eq:cuboid_constraint} \end{equation} -이닀. ν•œνŽΈ λΆ€ν”ΌλŠ” -\begin{equation} +이닀. ν•œνŽΈ λΆ€ν”Ό $V$λŠ” +\begin{equation*} V(a, b, c) = abc -\end{equation} +\end{equation*} 이닀. \subsection{μ œμ•½μ‘°κ±΄μ„ μ΄μš©ν•œ λ³€μˆ˜ 제거} λ°‘λ©΄μ˜ 넓이λ₯Ό $p$라 두면 $p = ab$이닀. 식 \ref{eq:cuboid_constraint}λŠ” -\begin{equation} +\begin{equation*} p + 2c(a + b) = k -\end{equation} +\end{equation*} 이 되며, 이λ₯Ό $c$에 λŒ€ν•΄ ν’€λ©΄ \begin{equation} c = \frac{k - p}{2(a + b)} @@ -128,42 +131,63 @@ 식 \ref{eq:cuboid_volume}μ—μ„œ $k$와 $p$κ°€ κ³ μ •λ˜λ©΄, \begin{equation*} - V(a, b) = p \cdot \frac{k - p}{2(a + b)} + V(a, b) = p \cdot \frac{k - p}{2(a + b)} = \frac{p \cdot (k - p)}{2(a + b)} \end{equation*} -μ΄λ―€λ‘œ, $V$λ₯Ό 크게 λ§Œλ“€κΈ° μœ„ν•΄μ„œλŠ” λΆ„λͺ¨μ˜ $(a + b)$κ°€ μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό ν•œλ‹€. +λΆ„μž 뢀뢄이 μƒμˆ˜κ°€ λœλ‹€. λ”°λΌμ„œ $V$λ₯Ό 크게 λ§Œλ“€κΈ° μœ„ν•΄μ„œλŠ” λΆ„λͺ¨μ˜ $(a + b)$κ°€ μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό ν•œλ‹€. 그런데 $ab = p$κ°€ κ³ μ •μΌλ•Œ μ‚°μˆ κΈ°ν•˜ 평균에 μ˜ν•΄ -\begin{equation} +\begin{equation*} a + b \geq 2 \sqrt{ab} = 2 \sqrt{p} -\end{equation} +\end{equation*} κ°€ μ„±λ¦½ν•˜κ³ , λ“±ν˜ΈλŠ” $a = b$일 λ•Œμ—λ§Œ μ„±λ¦½ν•œλ‹€. λ”°λΌμ„œ $ab = p$인 후보듀 쀑 $(a + b)$κ°€ μ΅œμ†Œκ°€ λ˜λŠ” κ²½μš°λŠ” $a = b$이며, -κ²°κ΅­ μ΅œμ ν•΄λŠ” λ°˜λ“œμ‹œ -\begin{equation} - a = b -\end{equation} -λ₯Ό λ§Œμ‘±ν•΄μ•Όν•œλ‹€. +κ²°κ΅­ μ΅œμ ν•΄λŠ” λ°˜λ“œμ‹œ $a = b$λ₯Ό λ§Œμ‘±ν•΄μ•Όν•œλ‹€. + +μœ„μ—μ„œ μ‚¬μš©ν•œ μ‚°μˆ κΈ°ν•˜ 평균(AM-GM) 방법 외에 νŽΈλ―ΈλΆ„μ„ μ΄μš©ν•˜λŠ” λ°©λ²•μœΌλ‘œλ„ $a = b$λ₯Ό λ„μΆœν•  수 μžˆλ‹€. +νŽΈλ―ΈλΆ„(partial derivative)μ΄λž€ 두 개 μ΄μƒμ˜ λ³€μˆ˜λ₯Ό κ°€μ§„ ν•¨μˆ˜μ—μ„œ, λ‚˜λ¨Έμ§€ λ³€μˆ˜λ₯Ό λͺ¨λ‘ μƒμˆ˜λ‘œ κ³ μ •ν•˜κ³  νŠΉμ • λ³€μˆ˜ ν•˜λ‚˜μ— λŒ€ν•΄μ„œλ§Œ λ―ΈλΆ„ν•˜λŠ” 것을 μ˜λ―Έν•œλ‹€. +예λ₯Ό λ“€μ–΄ $f(x, y) = x^2 y + 3xy^2$μ—μ„œ $y$λ₯Ό μƒμˆ˜λ‘œ μ·¨κΈ‰ν•˜κ³  $x$에 λŒ€ν•΄ λ―ΈλΆ„ν•˜λ©΄ +\begin{equation*} + \frac{\partial f}{\partial x} = 2xy + 3y^2 +\end{equation*} +이 λœλ‹€. + +식 \ref{eq:cuboid_volume}의 λΆ€ν”Ό $V(a, b) = \dfrac{ab(k - ab)}{2(a+b)}$에 νŽΈλ―ΈλΆ„μ„ μ μš©ν•˜λ©΄, +$b$λ₯Ό μƒμˆ˜λ‘œ 두고 $a$둜 λ―ΈλΆ„ν•œ $\dfrac{\partial V}{\partial a} = 0$을 μ •λ¦¬ν•˜λ©΄ +\begin{equation*} + \frac{\partial V}{\partial a} = 0 + \quad \Longrightarrow \quad + k = a^2 + 2ab +\end{equation*} +κ°€ 되고, λŒ€μΉ­μ μœΌλ‘œ $a$λ₯Ό μƒμˆ˜λ‘œ 두고 $b$둜 λ―ΈλΆ„ν•œ $\dfrac{\partial V}{\partial b} = 0$을 μ •λ¦¬ν•˜λ©΄ +\begin{equation*} + \frac{\partial V}{\partial b} = 0 + \quad \Longrightarrow \quad + k = 2ab + b^2 +\end{equation*} +κ°€ λœλ‹€. 두 식을 μ—°λ¦½ν•˜λ©΄ $a^2 = b^2$, 즉 $a = b$ ($a, b > 0$)κ°€ λ„μΆœλœλ‹€. + +이처럼 νŽΈλ―ΈλΆ„μ„ μ΄μš©ν•œ 접근은 두 방정식을 μ—°λ¦½ν•˜μ—¬ $a = b$λ₯Ό 직접 μ΄λŒμ–΄ λ‚Έλ‹€. 반면 AM-GM 방법은 "λΆ„λͺ¨ $(a+b)$λ₯Ό μ΅œμ†Œν™”ν•œλ‹€"λŠ” κΈ°ν•˜ν•™μ  직관을 λ°”νƒ•μœΌλ‘œ λ“±ν˜Έ 성립 쑰건을 λ°”λ‘œ μ½μ–΄λ‚Έλ‹€λŠ” μž₯점이 μžˆλ‹€. 두 방법 λͺ¨λ‘ 같은 결둠에 이λ₯΄μ§€λ§Œ, AM-GM은 μ™œ $a = b$μ—¬μ•Ό ν•˜λŠ”μ§€λ₯Ό 보닀 μ§κ΄€μ μœΌλ‘œ μ„€λͺ…ν•΄ μ€€λ‹€. \subsection{\texorpdfstring{μ΅œλŒ€ 뢀피와 $a$, $b$, $c$의 λΉ„μœ¨}{μ΅œλŒ€ 뢀피와 a, b, c의 λΉ„μœ¨}} 이제 $b$λ₯Ό $a$둜 두면, $p = ab = a^2$이고, 식 \ref{eq:cuboid_volume}을 μ •λ¦¬ν•˜λ©΄ λΆ€ν”ΌλŠ” -\begin{equation} +\begin{equation*} V(a) = p \cdot \frac{k - p}{2(a + b)} = \frac{ka - a^3}{4} -\end{equation} +\end{equation*} κ°€ λœλ‹€. μ΅œλŒ“κ°’μ΄ λ˜λŠ” μ§€μ μ—μ„œ λ„ν•¨μˆ˜μ˜ 값이 $0$이어야 ν•˜κΈ°μ—, $V'(a) = 0$을 ν’€λ©΄, -\begin{equation} +\begin{equation*} V'(a) = \frac{1}{4} \left( k - 3 a^2 \right) = 0 \quad \Longrightarrow \quad a = \sqrt{\frac{k}{3}} -\end{equation} +\end{equation*} 이 λœλ‹€. λ˜ν•œ $V''(a) = -\frac{6}{4}a$μ—μ„œ $a>0$이기 λ•Œλ¬Έμ— $V''(a) < 0$이 항상 μ„±λ¦½ν•œλ‹€. -λ”°λΌμ„œ κ·Έλ¦Ό \ref{fig:f_and_derivative}μ—μ„œλ„ λ³Ό 수 μžˆλ“―μ΄ $a < \sqrt{\frac{k}{3}}$μΌλ•ŒλŠ” $f'(a) > 0$μž„μœΌλ‘œ -$f(a)$κ°€ μ¦κ°€ν•˜λ©°, $a > \sqrt{\frac{k}{3}}$μΌλ•ŒλŠ” $f'(a) < 0$μž„μœΌλ‘œ $f(a)$κ°€ κ°μ†Œν•˜κΈ° λ•Œλ¬Έμ—, -$a = \sqrt{\frac{k}{3}}$인 지점이 $f(a)$의 μ΅œλŒ“κ°’μž„μ„ μ•Œ 수 μžˆλ‹€. +λ”°λΌμ„œ κ·Έλ¦Ό \ref{fig:f_and_derivative}μ—μ„œλ„ λ³Ό 수 μžˆλ“―μ΄ $a < \sqrt{\frac{k}{3}}$μΌλ•ŒλŠ” $V'(a) > 0$μž„μœΌλ‘œ +$V(a)$κ°€ μ¦κ°€ν•˜λ©°, $a > \sqrt{\frac{k}{3}}$μΌλ•ŒλŠ” $V'(a) < 0$μž„μœΌλ‘œ $V(a)$κ°€ κ°μ†Œν•˜κΈ° λ•Œλ¬Έμ—, +$a = \sqrt{\frac{k}{3}}$인 지점이 $V(a)$의 μ΅œλŒ“κ°’μž„μ„ μ•Œ 수 μžˆλ‹€. \begin{figure}[H] \centering @@ -183,30 +207,30 @@ $a = \sqrt{\frac{k}{3}}$인 지점이 $f(a)$의 μ΅œλŒ“κ°’μž„μ„ μ•Œ 수 μžˆλ‹€. \pgfmathsetmacro{\aopt}{sqrt(\k / 3)} \addplot[blue] {(\k*a - a^3) / 4}; - \addlegendentry{$f(a)$} + \addlegendentry{$V(a)$} \addplot[red] {1/4 * (\k - 3*a^2)}; - \addlegendentry{$f'(a)$} + \addlegendentry{$V'(a)$} \addplot[green] {-6/4*a}; - \addlegendentry{$f''(a)$} + \addlegendentry{$V''(a)$} \addplot[dashed, black] coordinates {(\aopt,-2) (\aopt,2)}; \addlegendentry{$a=\sqrt{\frac{k}{3}}$} \end{axis} \end{tikzpicture} - \caption{ν•¨μˆ˜ $f(a)$와 λ„ν•¨μˆ˜ 및 μ΄κ³„λ„ν•¨μˆ˜ κ·Έλž˜ν”„} + \caption{ν•¨μˆ˜ $V(a)$와 λ„ν•¨μˆ˜ 및 μ΄κ³„λ„ν•¨μˆ˜ κ·Έλž˜ν”„} \label{fig:f_and_derivative} \end{figure} λ„μΆœν•œ $a$값을 식 \ref{eq:cuboid_c}에 λŒ€μž…ν•˜λ©΄, -\begin{equation} +\begin{equation*} c = \frac{k - p}{2(a + b)} = \frac{k - \left( \sqrt{\frac{k}{3}} \right) ^ 2}{4 \sqrt{\frac{k}{3}}} = \frac{1}{2} \sqrt{\frac{k}{3}} = \frac{1}{2} a -\end{equation} +\end{equation*} 이 λœλ‹€. λ”°λΌμ„œ 졜적 λΉ„μœ¨μ€ \begin{equation} \boxed{a = b = 2c} @@ -221,11 +245,14 @@ $a = \sqrt{\frac{k}{3}}$인 지점이 $f(a)$의 μ΅œλŒ“κ°’μž„μ„ μ•Œ 수 μžˆλ‹€. \end{equation} 이 λœλ‹€. +μ§κ΄€μ μœΌλ‘œ μ •μœ‘λ©΄μ²΄(높이와 밑변이 같은 경우)κ°€ λΆ€ν”Όλ₯Ό μ΅œλŒ€ν™”ν•  것이라 μ˜ˆμƒν•˜μ˜€μœΌλ‚˜, μœ—λ©΄μ΄ μ—†λŠ” 쑰건으둜 인해 높이 $c$κ°€ λ°‘λ³€ $a$의 절반이 λ˜λŠ” ν˜•νƒœκ°€ μ΅œμ μž„μ„ μ•Œ 수 μžˆμ—ˆλ‹€. μ΄λŠ” μœ—λ©΄μ΄ μ—†μ–΄ μ ˆμ•½λœ 겉넓이λ₯Ό λ°‘λ©΄ ν™•μž₯에 더 μœ λ¦¬ν•˜κ²Œ μ‚¬μš©ν•  수 있기 λ•Œλ¬ΈμœΌλ‘œ ν•΄μ„λœλ‹€. + \subsection{해석 및 λ‹€μŒ 단계} 직윑면체의 λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ λ•Œ $a = b$μ΄λ―€λ‘œ 밑면은 μ •μ‚¬κ°ν˜•μ΄ λœλ‹€. 이λ₯Ό λ°”νƒ•μœΌλ‘œ 밑면이 μ •$n$κ°ν˜•μΈ κΈ°λ‘₯으둜 ν™•μž₯ν•˜μ—¬, λ³€μ˜ 수 $n$이 μ¦κ°€ν• μˆ˜λ‘ μ΅œλŒ€ λΆ€ν”Όκ°€ μ–΄λ–»κ²Œ λ³€ν•˜λŠ”μ§€ λΆ„μ„ν•œλ‹€. +μ΄λ•Œ 변이 λ§Žμ•„μ§ˆμˆ˜λ‘ 밑면이 원에 κ°€κΉŒμ›Œμ§€λ―€λ‘œ, λΆ€ν”Όκ°€ 단쑰 μ¦κ°€ν•˜μ—¬ 원기λ‘₯이 졜적일 것이라 μ˜ˆμƒν•˜μ˜€λ‹€. 이λ₯Ό μˆ˜ν•™μ μœΌλ‘œ μ—„λ°€νžˆ ν™•μΈν•œλ‹€. \section{μ •λ‹€κ°ν˜• κΈ°λ‘₯의 μ΅œμ ν™”} @@ -269,15 +296,15 @@ $n$개의 μ΄λ“±λ³€μ‚Όκ°ν˜•μœΌλ‘œ λΆ„ν• λœλ‹€. 이닀. μ΄λŸ¬ν•œ μ‚Όκ°ν˜•μ΄ $n$개 μ‘΄μž¬ν•˜λ―€λ‘œ, λ°‘λ©΄μ˜ 전체 넓이 $A$λŠ” -\begin{equation} +\begin{equation*} A = \frac{na^2}{4 \tan{(\pi / n)}} -\end{equation} +\end{equation*} 이 λœλ‹€. -μ΄λ•Œ κ³„μ‚°μ˜ 편의λ₯Ό μœ„ν•΄ -\begin{equation} +μ΄λ•Œ κ³„μ‚°μ˜ 편의λ₯Ό μœ„ν•΄ ($t$λŠ” $n$에 따라 κ²°μ •λ˜λŠ” μ–‘μˆ˜ μƒμˆ˜μ΄λ‹€) +\begin{equation*} t = \frac{n}{4 \tan(\pi / n)} -\end{equation} +\end{equation*} 둜 두면, μœ„ 식은 λ‹€μŒκ³Ό 같이 κ°„λ‹¨νžˆ ν‘œν˜„ν•  수 μžˆλ‹€. \begin{equation} A = ta^2 @@ -292,10 +319,10 @@ $n$개의 μ΄λ“±λ³€μ‚Όκ°ν˜•μœΌλ‘œ λΆ„ν• λœλ‹€. \end{equation} 둜 λ‚˜νƒ€λ‚Ό 수 μžˆλ‹€. -κΈ°λ‘₯의 κ²‰λ„“μ΄λŠ” μœ—λ©΄μ΄ μ—†κΈ° λ•Œλ¬Έμ— λ°‘λ©΄μ˜ 넓이 $A$와 μ˜†λ©΄μ˜ 넓이$Pb$의 합이닀. -\begin{equation} +κΈ°λ‘₯의 겉넓이 $S$λŠ” μœ—λ©΄μ΄ μ—†κΈ° λ•Œλ¬Έμ— λ°‘λ©΄μ˜ 넓이 $A$와 μ˜†λ©΄μ˜ 넓이 $Pb$의 합이닀. +\begin{equation*} S(a, b, n) = A + Pb -\end{equation} +\end{equation*} λ”°λΌμ„œ μ œμ•½μ‘°κ±΄μ€ \begin{equation} S(a, b, n) = k @@ -303,7 +330,7 @@ $n$개의 μ΄λ“±λ³€μ‚Όκ°ν˜•μœΌλ‘œ λΆ„ν• λœλ‹€. A + Pb = k \label{eq:regular_polygon_constraint} \end{equation} -이닀. ν•œνŽΈ λΆ€ν”ΌλŠ” +이닀. ν•œνŽΈ λΆ€ν”Ό $V$λŠ” \begin{equation} V(a, b, n) = Ab \label{eq:regular_polygon_v0} @@ -331,9 +358,8 @@ $b$λŠ” 더 이상 λ…λ¦½μ μœΌλ‘œ μ„ νƒλ˜λŠ” λ³€μˆ˜κ°€ μ•„λ‹ˆλ‹€. 결과적으둜 λΆ€ν”ΌλŠ” 식 \ref{eq:regular_polygon_v1}와 같이 $(a,n)$에 μ˜ν•΄ κ²°μ •λœλ‹€. λ”°λΌμ„œ μ œμ•½μ‘°κ±΄μ„ λ§Œμ‘±ν•˜λ©΄μ„œ λΆ€ν”Όλ₯Ό μ΅œλŒ€ν™”ν•˜λŠ” $(a, n)$ μŒμ„ μ°Ύμ•„μ•Ό ν•œλ‹€. -κ·ΈλŸ¬λ‚˜ μš°μ„  κ³ μ •λœ $n$μ—μ„œ μ΅œλŒ€ λΆ€ν”Όκ°€ λ˜λŠ” $a$λ₯Ό μ°ΎλŠ” 것뢀터 μ§„ν–‰ν•œλ‹€. -즉, $n$을 $3$μ΄μƒμ˜ μž„μ˜μ˜ μ •μˆ˜ $n_0$라고 ν•  λ•Œ, -λ¨Όμ € $n=n_0$λ₯Ό μƒμˆ˜λ‘œ 두고 $a$에 λŒ€ν•΄ λ‹€μŒμ˜ 1λ³€μˆ˜ ν•¨μˆ˜ +μš°μ„  κ³ μ •λœ $n$μ—μ„œ μ΅œλŒ€ λΆ€ν”Όκ°€ λ˜λŠ” $a$λ₯Ό μ°ΎλŠ” 것뢀터 μ§„ν–‰ν•œλ‹€. +즉, $n$을 $3$μ΄μƒμ˜ μž„μ˜μ˜ μ •μˆ˜ $n_0$라고 ν•  λ•Œ, λ¨Όμ € $n=n_0$λ₯Ό μƒμˆ˜λ‘œ 두고 $a$에 λŒ€ν•΄ λ‹€μŒμ˜ 1λ³€μˆ˜ ν•¨μˆ˜ \[ V_{n_0}(a):=V(a,n_0) \] @@ -341,43 +367,15 @@ $b$λŠ” 더 이상 λ…λ¦½μ μœΌλ‘œ μ„ νƒλ˜λŠ” λ³€μˆ˜κ°€ μ•„λ‹ˆλ‹€. \subsection{\texorpdfstring{$n = n_0$μ—μ„œμ˜ $a$에 λŒ€ν•œ 졜적 쑰건}{n = n0μ—μ„œμ˜ a에 λŒ€ν•œ 졜적 쑰건}} -$n=n_0$λ₯Ό κ³ μ •ν•˜λ©΄ $t$와 $n_0$λŠ” μƒμˆ˜μ΄λ©°, +$n=n_0$λ₯Ό κ³ μ •ν•˜λ©΄ $t$와 $n_0$λŠ” μƒμˆ˜μ΄λ©°, $V_{n_0}(a)=\frac{t}{n_0}(ka - ta^3)$λŠ” $a$에 λŒ€ν•œ ν•¨μˆ˜κ°€ λœλ‹€. +μ•žμ„œ μ§μœ‘λ©΄μ²΄μ—μ„œμ™€ λ™μΌν•œ λ°©λ²•μœΌλ‘œ $V_{n_0}'(a)=0$을 ν’€λ©΄, \begin{equation} - V_{n_0}(a)=\frac{t}{n_0} \left( ka - ta^3 \right) -\end{equation} -λŠ” $a$에 λŒ€ν•œ λ―ΈλΆ„ κ°€λŠ₯ν•œ ν•¨μˆ˜κ°€ λœλ‹€. -$V_{n_0}(a)$κ°€ μ΅œλŒ€κ°€ λ˜λŠ” μ§€μ μ—μ„œλŠ” ν•΄λ‹Ή ν•¨μˆ˜μ˜ λ„ν•¨μˆ˜μ˜ 값이 $0$이 λ˜μ–΄μ•Ό ν•˜κΈ° λ•Œλ¬Έμ— -\begin{equation} - V_{n_0}'(a)=0 -\end{equation} -이 성립해야 ν•œλ‹€. -이λ₯Ό κ³„μ‚°ν•˜λ©΄, -\begin{equation} - V_{n_0}'(a)=\frac{t}{n_0} \left( k - 3ta^2 \right) -\end{equation} -μ΄λ―€λ‘œ, -\begin{equation} - \frac{t}{n_0} \left( k - 3ta^2 \right) = 0 + \frac{t}{n_0}(k - 3ta^2) = 0 \quad \Longrightarrow \quad - k - 3ta^2 = 0 - \quad \Longrightarrow \quad - a = \sqrt{\frac{k}{3t}}. - \label{eq:regular_polygon_a_opt} -\end{equation} -λ₯Ό λ„μΆœν•  수 μžˆλ‹€. - -λ˜ν•œ \ref{eq:regular_polygon_a_opt}μ—μ„œ κ΅¬ν•œ $a$κ°€ $V_{n_0}(a)$κ°€ μ΅œλŒ€κ°€ λ˜λŠ” 지점이 되기 μœ„ν•΄μ„œλŠ” -ν•΄λ‹Ή ν•¨μˆ˜μ˜ μ΄κ³„λ„ν•¨μˆ˜κ°€ ν•΄λ‹Ή μ§€μ μ—μ„œ μŒμˆ˜μ—¬μ•Ό ν•œλ‹€. -\[ - V_{n_0}''(a)=\frac{t}{n_0}(-6ta) < 0 \quad (a>0) -\] -μ΄λ―€λ‘œ 식 \ref{eq:regular_polygon_a_opt}μ—μ„œ 얻은 μž„κ³„μ μ€ μ‹€μ œλ‘œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό μ€€λ‹€λŠ” 것을 μ•Œ 수 μžˆλ‹€. -λ”°λΌμ„œ μ •$n_0$각기λ‘₯μ—μ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό λ§Œλ“€κΈ° μœ„ν•œ λ³€μ˜ κΈΈμ΄λŠ” -\begin{equation} a = \sqrt{\frac{k}{3t}} \label{eq:regular_polygon_a} \end{equation} -둜 κ²°μ •λœλ‹€. +이 λœλ‹€. λ˜ν•œ $V_{n_0}''(a)=\frac{t}{n_0}(-6ta)<0$ ($a>0$)μ΄λ―€λ‘œ 이 μž„κ³„μ μ΄ μ΅œλŒ“κ°’μž„μ„ μ•Œ 수 μžˆλ‹€. \subsection{졜적 μ‘°κ±΄μ—μ„œμ˜ A와 b} @@ -389,27 +387,32 @@ $V_{n_0}(a)$κ°€ μ΅œλŒ€κ°€ λ˜λŠ” μ§€μ μ—μ„œλŠ” ν•΄λ‹Ή ν•¨μˆ˜μ˜ λ„ν•¨μˆ˜μ˜ $A$κ°€ $k$둜만 λ‚˜νƒ€λ‚΄μ–΄μ§„λ‹€λŠ” 것은 $n$의 값에 상관없이 μ •λ‹€κ°ν˜• κΈ°λ‘₯의 λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ λ•Œ, λ°‘λ©΄μ˜ λ„“μ΄λŠ” μΌμ •ν•˜λ‹€λŠ” 것을 μ˜λ―Έν•œλ‹€. μ΄λŠ” μœ„μ—μ„œ 직윑면체의 λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ˜λŠ” κ²½μš°μ—μ„œλ„ 두 λ°‘λ©΄μ˜ 길이가 λͺ¨λ‘ $\sqrt{\frac{k}{3}}$κ°€ λ˜μ–΄, λ°‘λ©΄μ˜ 넓이가 $\frac{k}{3}$이 λ˜μ—ˆκΈ° λ•Œλ¬Έμ— 이 κ²°κ³Όλ₯Ό λ§Œμ‘±ν•œλ‹€κ³  ν•  수 μžˆλ‹€. +이 κ²°κ³ΌλŠ” μ²˜μŒμ— μ˜ˆμƒν•˜μ§€ λͺ»ν•œ κ²ƒμ΄μ—ˆλ‹€. λ°‘λ©΄μ˜ ν˜•νƒœκ°€ μ‚Όκ°ν˜•μ΄λ“  μ •μ‚¬κ°ν˜•μ΄λ“  원이든 관계없이, μ΅œλŒ€ λΆ€ν”Ό μ‘°κ±΄μ—μ„œλŠ” κ²‰λ„“μ΄μ˜ μ •ν™•νžˆ $\frac{1}{3}$이 항상 밑면에 λ°°λΆ„λœλ‹€λŠ” 보편적 원리가 성립함을 λ°œκ²¬ν•˜μ˜€λ‹€. κΈ°λ‘₯의 λ†’μ΄λŠ” 식 \ref{eq:regular_polygon_b}에 κ΅¬ν•œ $A$값을 λŒ€μž…ν•˜λ©΄, -\begin{equation} +\begin{equation*} b = \frac{k - A}{P} = \frac{2k}{3P} +\end{equation*} +κ°€ λœλ‹€. λ”°λΌμ„œ μ΅œλŒ€ λΆ€ν”Ό μ‘°κ±΄μ—μ„œμ˜ λ°‘λ©΄ 넓이와 높이λ₯Ό μ •λ¦¬ν•˜λ©΄ +\begin{equation} + \boxed{A = \frac{k}{3}, \quad b = \frac{2k}{3P}} \end{equation} -κ°€ λœλ‹€. +이닀. \subsection{졜적 μ‘°κ±΄μ—μ„œμ˜ λΆ€ν”Ό} κΈ°λ‘₯의 λΆ€ν”ΌλŠ” $A$와 $b$의 곱이기에, μ΅œλŒ€ λΆ€ν”ΌλŠ” \begin{equation} - V_{max} = Ab = \frac{2k^2}{9P} + V_{\max} = Ab = \frac{2k^2}{9P} \end{equation} κ°€ λœλ‹€. μ—¬κΈ°μ„œ $k$λŠ” κ³ μ •λœ μ–‘μˆ˜μ΄λ©°, $P > 0$이닀. λ”°λΌμ„œ $V_{\max}$λŠ” $P$에 λŒ€ν•˜μ—¬ -\begin{equation} +\begin{equation*} V_{\max}(P) = \frac{C}{P} - \quad (C = \frac{2k^2}{9} > 0) -\end{equation} + \quad \left(C = \frac{2k^2}{9} > 0\right) +\end{equation*} 와 같은 ν˜•νƒœλ₯Ό κ°€μ§„λ‹€. ν•¨μˆ˜ $V_{\max}(P)$λŠ” $P > 0$μ—μ„œ 단쑰 κ°μ†Œ ν•¨μˆ˜μ΄λ―€λ‘œ, @@ -417,9 +420,9 @@ $P$κ°€ μž‘μ•„μ§ˆμˆ˜λ‘ $V_{\max}$λŠ” μ¦κ°€ν•œλ‹€. λ”°λΌμ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό μ–»κΈ° μœ„ν•΄μ„œλŠ” $P$κ°€ μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό ν•œλ‹€. $P$λŠ” 식 \ref{eq:regular_polygon_P}에 $a$값을 λŒ€μž…ν•˜λ©΄ -\begin{equation} +\begin{equation*} P(n) = na = n \sqrt{\frac{k}{3t}} = 2 \sqrt{\frac{kn \tan{(\pi / n)}}{3}} -\end{equation} +\end{equation*} 이 λœλ‹€. μ΄λ•Œ 식 \ref{eq:regular_polygon_a}μ—μ„œ $a$λ₯Ό $n$으둜 λ‚˜νƒ€λ‚΄μ—ˆκΈ° λ•Œλ¬Έμ—, $P$λŠ” $n$에 λŒ€ν•œ μΌλ³€μˆ˜ ν•¨μˆ˜κ°€ λœλ‹€. @@ -430,43 +433,35 @@ $P(n)$이 μ΅œμ†Œκ°€ 되기 μœ„ν•΄μ„œλŠ” $n \tan{(\pi / n)}$이 μ΅œμ†Œκ°€ λ˜μ–΄ 라고 ν•˜λ©΄, $Q(n)$이 μ΅œμ†Œκ°€ λ λ•Œμ˜ $n$이 κΈ°λ‘₯의 λΆ€ν”Όλ₯Ό μ΅œλŒ€ν™” ν•œλ‹€. μ—¬κΈ°μ„œ $n$은 μ •$n$κ°ν˜•μ˜ λ³€μ˜ 개수λ₯Ό μ˜λ―Έν•˜λ―€λ‘œ 본래 μ •μˆ˜ $n \geq 3$에 ν•œμ •λœλ‹€. κ·ΈλŸ¬λ‚˜ ν•¨μˆ˜ $Q(n)$의 증가 및 κ°μ†Œλ₯Ό λΆ„μ„ν•˜κΈ° μœ„ν•˜μ—¬, $n$을 μ‹€μˆ˜ $n > 2$둜 ν™•μž₯ν•˜μ—¬ 미뢄을 μˆ˜ν–‰ν•œλ‹€. -μ΄λŸ¬ν•œ 연속적 ν™•μž₯은 ν•¨μˆ˜μ˜ κ²½ν–₯성을 νŒŒμ•…ν•˜κΈ° μœ„ν•œ 것이며, ν‘Έν›„ 해석은 λ‹€μ‹œ μ •μˆ˜ $n$에 λŒ€ν•΄ 이루어진닀. +μ΄λŸ¬ν•œ 연속적 ν™•μž₯은 ν•¨μˆ˜μ˜ κ²½ν–₯성을 νŒŒμ•…ν•˜κΈ° μœ„ν•œ 것이며, μΆ”ν›„ 해석은 λ‹€μ‹œ μ •μˆ˜ $n$에 λŒ€ν•΄ 이루어진닀. + +이 방법이 μœ νš¨ν•˜λ €λ©΄ 연속 λ²”μœ„μ—μ„œ λ„μΆœν•œ 단쑰성이 이산적인 μ •μˆ˜ λ²”μœ„μ—μ„œλ„ λ™μΌν•˜κ²Œ μœ μ§€λ˜μ–΄μ•Ό ν•œλ‹€. λ§Œμ•½ $Q(n)$이 $n > 2$인 μ‹€μˆ˜ λ²”μœ„μ—μ„œ 극값을 κ°€μ§„λ‹€λ©΄, 연속 λΆ„μ„μ—μ„œ 얻은 단쑰 κ°μ†Œ 결둠이 μ •μˆ˜ $n$μ—λŠ” μ μš©λ˜μ§€ μ•Šμ„ 수 μžˆλ‹€. κ·ΈλŸ¬λ‚˜ λ³Έ νƒκ΅¬μ—μ„œλŠ” $Q'(n) < 0$을 λͺ¨λ“  $n > 2$에 λŒ€ν•΄ μ—„λ°€νžˆ 증λͺ…ν•˜μ˜€μœΌλ―€λ‘œ, 이 λ°©λ²•μ˜ 적용이 μ •λ‹Ήν™”λœλ‹€. $Q(n)$이 μ΅œμ†Œκ°€ 될 수 μžˆλŠ” 후보 지점은 κ·Ήκ°’κ³Ό 경계값이 μžˆλ‹€. λ¨Όμ € 극값이 μ΅œμ†Ÿκ°’μ΄ 되기 μœ„ν•΄μ„œλŠ” ν•΄λ‹Ή μ§€μ μ—μ„œ λ„ν•¨μˆ˜κ°€ $0$이어야 ν•˜λ©°, μ΄κ³„λ„ν•¨μˆ˜κ°€ μ–‘μˆ˜μ—¬μ•Ό ν•œλ‹€. $Q(n)$의 λ„ν•¨μˆ˜λŠ” 곱의 미뢄법에 따라 -\begin{equation} +\begin{equation*} Q'(n) = \tan\left(\frac{\pi}{n}\right) - \frac{\pi}{n}\sec^2\left(\frac{\pi}{n}\right) -\end{equation} +\end{equation*} κ°€ λœλ‹€. -μ΄λ•Œ 계산을 κ°„λ‹¨νžˆ ν•˜κΈ° μœ„ν•˜μ—¬ -\begin{equation} - \theta=\frac{\pi}{n} -\end{equation} -이라 두면, -\begin{equation} +μ΄λ•Œ 계산을 κ°„λ‹¨νžˆ ν•˜κΈ° μœ„ν•˜μ—¬ $\theta=\frac{\pi}{n}$이라 두면, +\begin{equation*} Q'(n) = \tan{\theta} - \theta \sec^2{\theta} -\end{equation} +\end{equation*} 둜 ν‘œν˜„ν•  수 μžˆκΈ°μ—, -\begin{equation} +\begin{equation*} \tan{\theta} - \theta \sec^2{\theta} = 0 \quad \Longrightarrow \quad \sin{\theta} \cos{\theta} = \theta -\end{equation} +\end{equation*} -배각곡식에 λ”°λ₯΄λ©΄ -\begin{equation} - 2 \sin{\theta} \cos{\theta} = \sin{2 \theta} - \quad \Longrightarrow \quad - \sin{\theta} \cos{\theta} = \frac{\sin{2 \theta}}{2} -\end{equation} -κ°€ μ„±λ¦½ν•œλ‹€. +배각곡식 $2\sin\theta\cos\theta = \sin 2\theta$에 λ”°λ₯΄λ©΄ $\sin\theta\cos\theta = \frac{\sin 2\theta}{2}$κ°€ μ„±λ¦½ν•œλ‹€. λ”°λΌμ„œ 극값이 μ‘΄μž¬ν•˜κΈ° μœ„ν•΄μ„œλŠ” -\begin{equation} +\begin{equation*} \theta = \frac{\sin 2\theta}{2} -\end{equation} +\end{equation*} λ₯Ό λ§Œμ‘±ν•΄μ•Ό ν•œλ‹€. 이λ₯Ό λΆ„μ„ν•˜κΈ° μœ„ν•˜μ—¬ λ‹€μŒκ³Ό 같은 ν•¨μˆ˜λ₯Ό μ •μ˜ν•œλ‹€. \begin{equation} @@ -474,50 +469,27 @@ $Q(n)$의 λ„ν•¨μˆ˜λŠ” 곱의 미뢄법에 따라 \quad (\theta \leq \frac{\pi}{3}) \end{equation} 그러면 μœ„ 방정식은 $f(\theta)=0$을 ν‘ΈλŠ” λ¬Έμ œμ™€ κ°™λ‹€. -이 ν•¨μˆ˜μ˜ λ„ν•¨μˆ˜λŠ” -\begin{equation} - f'(\theta) = \cos{2 \theta} - 1 -\end{equation} -이닀. -μ‚Όκ°ν•¨μˆ˜μ˜ μ„±μ§ˆμ— μ˜ν•΄ $\cos{2 \theta} \leq 1$이며, -$\theta > 0$일 λ•ŒλŠ” $\cos 2\theta < 1$μ΄λ―€λ‘œ -\begin{equation} - f'(\theta) < 0 \quad (\theta > 0) -\end{equation} -κ°€ μ„±λ¦½ν•œλ‹€. λ”°λΌμ„œ $f(\theta)$λŠ” $\theta>0$μ—μ„œ 단쑰 κ°μ†Œν•œλ‹€. -λ˜ν•œ -\begin{equation} - f(0) = 0 -\end{equation} -μ΄λ―€λ‘œ, $f(\theta) = 0$을 λ§Œμ‘±ν•˜λŠ” ν•΄λŠ” $\theta = 0$ ν•˜λ‚˜λΏμ΄λ‹€. +이 ν•¨μˆ˜μ˜ λ„ν•¨μˆ˜λŠ” $f'(\theta) = \cos 2\theta - 1$이닀. +μ‚Όκ°ν•¨μˆ˜μ˜ μ„±μ§ˆμ— μ˜ν•΄ $\cos 2\theta \leq 1$이며, $\theta > 0$일 λ•ŒλŠ” $\cos 2\theta < 1$μ΄λ―€λ‘œ $f'(\theta) < 0$ ($\theta > 0$)κ°€ μ„±λ¦½ν•œλ‹€. λ”°λΌμ„œ $f(\theta)$λŠ” $\theta>0$μ—μ„œ 단쑰 κ°μ†Œν•œλ‹€. +λ˜ν•œ $f(0) = 0$μ΄λ―€λ‘œ, $f(\theta) = 0$을 λ§Œμ‘±ν•˜λŠ” ν•΄λŠ” $\theta = 0$ ν•˜λ‚˜λΏμ΄λ‹€. κ·ΈλŸ¬λ‚˜ $\theta=\frac{\pi}{n}$의 μ •μ˜μ— μ˜ν•΄ $\theta=0$이 되렀면 $n \to \infty$이어야 ν•˜λ©°, μœ ν•œν•œ $n$μ—μ„œλŠ” $\theta>0$이닀. λ”°λΌμ„œ $Q(n)$은 μœ ν•œν•œ $n$μ—μ„œ λ‚΄λΆ€ 극값을 κ°–μ§€ μ•ŠλŠ”λ‹€. +더 λ‚˜μ•„κ°€, μœ„μ—μ„œ $f(\theta) < 0$ ($\theta > 0$)μž„μ„ λ³΄μ˜€μœΌλ―€λ‘œ $\sin\theta\cos\theta < \theta$κ°€ μ„±λ¦½ν•œλ‹€. 이λ₯Ό μ΄μš©ν•˜λ©΄ +\begin{equation*} + Q'(n) = \tan\theta - \theta\sec^2\theta = \frac{\sin\theta\cos\theta - \theta}{\cos^2\theta} < 0 +\end{equation*} +μž„μ„ 직접 확인할 수 μžˆλ‹€. λΆ„μž $\sin\theta\cos\theta - \theta < 0$이고 λΆ„λͺ¨ $\cos^2\theta > 0$μ΄λ―€λ‘œ, $Q(n)$은 $n \geq 3$ μ „μ²΄μ—μ„œ μˆœκ°μ†Œ ν•¨μˆ˜μ΄λ‹€. + λ‚΄λΆ€μ—μ„œ 극값이 μ‘΄μž¬ν•˜μ§€ μ•ŠμœΌλ―€λ‘œ, μ΅œμ†Ÿκ°’μ€ κ²½κ³„μ—μ„œλ§Œ λ°œμƒν•  수 μžˆλ‹€. 본래 λ¬Έμ œμ—μ„œ $n$은 μ •$n$κ°ν˜•μ˜ λ³€μ˜ 개수λ₯Ό μ˜λ―Έν•˜λ―€λ‘œ μ •μˆ˜ $n \geq 3$에 ν•œμ •λœλ‹€. -λ”°λΌμ„œ $Q(n)$의 μ •μ˜μ—­μ€ -\begin{equation} - \{n \mid n \geq 3,\; n \in \mathbb{Z}\} -\end{equation} -이닀. +λ”°λΌμ„œ $Q(n)$의 μ •μ˜μ—­μ€ $\{n \mid n \geq 3,\; n \in \mathbb{Z}\}$이닀. -이 μ •μ˜μ—­μ—μ„œ κ°€μž₯ μž‘μ€ 값은 $n=3$μ΄λ―€λ‘œ, 쒌츑 경계값은 -\begin{equation} - Q(3) \simeq 5.196 -\end{equation} -이닀. +이 μ •μ˜μ—­μ—μ„œ κ°€μž₯ μž‘μ€ 값은 $n=3$μ΄λ―€λ‘œ, 쒌츑 경계값은 $Q(3) \simeq 5.196$이닀. 우츑 경계값을 κ΅¬ν•˜κΈ° μœ„ν•΄μ„œλŠ” κ·Ήν•œμ„ μ‚¬μš©ν•œλ‹€. -이 경우 λ¬΄ν•œλŒ€λ‘œμ˜ κ·Ήν•œκ°’μ„ κ΅¬ν•˜λŠ” 것보닀 $0$으둜의 κ·Ήν•œκ°’μ„ κ΅¬ν•˜λŠ”κ²Œ 더 쉽기 λ•Œλ¬Έμ— -\begin{equation} - u = \frac{1}{n} -\end{equation} -으둜 μ •μ˜ν•˜λ©΄, -\begin{equation} - \lim_{n \to \infty}{Q(n)} = \lim_{u \to 0}{Q(u)} -\end{equation} -이 λœλ‹€. 이λ₯Ό κ³„μ‚°ν•˜λ©΄ +이 경우 λ¬΄ν•œλŒ€λ‘œμ˜ κ·Ήν•œκ°’μ„ κ΅¬ν•˜λŠ” 것보닀 $0$으둜의 κ·Ήν•œκ°’μ„ κ΅¬ν•˜λŠ”κ²Œ 더 쉽기 λ•Œλ¬Έμ— $u = \frac{1}{n}$으둜 μ •μ˜ν•˜λ©΄ $\lim_{n \to \infty}Q(n) = \lim_{u \to 0}Q(u)$이 λœλ‹€. 이λ₯Ό κ³„μ‚°ν•˜λ©΄ \begin{equation} \lim_{u \to 0}{Q(u)} = \lim_{u \to 0}{\frac{1}{u} \tan{\pi u}} = \pi \cdot \lim_{u \to 0}{\frac{\tan{(\pi u)}}{\pi u}} @@ -528,13 +500,14 @@ $\theta=0$이 되렀면 $n \to \infty$이어야 ν•˜λ©°, $\lim_{u \to 0}{Q(u)} < Q(3)$이기 λ•Œλ¬Έμ— $Q(u)$의 μ΅œμ†Ÿκ°’μ€ $n$이 λ¬΄ν•œλŒ€μΌλ•Œμž„μ„ μ•Œ 수 μžˆλ‹€. μ΄λŠ” λ°‘λ©΄μ˜ 변이 더 λ§Žμ„μˆ˜λ‘ κΈ°λ‘₯의 λΆ€ν”Όκ°€ 더 컀짐을 μ˜λ―Έν•œλ‹€. -μ •λ‹€κ°ν˜•μ—μ„œ λ³€μ˜ λŒ€μˆ˜κ°€ λŠμž„μ—†μ΄ λŠ˜μ–΄λ‚˜λ©΄ 원이 되기 λ•Œλ¬Έμ— -λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ˜λŠ” 밑면은 원이 되며, μ΄λŠ” 원기λ‘₯이 λœλ‹€. +μ •λ‹€κ°ν˜•μ—μ„œ λ³€μ˜ λŒ€μˆ˜κ°€ λŠμž„μ—†μ΄ λŠ˜μ–΄λ‚˜λ©΄ 원이 되기 λ•Œλ¬Έμ— λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ˜λŠ” 밑면은 원이 되며, μ΄λŠ” 원기λ‘₯이 λœλ‹€. λ”°λΌμ„œ μ΄λ²ˆμ—λŠ” μ œμ•½μ‘°κ±΄ν•˜μ—μ„œ κ°€μž₯ λΆ€ν”Όκ°€ 큰 원기λ‘₯에 λŒ€ν•΄μ„œ νƒκ΅¬ν•œλ‹€. \section{원기λ‘₯의 μ΅œμ ν™”} -원기λ‘₯ κΈ°λ‘₯(μœ—λ©΄ μ—†μŒ) λΆ„μ„μ—μ„œλŠ” λ°‘λ©΄ μ›μ˜ λ°˜μ§€λ¦„μ„ $a$, κΈ°λ‘₯의 높이λ₯Ό $b$라 ν•œλ‹€. +μ•žμ„œ μ •λ‹€κ°ν˜• λΆ„μ„μ—μ„œ $Q(n)$이 μˆœκ°μ†Œν•˜λ©° $V_{\max}(n)$이 단쑰 증가함을 λ³΄μ˜€κ³ , κ·Έ κ·Ήν•œμ΄ $\frac{k^{3/2}}{3\sqrt{3\pi}}$μž„μ„ ν™•μΈν•˜μ˜€λ‹€. 이제 κ·Ήν•œμ˜ λ„ν˜•μΈ 원기λ‘₯의 μ΅œλŒ€ λΆ€ν”Όλ₯Ό 직접 κ³„μ‚°ν•˜μ—¬ 이 κ°’κ³Ό μΌμΉ˜ν•˜λŠ”μ§€ κ²€μ¦ν•œλ‹€. + +원기λ‘₯(μœ—λ©΄ μ—†μŒ) λΆ„μ„μ—μ„œλŠ” λ°‘λ©΄ μ›μ˜ λ°˜μ§€λ¦„μ„ $a$, κΈ°λ‘₯의 높이λ₯Ό $b$라 ν•œλ‹€. μ΄λ•Œ, $a$와 $b$λŠ” κΈΈμ΄μ΄λ―€λ‘œ $a > 0$, $b > 0$이닀. κ·Έλ¦Ό \ref{fig:cylinder}은 ν•΄λ‹Ή κΈ°λ‘₯을 λ‚˜νƒ€λ‚΄λ©°, 길이 λΉ„μœ¨μ€ μ‹€μ œ 해와 λ¬΄κ΄€ν•˜λ‹€. @@ -545,17 +518,17 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 λ•Œλ¬Έμ— $Q(u)$의 μ΅œμ†Ÿκ°’μ€ $n$이 무 \label{fig:cylinder} \end{figure} -μœ—λ©΄μ΄ μ—†μœΌλ―€λ‘œ κ²‰λ„“μ΄λŠ” λ°‘λ©΄κ³Ό μ˜†λ©΄μ˜ 합이닀 -\begin{equation} +μœ—λ©΄μ΄ μ—†μœΌλ―€λ‘œ 겉넓이 $S$λŠ” λ°‘λ©΄κ³Ό μ˜†λ©΄μ˜ 합이닀: +\begin{equation*} S(a, b) = \pi a^2 + 2 \pi ab -\end{equation} -κ²‰λ„“μ΄λŠ” k둜 κ³ μ •λ˜μ–΄ 있기 λ•Œλ¬Έμ— μ œμ•½μ‘°κ±΄μ€ +\end{equation*} +κ²‰λ„“μ΄λŠ” $k$둜 κ³ μ •λ˜μ–΄ 있기 λ•Œλ¬Έμ— μ œμ•½μ‘°κ±΄μ€ \begin{equation} - S(a, b) = k \quad \Longleftrightarrow \pi a^2 + 2 \pi ab = k + S(a, b) = k \quad \Longleftrightarrow \quad \pi a^2 + 2 \pi ab = k \label{eq:cylinder_constraint} \end{equation} 이닀. -λΆ€ν”ΌλŠ” +ν•œνŽΈ λΆ€ν”Ό $V$λŠ” \begin{equation} V(a, b) = \pi a^2 b \label{eq:cylinder_V} @@ -569,37 +542,25 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 λ•Œλ¬Έμ— $Q(u)$의 μ΅œμ†Ÿκ°’μ€ $n$이 무 \end{equation} κ°€ λœλ‹€. 이λ₯Ό μ΄μš©ν•΄ 식 \ref{eq:cylinder_V}에 λŒ€μž…ν•˜λ©΄ λΆ€ν”ΌλŠ” -\begin{equation} +\begin{equation*} V(a) = \frac{ak - \pi a^3}{2} -\end{equation} +\end{equation*} 이 λœλ‹€. -λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ 되기 μœ„ν•΄μ„œλŠ” $V(a)$의 λ„ν•¨μˆ˜κ°€ $0$이 되며, μ΄κ³„λ„ν•¨μˆ˜κ°€ 음수인 지점이어야 ν•œλ‹€. -λ„ν•¨μˆ˜λŠ” -\begin{equation} - V'(a) = \frac{k - 3 \pi a^2}{2} -\end{equation} -이기에, -\begin{equation} - V'(a) = 0 - \quad \Longleftrightarrow \quad - \frac{k - 3 \pi a^2}{2} = 0 - \quad \Longleftrightarrow \quad - a = \sqrt{\frac{k}{3 \pi}} -\end{equation} -이닀. -λ˜ν•œ, -\begin{equation} - V''(a) = -3 \pi a -\end{equation} -둜 $a > 0$이기에 $V''(a) < 0$이 항상 μ„±λ¦½ν•˜κΈ°μ— $a = \sqrt{\frac{k}{3 \pi}}$이 $V(a)$의 μ΅œλŒ“κ°μž„μ„ μ•Œ 수 μžˆλ‹€. +μ•žμ„œμ™€ λ™μΌν•œ λ°©λ²•μœΌλ‘œ $V'(a)=0$을 ν’€λ©΄, +\begin{equation*} + \frac{k - 3\pi a^2}{2} = 0 + \quad \Longrightarrow \quad + a = \sqrt{\frac{k}{3\pi}} +\end{equation*} +이 λœλ‹€. $V''(a) = -3\pi a < 0$ ($a > 0$)μ΄λ―€λ‘œ 이 μž„κ³„μ μ΄ μ΅œλŒ“κ°’μ΄λ‹€. μ΄λ•Œμ˜ $b$λŠ” 식 \ref{eq:cylinder_b}에 κ΅¬ν•œ $a$값을 λŒ€μž…ν•˜λ©΄, -\begin{equation} +\begin{equation*} b = \frac{k - \pi a^2}{2 \pi a} = \frac{k - \pi \frac{k}{3 \pi}}{2 \pi \sqrt{\frac{k}{3 \pi}}} = \sqrt{\frac{k}{3 \pi}} -\end{equation} +\end{equation*} κ°€ λœλ‹€. λ”°λΌμ„œ 졜적 λΉ„μœ¨μ€ \begin{equation} @@ -609,7 +570,7 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 λ•Œλ¬Έμ— $Q(u)$의 μ΅œμ†Ÿκ°’μ€ $n$이 무 μ΅œλŒ€ λΆ€ν”ΌλŠ” \begin{equation} - V_{max} = \pi a^2 b = \pi a^3 = \pi \left( \frac{k}{3 \pi} \right)^{3/2} + V_{\max} = \pi a^2 b = \pi a^3 = \pi \left( \frac{k}{3 \pi} \right)^{3/2} \end{equation} 이닀. @@ -621,7 +582,7 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 λ•Œλ¬Έμ— $Q(u)$의 μ΅œμ†Ÿκ°’μ€ $n$이 무 \begin{tikzpicture} \begin{axis}[ axis lines=middle, - xlabel=$n$, ylabel=$V_{max}$, + xlabel=$n$, ylabel=$V_{\max}$, legend style={draw=none, fill=none} ] @@ -636,18 +597,16 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 λ•Œλ¬Έμ— $Q(u)$의 μ΅œμ†Ÿκ°’μ€ $n$이 무 (\n, {(\k^(3/2)) / (3*sqrt(3*\n*tan(deg(pi/\n))))}) }; } - \addlegendentry{$V_{max}(n)$} + \addlegendentry{$V_{\max}(n)$} \end{axis} \end{tikzpicture} - \caption{ν•¨μˆ˜ $V_{max}(n)$} + \caption{ν•¨μˆ˜ $V_{\max}(n)$} \label{fig:v_max_n} \end{figure} -κ·Έλ¦Ό \ref{fig:v_max_n}은 μ •$n$κ°ν˜• κΈ°λ‘₯μ—μ„œ μ΅œλŒ€ λΆ€ν”Ό $V_{\max}(n)$이 -λ³€μ˜ 개수 $n$에 따라 μ–΄λ–»κ²Œ λ³€ν™”ν•˜λŠ”μ§€λ₯Ό λ‚˜νƒ€λ‚Έ 것이닀. -μ΄λ•Œ $k$λŠ” μΌμ •ν•œ κ°’μœΌλ‘œ κ³ μ •λ˜μ–΄ 있으며, -각 $n$에 λŒ€ν•΄ μ•žμ„œ λ„μΆœν•œ 졜적 쑰건을 μ μš©ν•˜μ—¬ κ³„μ‚°ν•œ 값이닀. +κ·Έλ¦Ό \ref{fig:v_max_n}은 μ •$n$κ°ν˜• κΈ°λ‘₯μ—μ„œ μ΅œλŒ€ λΆ€ν”Ό $V_{\max}(n)$이 λ³€μ˜ 개수 $n$에 따라 μ–΄λ–»κ²Œ λ³€ν™”ν•˜λŠ”μ§€λ₯Ό λ‚˜νƒ€λ‚Έ 것이닀. +μ΄λ•Œ $k$λŠ” μΌμ •ν•œ κ°’μœΌλ‘œ κ³ μ •λ˜μ–΄ 있으며, 각 $n$에 λŒ€ν•΄ μ•žμ„œ λ„μΆœν•œ 졜적 쑰건을 μ μš©ν•˜μ—¬ κ³„μ‚°ν•œ 값이닀. κ·Έλž˜ν”„λ₯Ό 톡해 λ‹€μŒκ³Ό 같은 νŠΉμ§•μ„ κ΄€μ°°ν•  수 μžˆλ‹€. @@ -657,87 +616,45 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 λ•Œλ¬Έμ— $Q(u)$의 μ΅œμ†Ÿκ°’μ€ $n$이 무 \end{itemize} μ΄λŸ¬ν•œ κ²½ν–₯은 μ•žμ„œ μœ λ„ν•œ 식을 톡해 μ΄λ‘ μ μœΌλ‘œλ„ μ„€λͺ…ν•  수 μžˆλ‹€. -μ΅œλŒ€ λΆ€ν”ΌλŠ” -\begin{equation} - V_{\max}(n) = \frac{2k^2}{9P(n)} -\end{equation} -이며, λ‘˜λ ˆλŠ” -\begin{equation} - P(n) = 2 \sqrt{\frac{kn \tan(\pi / n)}{3}} -\end{equation} -μ΄λ―€λ‘œ, -\begin{equation} - V_{\max}(n) - = \frac{2k^2}{9 \cdot 2 \sqrt{\frac{kn \tan(\pi / n)}{3}}} - = \frac{k^{3/2}}{3 \sqrt{3}} \cdot \frac{1}{\sqrt{n \tan(\pi / n)}} -\end{equation} -둜 λ‚˜νƒ€λ‚Ό 수 μžˆλ‹€. +μ•žμ„œ λ„μΆœν•œ $V_{\max}(n) = \frac{2k^2}{9P(n)}$κ³Ό $P(n) = 2\sqrt{\frac{kn\tan(\pi/n)}{3}}$λ₯Ό κ²°ν•©ν•˜λ©΄, +\begin{equation*} + V_{\max}(n) = \frac{k^{3/2}}{3\sqrt{3}} \cdot \frac{1}{\sqrt{Q(n)}} +\end{equation*} +둜 λ‚˜νƒ€λ‚Ό 수 μžˆλ‹€. λ”°λΌμ„œ $V_{\max}(n)$의 증감은 $Q(n) = n\tan\!\left(\frac{\pi}{n}\right)$의 변화에 μ˜ν•΄ κ²°μ •λœλ‹€. -λ”°λΌμ„œ $V_{\max}(n)$의 증감은 ν•¨μˆ˜ -\begin{equation} - Q(n) = n \tan\left(\frac{\pi}{n}\right) -\end{equation} -의 변화에 μ˜ν•΄ κ²°μ •λœλ‹€. - -μ•žμ„œ $Q(n)$이 $n$이 증가함에 따라 κ°μ†Œν•˜λ©°, -\begin{equation} - \lim_{n \to \infty} Q(n) = \pi -\end{equation} -μž„μ„ λ³΄μ˜€λ‹€. -λ”°λΌμ„œ -\begin{equation} +μ•žμ„œ $Q(n)$이 μˆœκ°μ†Œν•˜λ©° $\lim_{n \to \infty} Q(n) = \pi$μž„μ„ λ³΄μ˜€λ‹€. λ”°λΌμ„œ +\begin{equation*} \lim_{n \to \infty} V_{\max}(n) = \frac{k^{3/2}}{3 \sqrt{3\pi}} -\end{equation} +\end{equation*} κ°€ λœλ‹€. 이 값은 이후 원기λ‘₯의 μ΅œλŒ€ λΆ€ν”Όλ‘œ λ„μΆœλœ κ°’κ³Ό μ •ν™•νžˆ μΌμΉ˜ν•œλ‹€. +μ΄λŸ¬ν•œ μΌμΉ˜λŠ” μ •λ‹€κ°ν˜•μ΄ $n \to \infty$일 λ•Œ μ›μœΌλ‘œ μˆ˜λ ΄ν•œλ‹€λŠ” κΈ°ν•˜ν•™μ  μ‚¬μ‹€μ˜ μˆ˜ν•™μ  ν‘œν˜„μ΄λ‹€. 탐ꡬ μ΄ˆλ°˜μ— μ„Έμš΄ μ˜ˆμƒβ€”λ³€μ΄ λ§Žμ„μˆ˜λ‘ λΆ€ν”Όκ°€ 크고 원기λ‘₯이 μ΅œμ β€”μ΄ κ·Ήν•œ 뢄석과 원기λ‘₯ 직접 계산 λͺ¨λ‘μ—μ„œ μΌκ΄€λ˜κ²Œ κ²€μ¦λœ μ…ˆμ΄λ‹€. -λ˜ν•œ $n$이 μ¦κ°€ν• μˆ˜λ‘ 증가폭이 점점 μž‘μ•„μ§€λŠ” μ΄μœ λŠ” -$\tan(\pi/n)$이 $n$이 컀질수둝 $\frac{\pi}{n}$에 κ°€κΉŒμ›Œμ§€κΈ° λ•Œλ¬Έμ΄λ‹€. -즉, -\begin{equation} - \tan\left(\frac{\pi}{n}\right) \approx \frac{\pi}{n} - \quad (n \to \infty) -\end{equation} -μ΄λ―€λ‘œ, -\begin{equation} - Q(n) \approx \pi -\end{equation} -둜 λΉ λ₯΄κ²Œ μˆ˜λ ΄ν•˜κ²Œ 되고, -이에 따라 $V_{\max}(n)$ μ—­μ‹œ 일정 값에 점점 κ°€κΉŒμ›Œμ§€κ²Œ λœλ‹€. +λ˜ν•œ $n$이 μ¦κ°€ν• μˆ˜λ‘ 증가폭이 점점 μž‘μ•„μ§€λŠ” μ΄μœ λŠ” $\tan(\pi/n)$이 $n$이 컀질수둝 $\frac{\pi}{n}$에 κ°€κΉŒμ›Œμ§€κΈ° λ•Œλ¬Έμ΄λ‹€. +즉, $\tan\!\left(\frac{\pi}{n}\right) \approx \frac{\pi}{n}$ ($n \to \infty$)μ΄λ―€λ‘œ $Q(n) \approx \pi$둜 λΉ λ₯΄κ²Œ μˆ˜λ ΄ν•˜κ²Œ 되고, 이에 따라 $V_{\max}(n)$ μ—­μ‹œ 일정 값에 점점 κ°€κΉŒμ›Œμ§€κ²Œ λœλ‹€. -κΈ°ν•˜ν•™μ μœΌλ‘œ ν•΄μ„ν•˜λ©΄, -μ •λ‹€κ°ν˜•μ€ λ³€μ˜ κ°œμˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ 원에 κ°€κΉŒμ›Œμ§€λ©°, -같은 넓이λ₯Ό κ°€μ§ˆ λ•Œ λ‘˜λ ˆκ°€ 점점 κ°μ†Œν•œλ‹€. -이둜 인해 λ™μΌν•œ 겉넓이 μ‘°κ±΄μ—μ„œ 더 λ§Žμ€ 면적을 밑면에 ν• λ‹Ήν•  수 있게 되고, -결과적으둜 λΆ€ν”Όκ°€ μ¦κ°€ν•˜κ²Œ λœλ‹€. +κΈ°ν•˜ν•™μ μœΌλ‘œ ν•΄μ„ν•˜λ©΄, μ •λ‹€κ°ν˜•μ€ λ³€μ˜ κ°œμˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ 원에 κ°€κΉŒμ›Œμ§€λ©°, 같은 넓이λ₯Ό κ°€μ§ˆ λ•Œ λ‘˜λ ˆκ°€ 점점 κ°μ†Œν•œλ‹€. +이둜 인해 λ™μΌν•œ 겉넓이 μ‘°κ±΄μ—μ„œ 더 λ§Žμ€ 면적을 밑면에 ν• λ‹Ήν•  수 있게 되고, 결과적으둜 λΆ€ν”Όκ°€ μ¦κ°€ν•˜κ²Œ λœλ‹€. -κ·ΈλŸ¬λ‚˜ $n$이 μΆ©λΆ„νžˆ 컀진 μ΄ν›„μ—λŠ” -μ •λ‹€κ°ν˜•κ³Ό 원 μ‚¬μ΄μ˜ 차이가 맀우 μž‘μ•„μ§€κΈ° λ•Œλ¬Έμ—, -λΆ€ν”Ό 증가 μ—­μ‹œ 거의 λ°œμƒν•˜μ§€ μ•ŠλŠ”λ‹€. +κ·ΈλŸ¬λ‚˜ $n$이 μΆ©λΆ„νžˆ 컀진 μ΄ν›„μ—λŠ” μ •λ‹€κ°ν˜•κ³Ό 원 μ‚¬μ΄μ˜ 차이가 맀우 μž‘μ•„μ§€κΈ° λ•Œλ¬Έμ—, λΆ€ν”Ό 증가 μ—­μ‹œ 거의 λ°œμƒν•˜μ§€ μ•ŠλŠ”λ‹€. μ΄λŠ” κ·Έλž˜ν”„μ—μ„œ 증가가 점점 μ™„λ§Œν•΄μ§€λŠ” ν˜•νƒœλ‘œ λ‚˜νƒ€λ‚œλ‹€. -λ”°λΌμ„œ μ •$n$κ°ν˜• κΈ°λ‘₯의 μ΅œλŒ€ λΆ€ν”ΌλŠ” -$n$이 증가함에 따라 단쑰 μ¦κ°€ν•˜λ©°, -κ·Έ κ·Ήν•œμ€ 원기λ‘₯의 μ΅œλŒ€ 뢀피와 μΌμΉ˜ν•œλ‹€. +λ”°λΌμ„œ μ •$n$κ°ν˜• κΈ°λ‘₯의 μ΅œλŒ€ λΆ€ν”ΌλŠ” $n$이 증가함에 따라 단쑰 μ¦κ°€ν•˜λ©°, κ·Έ κ·Ήν•œμ€ 원기λ‘₯의 μ΅œλŒ€ 뢀피와 μΌμΉ˜ν•œλ‹€. \section{κ²°λ‘ } -λ³Έ μ—°κ΅¬λŠ” κ³ μ •λœ 겉넓이 $k$λ₯Ό κ°€μ§€λŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯ μ€‘μ—μ„œ -μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ„ νƒκ΅¬ν•˜μ˜€λ‹€. -μ΄λŠ” 3D ν”„λ¦°ν„°λ‘œ μ œν•œλœ ν•„λΌλ©˜νŠΈ 양을 μ‚¬μš©ν•  λ•Œ -κ°€μž₯ 효율적인 μ»¨ν…Œμ΄λ„ˆ ν˜•νƒœλ₯Ό μ°ΎλŠ” λ¬Έμ œλ‘œλΆ€ν„° μΆœλ°œν•˜μ˜€λ‹€. +λ³Έ μ—°κ΅¬λŠ” κ³ μ •λœ 겉넓이 $k$λ₯Ό κ°€μ§€λŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯ μ€‘μ—μ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ„ νƒκ΅¬ν•˜μ˜€λ‹€. +μ΄λŠ” 3D ν”„λ¦°ν„°λ‘œ μ œν•œλœ ν•„λΌλ©˜νŠΈ 양을 μ‚¬μš©ν•  λ•Œ κ°€μž₯ 효율적인 μ»¨ν…Œμ΄λ„ˆ ν˜•νƒœλ₯Ό μ°ΎλŠ” λ¬Έμ œλ‘œλΆ€ν„° μΆœλ°œν•˜μ˜€λ‹€. λ¨Όμ € 밑면이 μ§μ‚¬κ°ν˜•μΈ 직윑면체의 경우λ₯Ό λΆ„μ„ν•˜μ˜€λ‹€. -겉넓이 $S=ab+2ac+2bc=k$의 μ œμ•½μ‘°κ±΄ ν•˜μ—μ„œ -λΆ€ν”Ό $V=abc$λ₯Ό μ΅œλŒ€ν™”ν•œ κ²°κ³Ό, +겉넓이 $S=ab+2ac+2bc=k$의 μ œμ•½μ‘°κ±΄ ν•˜μ—μ„œ λΆ€ν”Ό $V=abc$λ₯Ό μ΅œλŒ€ν™”ν•œ κ²°κ³Ό, \[ a=b=2c \] 일 λ•Œ μ΅œλŒ€ λΆ€ν”Όκ°€ 됨을 λ„μΆœν•˜μ˜€λ‹€. -즉, 밑면은 μ •μ‚¬κ°ν˜•μ΄ 되며, -λ†’μ΄λŠ” λ°‘λ³€ 길이의 절반이 λœλ‹€. +즉, 밑면은 μ •μ‚¬κ°ν˜•μ΄ 되며, λ†’μ΄λŠ” λ°‘λ³€ 길이의 절반이 λœλ‹€. λ˜ν•œ μ΄λ•Œ λ°‘λ©΄μ˜ λ„“μ΄λŠ” 항상 \[ A=\frac{k}{3} @@ -745,9 +662,7 @@ $n$이 증가함에 따라 단쑰 μ¦κ°€ν•˜λ©°, 이 됨을 ν™•μΈν•˜μ˜€λ‹€. 이λ₯Ό μΌλ°˜ν™”ν•˜μ—¬ 밑면이 μ •$n$κ°ν˜•μΈ κΈ°λ‘₯을 λΆ„μ„ν•˜μ˜€λ‹€. -μ œμ•½μ‘°κ±΄μ„ μ΄μš©ν•˜μ—¬ 높이λ₯Ό μ†Œκ±°ν•˜κ³  -λΆ€ν”Όλ₯Ό $(a,n)$의 ν•¨μˆ˜λ‘œ ν‘œν˜„ν•œ λ’€, -κ³ μ •λœ $n$μ—μ„œ μ΅œλŒ€κ°€ λ˜λŠ” $a$λ₯Ό κ΅¬ν•˜μ˜€λ‹€. +μ œμ•½μ‘°κ±΄μ„ μ΄μš©ν•˜μ—¬ 높이λ₯Ό μ†Œκ±°ν•˜κ³  λΆ€ν”Όλ₯Ό $(a,n)$의 ν•¨μˆ˜λ‘œ ν‘œν˜„ν•œ λ’€, κ³ μ •λœ $n$μ—μ„œ μ΅œλŒ€κ°€ λ˜λŠ” $a$λ₯Ό κ΅¬ν•˜μ˜€λ‹€. κ·Έ κ²°κ³Ό, \[ a=\sqrt{\frac{k}{3t}} @@ -757,57 +672,108 @@ $n$이 증가함에 따라 단쑰 μ¦κ°€ν•˜λ©°, A=\frac{k}{3} \] 이 됨을 ν™•μΈν•˜μ˜€λ‹€. -μ΄λŠ” λ°‘λ©΄μ˜ ν˜•νƒœμ™€ λ¬΄κ΄€ν•˜κ²Œ -μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” κ²½μš°μ—λŠ” -전체 κ²‰λ„“μ΄μ˜ $\frac{1}{3}$이 밑면에, -$\frac{2}{3}$이 μ˜†λ©΄μ— λ°°λΆ„λœλ‹€λŠ” 것을 μ˜λ―Έν•œλ‹€. +μ΄λŠ” λ°‘λ©΄μ˜ ν˜•νƒœμ™€ λ¬΄κ΄€ν•˜κ²Œ, μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” κ²½μš°μ—λŠ” 전체 κ²‰λ„“μ΄μ˜ $\frac{1}{3}$이 밑면에, $\frac{2}{3}$이 μ˜†λ©΄μ— λ°°λΆ„λœλ‹€λŠ” 것을 μ˜λ―Έν•œλ‹€. -λ‹€μŒμœΌλ‘œ λΆ€ν”Ό μ΅œλŒ€ν™”λ₯Ό μœ„ν•΄μ„œλŠ” -λ‘˜λ ˆ $P$κ°€ μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό 함을 λ³΄μ˜€κ³ , -이λ₯Ό μœ„ν•΄ ν•¨μˆ˜ +λ‹€μŒμœΌλ‘œ λΆ€ν”Ό μ΅œλŒ€ν™”λ₯Ό μœ„ν•΄μ„œλŠ” λ‘˜λ ˆ $P$κ°€ μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό 함을 λ³΄μ˜€κ³ , 이λ₯Ό μœ„ν•΄ ν•¨μˆ˜ \[ Q(n)=n\tan\left(\frac{\pi}{n}\right) \] 을 λΆ„μ„ν•˜μ˜€λ‹€. -λ‚΄λΆ€ 극값이 μ‘΄μž¬ν•˜μ§€ μ•ŠμŒμ„ 증λͺ…ν•˜κ³ , -경계 뢄석을 톡해 +λ‚΄λΆ€ 극값이 μ‘΄μž¬ν•˜μ§€ μ•ŠμŒμ„ 증λͺ…ν•˜κ³ , 경계 뢄석을 톡해 \[ \lim_{n\to\infty}Q(n)=\pi \] μž„μ„ λ„μΆœν•˜μ˜€λ‹€. -μ΄λŠ” λ³€μ˜ κ°œμˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ λ‘˜λ ˆκ°€ κ°μ†Œν•˜λ©°, -κ·Έ κ·Ήν•œμ΄ μ›μ˜ κ²½μš°μ™€ μΌμΉ˜ν•¨μ„ μ˜λ―Έν•œλ‹€. +μ΄λŠ” λ³€μ˜ κ°œμˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ λ‘˜λ ˆκ°€ κ°μ†Œν•˜λ©°, κ·Έ κ·Ήν•œμ΄ μ›μ˜ κ²½μš°μ™€ μΌμΉ˜ν•¨μ„ μ˜λ―Έν•œλ‹€. -λ”°λΌμ„œ κ³ μ •λœ 겉넓이 쑰건 ν•˜μ—μ„œ -μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ€ -밑면이 원인 원기λ‘₯μž„μ„ μ•Œ 수 μžˆλ‹€. -μ΄λŠ” μ •λ‹€κ°ν˜•μ΄ λ³€μ˜ 개수λ₯Ό λ¬΄ν•œνžˆ 늘릴 λ•Œ -원이 λœλ‹€λŠ” κΈ°ν•˜ν•™μ  사싀과도 μΌμΉ˜ν•œλ‹€. +λ”°λΌμ„œ κ³ μ •λœ 겉넓이 쑰건 ν•˜μ—μ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ€ 밑면이 원인 원기λ‘₯μž„μ„ μ•Œ 수 μžˆλ‹€. +μ΄λŠ” μ •λ‹€κ°ν˜•μ΄ λ³€μ˜ 개수λ₯Ό λ¬΄ν•œνžˆ 늘릴 λ•Œ 원이 λœλ‹€λŠ” κΈ°ν•˜ν•™μ  사싀과도 μΌμΉ˜ν•œλ‹€. + +μ„Έ λ„ν˜•μ— λŒ€ν•œ 졜적 쑰건과 μ΅œλŒ€ λΆ€ν”Όλ₯Ό ν‘œ \ref{tab:summary}에 μ •λ¦¬ν•œλ‹€. + +\begin{table}[H] + \centering + \begin{tabular}{@{}lccc@{}} + \toprule + λ„ν˜• & 졜적 λΉ„μœ¨ & λ°‘λ©΄ 넓이 & μ΅œλŒ€ λΆ€ν”Ό \\ + \midrule + 직윑면체 & $a = b = 2c$ & $\dfrac{k}{3}$ & $\dfrac{1}{2}\!\left(\dfrac{k}{3}\right)^{\!3/2}$ \\[8pt] + μ •$n$κ°ν˜• κΈ°λ‘₯ & $A = \dfrac{k}{3},\; b = \dfrac{2k}{3P}$ & $\dfrac{k}{3}$ & $\dfrac{2k^2}{9P(n)}$ \\[8pt] + 원기λ‘₯ & $a = b$ & $\dfrac{k}{3}$ & $\pi\!\left(\dfrac{k}{3\pi}\right)^{\!3/2}$ \\ + \bottomrule + \end{tabular} + \caption{λ„ν˜•λ³„ μ΅œλŒ€ λΆ€ν”Ό 쑰건 비ꡐ} + \label{tab:summary} +\end{table} + +ν‘œμ—μ„œ λ°‘λ©΄ 넓이가 μ„Έ 경우 λͺ¨λ‘ $\frac{k}{3}$둜 동일함을 ν•œλˆˆμ— 확인할 수 μžˆλ‹€. + +\subsection*{ꡬ체적 수치 μ˜ˆμ‹œ} + +νƒκ΅¬μ˜ 좜발점인 3D ν”„λ¦°νŒ… 상황에 μ‹€μ œ 수치λ₯Ό λŒ€μž…ν•˜μ—¬ κ²°κ³Όλ₯Ό ꡬ체적으둜 ν™•μΈν•œλ‹€. +ν•„λΌλ©˜νŠΈ 양이 겉넓이 $k = 600\,\text{cm}^2$에 ν•΄λ‹Ήν•œλ‹€κ³  ν•˜μž. + +\textbf{원기λ‘₯:} +\begin{equation*} + a = \sqrt{\frac{k}{3\pi}} = \sqrt{\frac{600}{3\pi}} \approx 7.98\,\text{cm}, \qquad b = a \approx 7.98\,\text{cm} +\end{equation*} +\begin{equation*} + V_{\max} = \pi a^3 = \pi\!\left(\frac{600}{3\pi}\right)^{\!3/2} \approx 1596\,\text{cm}^3 +\end{equation*} + +\textbf{직윑면체:} +\begin{equation*} + a = b = \sqrt{\frac{k}{3}} = \sqrt{200} \approx 14.14\,\text{cm}, \qquad c = \frac{a}{2} \approx 7.07\,\text{cm} +\end{equation*} +\begin{equation*} + V_{\max} = a^2 c = \frac{1}{2}\!\left(\frac{k}{3}\right)^{\!3/2} \approx 1414\,\text{cm}^3 +\end{equation*} + +μ •$n$κ°ν˜• κΈ°λ‘₯의 경우 $n = 3, 4, 6$ 및 원기λ‘₯에 λŒ€ν•œ κ²°κ³Όλ₯Ό ν‘œ \ref{tab:numerical}에 μ •λ¦¬ν•œλ‹€. +μ΄λ•Œ λͺ¨λ“  κ²½μš°μ—μ„œ λ°‘λ©΄ 넓이 $A = k/3 = 200\,\text{cm}^2$둜 μΌμ •ν•˜λ‹€. + +\begin{table}[H] + \centering + \begin{tabular}{@{}ccccc@{}} + \toprule + $n$ & λ°‘λ©΄ & $a\,(\text{cm})$ & $b\,(\text{cm})$ & $V_{\max}\,(\text{cm}^3)$ \\ + \midrule + 3 & μ •μ‚Όκ°ν˜• & 21.49 & 6.20 & 1241 \\ + 4 & μ •μ‚¬κ°ν˜• & 14.14 & 7.07 & 1414 \\ + 6 & μ •μœ‘κ°ν˜• & 8.77 & 7.60 & 1520 \\ + $\infty$ & 원 & 7.98 & 7.98 & 1596 \\ + \bottomrule + \end{tabular} + \caption{$k = 600\,\text{cm}^2$일 λ•Œ μ •$n$κ°ν˜• κΈ°λ‘₯의 졜적 치수 및 μ΅œλŒ€ λΆ€ν”Ό} + \label{tab:numerical} +\end{table} + +$n$이 μ¦κ°€ν• μˆ˜λ‘ λΆ€ν”Όκ°€ 단쑰 μ¦κ°€ν•˜μ—¬ 원기λ‘₯μ—μ„œ μ΅œλŒ“κ°’μ— μˆ˜λ ΄ν•¨μ„ 수치둜 확인할 수 μžˆλ‹€. +직윑면체($n = 4$)와 원기λ‘₯을 λΉ„κ΅ν•˜λ©΄ 같은 ν•„λΌλ©˜νŠΈ μ–‘μœΌλ‘œ μ•½ $\frac{1596 - 1414}{1414} \approx 12.9\%$ 더 큰 λΆ€ν”Όλ₯Ό 얻을 수 μžˆλ‹€. +μ΄λŠ” μ†Œκ·œλͺ¨ λΆ€ν’ˆ 정리 μ»¨ν…Œμ΄λ„ˆλ₯Ό 섀계할 λ•Œ λ°‘λ©΄ ν˜•νƒœμ˜ 선택이 μ‹€μ§ˆμ μΈ 차이λ₯Ό λ§Œλ“€μ–΄λƒ„μ„ 보여쀀닀. λ³Έ 연ꡬλ₯Ό 톡해 λ„μΆœλœ κ°€μž₯ μ€‘μš”ν•œ 톡찰은 λ‹€μŒκ³Ό κ°™λ‹€. \begin{itemize} \item μ΅œλŒ€ λΆ€ν”Όκ°€ λ˜λŠ” 경우, λ°‘λ©΄ λ„“μ΄λŠ” 항상 $\frac{k}{3}$이닀. \item μ΅œλŒ€ λΆ€ν”ΌλŠ” μ˜†λ©΄ λ‘˜λ ˆκ°€ μ΅œμ†Œκ°€ λ˜λŠ” λ„ν˜•μ—μ„œ λ°œμƒν•œλ‹€. - \item μ •λ‹€κ°ν˜•μ€ λ³€μ˜ μˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ 원에 μˆ˜λ ΄ν•˜λ©°, - 원기λ‘₯이 졜적 ꡬ쑰가 λœλ‹€. + \item μ •λ‹€κ°ν˜•μ€ λ³€μ˜ μˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ 원에 μˆ˜λ ΄ν•˜λ©°, 원기λ‘₯이 졜적 ꡬ쑰가 λœλ‹€. \end{itemize} -이 κ²°κ³ΌλŠ” κΈ°ν•˜ν•™μ  λŒ€μΉ­μ„±μ΄ -μ΅œμ ν™”λ₯Ό μœ λ„ν•œλ‹€λŠ” 일반적인 원리와도 μ—°κ²°λœλ‹€. -λŒ€μΉ­μ„±μ΄ λ†’μ•„μ§ˆμˆ˜λ‘ λ™μΌν•œ κ²‰λ„“μ΄λ‘œ -더 큰 λ‚΄λΆ€ 곡간을 확보할 수 μžˆλ‹€. +이 κ²°κ³ΌλŠ” κΈ°ν•˜ν•™μ  λŒ€μΉ­μ„±μ΄ μ΅œμ ν™”λ₯Ό μœ λ„ν•œλ‹€λŠ” 일반적인 원리와도 μ—°κ²°λœλ‹€. +λŒ€μΉ­μ„±μ΄ λ†’μ•„μ§ˆμˆ˜λ‘ λ™μΌν•œ κ²‰λ„“μ΄λ‘œ 더 큰 λ‚΄λΆ€ 곡간을 확보할 수 μžˆλ‹€. -λ‹€λ§Œ λ³Έ μ—°κ΅¬μ—μ„œλŠ” λ²½ λ‘κ»˜λ₯Ό λ¬΄μ‹œν•˜κ³  -겉넓이와 ν•„λΌλ©˜νŠΈ μ‚¬μš©λŸ‰μ΄ μ™„μ „νžˆ λΉ„λ‘€ν•œλ‹€κ³  κ°€μ •ν•˜μ˜€λ‹€. -μ‹€μ œ 3D ν”„λ¦°νŒ…μ—μ„œλŠ” 재료의 밀도, -적측 ꡬ쑰, λ‚΄λΆ€ 채움λ₯ (infill) λ“±μ˜ μš”μ†Œκ°€ 영ν–₯을 쀄 수 μžˆλ‹€. -ν–₯ν›„ μ—°κ΅¬μ—μ„œλŠ” μ΄λŸ¬ν•œ ν˜„μ‹€μ  μš”μ†Œλ₯Ό ν¬ν•¨ν•˜μ—¬ -보닀 μ‹€μ œμ μΈ λͺ¨λΈλ‘œ ν™•μž₯ν•  수 μžˆμ„ 것이닀. +λ‹€λ§Œ λ³Έ μ—°κ΅¬μ—λŠ” λͺ‡ κ°€μ§€ ν•œκ³„μ™€ ν™•μž₯ κ°€λŠ₯성이 μžˆλ‹€. +첫째, λ²½ λ‘κ»˜λ₯Ό λ¬΄μ‹œν•˜μ—¬ 겉넓이와 ν•„λΌλ©˜νŠΈ μ‚¬μš©λŸ‰μ΄ μ™„μ „νžˆ λΉ„λ‘€ν•œλ‹€κ³  κ°€μ •ν•˜μ˜€λ‹€. μ‹€μ œ 3D ν”„λ¦°νŒ…μ—μ„œλŠ” μΌμ •ν•œ λ²½ λ‘κ»˜κ°€ μœ μ§€λ˜λ―€λ‘œ, λ„ν˜•μ˜ λͺ¨μ„œλ¦¬λ‚˜ 꼭짓점 처리 방식에 따라 ν•„λΌλ©˜νŠΈ μ‚¬μš©λŸ‰ 계산이 λ‹¬λΌμ§ˆ 수 μžˆλ‹€. +λ‘˜μ§Έ, λ³Έ μ—°κ΅¬λŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯에 ν•œμ •λ˜μ—ˆλ‹€. μœ—λ©΄μ΄ μžˆλŠ” λ‹«νžŒ 원기λ‘₯의 경우 겉넓이 곡식이 $S = 2\pi a^2 + 2\pi ab$둜 λ°”λ€Œλ©°, λ™μΌν•œ λ°©λ²•μœΌλ‘œ μ΅œμ ν™”ν•˜λ©΄ +\[ + V'(a) = \frac{k}{2} - 3\pi a^2 = 0 \quad \Longrightarrow \quad a = \sqrt{\frac{k}{6\pi}}, \quad b = 2a +\] +κ°€ λ˜μ–΄ 높이가 지름과 같은 ν˜•νƒœκ°€ μ΅œμ μ΄λ‹€. μ΄λŠ” μ—΄λ¦° 원기λ‘₯의 졜적 λΉ„μœ¨ $a = b$와 λ‹€λ₯΄λ©°, μœ—λ©΄μ˜ 쑴재 μ—¬λΆ€κ°€ 졜적 ꡬ쑰에 직접적인 영ν–₯을 μ€Œμ„ 보여쀀닀. +μ…‹μ§Έ, μ‹€μ œ 3D ν”„λ¦°νŒ…μ—μ„œλŠ” 재료의 밀도, 적측 ꡬ쑰, λ‚΄λΆ€ 채움λ₯ (infill) λ“±μ˜ μš”μ†Œκ°€ 영ν–₯을 쀄 수 있으며, ν–₯ν›„ μ—°κ΅¬μ—μ„œλŠ” μ΄λŸ¬ν•œ ν˜„μ‹€μ  μš”μ†Œλ₯Ό ν¬ν•¨ν•˜μ—¬ 보닀 μ •λ°€ν•œ λͺ¨λΈλ‘œ ν™•μž₯ν•  수 μžˆμ„ 것이닀. -결둠적으둜, -κ³ μ •λœ 겉넓이λ₯Ό κ°€μ§€λŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯ 쀑 -μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ€ 원기λ‘₯이며, -μ΄λŠ” λŒ€μΉ­μ„±κ³Ό λ‘˜λ ˆ μ΅œμ†Œν™”μ˜ 원리에 μ˜ν•΄ μ„€λͺ…λœλ‹€. +결둠적으둜, κ³ μ •λœ 겉넓이λ₯Ό κ°€μ§€λŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯ 쀑 μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ€ 원기λ‘₯이며, μ΄λŠ” λŒ€μΉ­μ„±κ³Ό λ‘˜λ ˆ μ΅œμ†Œν™”μ˜ 원리에 μ˜ν•΄ μ„€λͺ…λœλ‹€. + +\printbibliography \end{document} \ No newline at end of file diff --git a/latex/main.tex.backup b/latex/main.tex.backup new file mode 100644 index 0000000..526a66c --- /dev/null +++ b/latex/main.tex.backup @@ -0,0 +1,821 @@ +\documentclass[11pt]{article} +\usepackage{setspace} +\usepackage{kotex} +\usepackage[toc, page]{appendix} +\usepackage{hyperref} +\hypersetup{ + pdfborder={0 0 0} +} +\usepackage{float} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{siunitx} +\usepackage{graphicx} +\graphicspath{{images/}} +\usepackage[style=apa,backend=biber]{biblatex} +\addbibresource{references.bib} + +\usepackage{pgfplots} +\pgfplotsset{compat=1.18} + +\setlength{\parindent}{0pt} +\setlength{\parskip}{0.6em} +\doublespacing + +\begin{document} + +\begin{titlepage} + \centering + \vspace*{\fill} + {\Large κ³ μ •λœ 겉넓이λ₯Ό κ°–λŠ” κΈ°λ‘₯의 μ΅œλŒ€ 뢀피에 λŒ€ν•œ 탐ꡬ} + + \vspace{2em} + + {\large 29νŽ˜μ΄μ§€} + \vspace*{\fill} +\end{titlepage} + +\setcounter{page}{1} + +\newpage + +\tableofcontents + +\newpage + +\section{μ„œλ‘ } + +λ³Έ μ—°κ΅¬λŠ” 3D ν”„λ¦°ν„°λ‘œ μ •ν•΄μ§„ ν•„λΌλ©˜νŠΈ μ–‘μœΌλ‘œ μ΅œλŒ€ν•œμ˜ λΆ€ν”Όλ₯Ό κ°€μ§„ μ»¨ν…Œμ΄λ„ˆλ₯Ό κ΅¬ν•˜λŠ” 것에 λŒ€ν•΄ νƒκ΅¬ν•œλ‹€. +λ³Έ μ €μžλŠ” ν‰μ†Œ λͺ¨λΈλ§μ΄λ‚˜ 3D ν”„λ¦°νŒ…μ— 관심이 λ§Žμ•„ ν•„μš”ν•œ μ œν’ˆμ΄ μžˆμ„λ•Œ 직접 λ””μžμΈν•˜μ—¬ 좜λ ₯ν•˜λŠ” 것을 μ’‹μ•„ν•œλ‹€. +μ΅œκ·Όμ—λŠ” λ‘œλ΄‡ κ°œλ°œμ„ μœ„ν•΄ λŒ€λŸ‰μ˜ λ‚˜μ‚¬μ™€ λ„ˆνŠΈλ₯Ό μ£Όλ¬Έν–ˆλ‹€. +λ‹€μ–‘ν•œ μ’…λ₯˜μ˜ λΆ€ν’ˆλ“€μ΄ λΉ„λ‹λ΄‰νˆ¬μ— 담겨 μ™”λ‹€. +이듀을 λΆ„λ₯˜ν•˜μ—¬ μ •λ¦¬ν•˜μ—¬ ν•„μš”ν• λ•Œμ— μ‚¬μš©ν•˜κΈ° μœ„ν•΄ λΆ€ν’ˆ μ’…λ₯˜λ³„λ‘œ μ»¨ν…Œμ΄λ„ˆλ₯Ό λ§Œλ“€κΈ°λ‘œ ν–ˆλ‹€. +μ»¨ν…Œμ΄λ„ˆλ₯Ό λ””μžμΈν•˜λŠ” 것은 맀우 쉽닀. +κ·ΈλŸ¬λ‚˜ 적은 ν•„λΌλ©˜νŠΈλ‘œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§„ μ»¨ν…Œμ΄λ„ˆλ₯Ό λ§Œλ“€ 수 μžˆλŠ” λ°©μ•ˆμ— λŒ€ν•΄μ„œλŠ” 생각해보지 μ•Šμ•˜λ‹€. +3D ν”„λ¦°νŒ…μ„ μžμ£Όν•˜κ³  λͺ¨λΈλ§μ΄λ‚˜ 좜λ ₯에 λ¬Έμ œκ°€ μžˆλŠ” 경우 λ‹€μ‹œ 좜λ ₯ν•΄μ•Όν•˜κΈ° λ•Œλ¬Έμ— ν‰μ†Œ ν•„λΌλ©˜νŠΈλ₯Ό λ‚­λΉ„ν•˜λŠ” κ²½μš°κ°€ λ§Žμ•˜λ‹€. +μ΄λ•Œλ¬Έμ— ν•„λΌλ©˜νŠΈλ₯Ό 자주 κ΅¬λ§€ν•΄μ•Όν•˜κΈ°λ„ ν–ˆμœΌλ©°, ν™˜κ²½μ—λ„ μ’‹μ§€ μ•Šμ•˜λ‹€. +λ”°λΌμ„œ μ •ν•΄μ§„ ν•„λΌλ©˜νŠΈ μ–‘μœΌλ‘œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§„ μ»¨ν…Œμ΄λ„ˆλ₯Ό κ΅¬ν•˜κ²Œ λ˜μ—ˆλ‹€. + +\section{연ꡬ 섀계} + +λ³Έ μ €μžκ°€ μ›ν•˜λŠ” μ»¨ν…Œμ΄λ„ˆλŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯이닀. + +ν•΄λ‹Ή μ»¨ν…Œμ΄λ„ˆλ₯Ό λ§Œλ“€λ•Œ μ‚¬μš©ν•  수 μžˆλŠ” ν•„λΌλ©˜νŠΈμ˜ 양을 μ œν•œν•œλ‹€. + +μ΄λ•Œ 속이 빈 μ»¨ν…Œμ΄λ„ˆλ₯Ό 좜λ ₯ν• λ•Œ ν•„μš”ν•œ ν•„λΌλ©˜νŠΈμ˜ 양은 μ»¨ν…Œμ΄λ„ˆμ˜ 겉넓이와 λΉ„λ‘€ν•œλ‹€. +벽의 λ‘κ»˜λŠ” λ¬΄μ‹œν•œλ‹€. + +κ·Έλ ‡λ‹€λ©΄ κ³ μ •λœ ν•„λΌλ©˜νŠΈ μ–‘μœΌλ‘œ μ œμž‘ν•  수 μžˆλŠ” μ΅œλŒ€ λΆ€ν”Όμ˜ μ»¨ν…Œμ΄λ„ˆλ₯Ό κ΅¬ν•˜λŠ” 것은, +κ³ μ •λœ 겉넓이λ₯Ό κ°€μ§„ μ΅œλŒ€ λΆ€ν”Όμ˜ μ»¨ν…Œμ΄λ„ˆλ₯Ό κ΅¬ν•˜λŠ” 것과 κ°™λ‹€. + +μš°μ„  밑면이 μ§μ‚¬κ°ν˜•μΈ κΈ°λ‘₯인 μ§μœ‘λ©΄μ²΄μ— λŒ€ν•΄μ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό 탐ꡬ해본 λ’€, +밑면이 μ •λ‹€κ°ν˜•μΈ κΈ°λ‘₯에 λŒ€ν•΄μ„œλ„ νƒκ΅¬ν•˜μ—¬ μ–΄λ–€ 밑면이 μ–΄λ–€ λ„ν˜•μΌλ•Œ λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ˜λŠ”μ§€ νƒκ΅¬ν•œλ‹€. + +κ²‰λ„“μ΄λŠ” $k$λΌλŠ” μƒμˆ˜λ‘œ κ³ μ •ν•œλ‹€. + +\section{직윑면체의 μ΅œμ ν™”} + +직윑면체(μœ—λ©΄ μ—†μŒ)의 κ°€λ‘œ, μ„Έλ‘œ, 높이λ₯Ό 각각 $a$, $b$, $c$둜 λ‘”λ‹€. +(이듀은 λͺ¨λ‘ κΈΈμ΄μ΄λ―€λ‘œ $a > 0$, $b > 0$, $c > 0$이닀.) +κ·Έλ¦Ό \ref{fig:cuboid}은 ν•΄λ‹Ή 직윑면체λ₯Ό λ‚˜νƒ€λ‚΄λ©°, 길이 λΉ„μœ¨μ€ μ‹€μ œ 해와 λ¬΄κ΄€ν•˜λ‹€. + +\begin{figure}[H] + \centering + \includegraphics[width=0.6\textwidth]{cuboid.png} + \caption{μœ—λ©΄μ΄ μ—†λŠ” 직윑면체} + \label{fig:cuboid} +\end{figure} + +λ³Έ μ—°κ΅¬μ—μ„œλŠ” μ‚¬μš©ν•  수 μžˆλŠ” ν•„λΌλ©˜νŠΈ 양이 κ³ μ •λ˜μ–΄ μžˆμœΌλ―€λ‘œ(λ‘κ»˜ λ¬΄μ‹œ), +μ»¨ν…Œμ΄λ„ˆμ˜ 겉넓이λ₯Ό μƒμˆ˜ $k$둜 κ³ μ •ν•œλ‹€. + +μœ—λ©΄μ΄ μ—†μœΌλ―€λ‘œ κ²‰λ„“μ΄λŠ” λ°”λ‹₯ $ab$와 λ„€ μ˜†λ©΄ $2bc+2ca$의 합이닀: + +\begin{equation} + S(a, b, c) = ab + 2bc + 2ca +\end{equation} + +λ”°λΌμ„œ μ œμ•½μ‘°κ±΄μ€ +\begin{equation} + S(a, b, c) = k + \quad \Longleftrightarrow \quad + ab + 2c(a + b) = k + \label{eq:cuboid_constraint} +\end{equation} +이닀. ν•œνŽΈ λΆ€ν”ΌλŠ” +\begin{equation} + V(a, b, c) = abc +\end{equation} +이닀. + +\subsection{μ œμ•½μ‘°κ±΄μ„ μ΄μš©ν•œ λ³€μˆ˜ 제거} + +λ°‘λ©΄μ˜ 넓이λ₯Ό $p$라 두면 $p = ab$이닀. +식 \ref{eq:cuboid_constraint}λŠ” +\begin{equation} + p + 2c(a + b) = k +\end{equation} +이 되며, 이λ₯Ό $c$에 λŒ€ν•΄ ν’€λ©΄ +\begin{equation} + c = \frac{k - p}{2(a + b)} + \label{eq:cuboid_c} +\end{equation} +이닀. λ”°λΌμ„œ λΆ€ν”ΌλŠ” +\begin{equation} + V(a, b) = ab \cdot c + = p \cdot \frac{k - p}{2(a + b)} + \label{eq:cuboid_volume} +\end{equation} +둜 μ“Έ 수 μžˆλ‹€. + +\subsection{\texorpdfstring{κ·ΉλŒ“κ°’μ—μ„œ $a = b$인 이유}{κ·ΉλŒ“κ°’μ—μ„œ a = b인 이유}} + +이제 μ–΄λ–€ 후보 $(a, b, c)$κ°€ λΆ€ν”Όλ₯Ό μ΅œλŒ€λ‘œ λ§Œλ“œλŠ” μ΅œμ ν•΄λΌκ³  κ°€μ •ν•˜μž. +κ·Έ ν›„λ³΄μ—μ„œμ˜ λ°‘λ©΄ 넓이 $p = ab$λŠ” μ–΄λ–€ νŠΉμ •ν•œ κ°’ $p_0$λ₯Ό κ°–λŠ”λ‹€. +μ΄λ•Œ $p$κ°€ λ™μΌν•œ λ‹€λ₯Έ μ§μ‚¬κ°ν˜•λ“€(즉 $ab = p_0$인 λ‹€λ₯Έ $(a, b)$)κ³Ό 비ꡐ해도, μ΅œμ ν•΄λΌλ©΄ λΆ€ν”Όκ°€ κ°€μž₯ μ»€μ•Όν•œλ‹€. + +식 \ref{eq:cuboid_volume}μ—μ„œ $k$와 $p$κ°€ κ³ μ •λ˜λ©΄, +\begin{equation*} + V(a, b) = p \cdot \frac{k - p}{2(a + b)} +\end{equation*} +μ΄λ―€λ‘œ, $V$λ₯Ό 크게 λ§Œλ“€κΈ° μœ„ν•΄μ„œλŠ” λΆ„λͺ¨μ˜ $(a + b)$κ°€ μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό ν•œλ‹€. +그런데 $ab = p$κ°€ κ³ μ •μΌλ•Œ μ‚°μˆ κΈ°ν•˜ 평균에 μ˜ν•΄ +\begin{equation} + a + b \geq 2 \sqrt{ab} = 2 \sqrt{p} +\end{equation} +κ°€ μ„±λ¦½ν•˜κ³ , λ“±ν˜ΈλŠ” $a = b$일 λ•Œμ—λ§Œ μ„±λ¦½ν•œλ‹€. +λ”°λΌμ„œ $ab = p$인 후보듀 쀑 $(a + b)$κ°€ μ΅œμ†Œκ°€ λ˜λŠ” κ²½μš°λŠ” $a = b$이며, +κ²°κ΅­ μ΅œμ ν•΄λŠ” λ°˜λ“œμ‹œ +\begin{equation} + a = b +\end{equation} +λ₯Ό λ§Œμ‘±ν•΄μ•Όν•œλ‹€. + +\subsection{\texorpdfstring{μ΅œλŒ€ 뢀피와 $a$, $b$, $c$의 λΉ„μœ¨}{μ΅œλŒ€ 뢀피와 a, b, c의 λΉ„μœ¨}} + +이제 $b$λ₯Ό $a$둜 두면, $p = ab = a^2$이고, 식 \ref{eq:cuboid_volume}을 μ •λ¦¬ν•˜λ©΄ λΆ€ν”ΌλŠ” +\begin{equation} + V(a) = p \cdot \frac{k - p}{2(a + b)} + = \frac{ka - a^3}{4} +\end{equation} +κ°€ λœλ‹€. + +μ΅œλŒ“κ°’μ΄ λ˜λŠ” μ§€μ μ—μ„œ λ„ν•¨μˆ˜μ˜ 값이 $0$이어야 ν•˜κΈ°μ—, $V'(a) = 0$을 ν’€λ©΄, +\begin{equation} + V'(a) = \frac{1}{4} \left( k - 3 a^2 \right) = 0 + \quad \Longrightarrow \quad + a = \sqrt{\frac{k}{3}} +\end{equation} +이 λœλ‹€. + +λ˜ν•œ $V''(a) = -\frac{6}{4}a$μ—μ„œ $a>0$이기 λ•Œλ¬Έμ— $V''(a) < 0$이 항상 μ„±λ¦½ν•œλ‹€. +λ”°λΌμ„œ κ·Έλ¦Ό \ref{fig:f_and_derivative}μ—μ„œλ„ λ³Ό 수 μžˆλ“―μ΄ $a < \sqrt{\frac{k}{3}}$μΌλ•ŒλŠ” $f'(a) > 0$μž„μœΌλ‘œ +$f(a)$κ°€ μ¦κ°€ν•˜λ©°, $a > \sqrt{\frac{k}{3}}$μΌλ•ŒλŠ” $f'(a) < 0$μž„μœΌλ‘œ $f(a)$κ°€ κ°μ†Œν•˜κΈ° λ•Œλ¬Έμ—, +$a = \sqrt{\frac{k}{3}}$인 지점이 $f(a)$의 μ΅œλŒ“κ°’μž„μ„ μ•Œ 수 μžˆλ‹€. + +\begin{figure}[H] + \centering + + \begin{tikzpicture} + \begin{axis}[ + variable=a, + domain=0:3, + samples=100, + axis lines=middle, + xlabel=$a$, ylabel=, + xtick=\empty, ytick=\empty, + ymin=-2, ymax=2, + legend style={draw=none, fill=none} + ] + \pgfmathsetmacro{\k}{5} + \pgfmathsetmacro{\aopt}{sqrt(\k / 3)} + + \addplot[blue] {(\k*a - a^3) / 4}; + \addlegendentry{$f(a)$} + + \addplot[red] {1/4 * (\k - 3*a^2)}; + \addlegendentry{$f'(a)$} + + \addplot[green] {-6/4*a}; + \addlegendentry{$f''(a)$} + + \addplot[dashed, black] coordinates {(\aopt,-2) (\aopt,2)}; + \addlegendentry{$a=\sqrt{\frac{k}{3}}$} + \end{axis} + \end{tikzpicture} + + \caption{ν•¨μˆ˜ $f(a)$와 λ„ν•¨μˆ˜ 및 μ΄κ³„λ„ν•¨μˆ˜ κ·Έλž˜ν”„} + \label{fig:f_and_derivative} +\end{figure} + +λ„μΆœν•œ $a$값을 식 \ref{eq:cuboid_c}에 λŒ€μž…ν•˜λ©΄, +\begin{equation} + c = \frac{k - p}{2(a + b)} + = \frac{k - \left( \sqrt{\frac{k}{3}} \right) ^ 2}{4 \sqrt{\frac{k}{3}}} + = \frac{1}{2} \sqrt{\frac{k}{3}} + = \frac{1}{2} a +\end{equation} +이 λœλ‹€. λ”°λΌμ„œ 졜적 λΉ„μœ¨μ€ +\begin{equation} + \boxed{a = b = 2c} +\end{equation} +이닀. + +λ§ˆμ§€λ§‰μœΌλ‘œ μ΅œλŒ€ λΆ€ν”ΌλŠ” +\begin{equation} + V_{\max} = abc = a^2 c + =\left( \frac{k}{3} \right) \left( \frac{1}{2} \sqrt{\frac{k}{3}} \right) + =\frac{1}{2} \left( \frac{k}{3} \right) ^ {3/2}. +\end{equation} +이 λœλ‹€. + +\subsection{해석 및 λ‹€μŒ 단계} + +직윑면체의 λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ λ•Œ $a = b$μ΄λ―€λ‘œ 밑면은 μ •μ‚¬κ°ν˜•μ΄ λœλ‹€. +이λ₯Ό λ°”νƒ•μœΌλ‘œ 밑면이 μ •$n$κ°ν˜•μΈ κΈ°λ‘₯으둜 ν™•μž₯ν•˜μ—¬, +λ³€μ˜ 수 $n$이 μ¦κ°€ν• μˆ˜λ‘ μ΅œλŒ€ λΆ€ν”Όκ°€ μ–΄λ–»κ²Œ λ³€ν•˜λŠ”μ§€ λΆ„μ„ν•œλ‹€. + +\section{μ •λ‹€κ°ν˜• κΈ°λ‘₯의 μ΅œμ ν™”} + +μ •λ‹€κ°ν˜• κΈ°λ‘₯(μœ—λ©΄ μ—†μŒ) λΆ„μ„μ—μ„œλŠ” λ°‘λ©΄μ˜ ν•œ λ³€μ˜ 길이λ₯Ό $a$, κΈ°λ‘₯의 높이λ₯Ό $b$라 ν•˜κ³ , λ°‘λ©΄μ˜ λ³€μ˜ 개수λ₯Ό $n$이라 ν•œλ‹€. +μ΄λ•Œ, $a$와 $b$λŠ” κΈΈμ΄μ΄λ―€λ‘œ $a>0$, $b>0$이며, 밑면은 μ΅œμ†Œ μ‚Όκ°ν˜•μ΄λ―€λ‘œ $n \geq 3$이닀. +κ·Έλ¦Ό \ref{fig:regular_polygon}은 ν•΄λ‹Ή κΈ°λ‘₯을 λ‚˜νƒ€λ‚΄λ©°, 길이 λΉ„μœ¨μ€ μ‹€μ œ 해와 λ¬΄κ΄€ν•˜λ‹€. + +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{regular_polygon.png} + \caption{μœ—λ©΄μ΄ μ—†λŠ” μ •λ‹€κ°ν˜• κΈ°λ‘₯} + \label{fig:regular_polygon} +\end{figure} + +μ •λ‹€κ°ν˜• κΈ°λ‘₯ λΆ„μ„μ—μ„œλ„ 벽의 λ‘κ»˜λŠ” λ¬΄μ‹œν•˜λ©°, κ²‰λ„“μ΄λŠ” μƒμˆ˜ $k$둜 κ³ μ •ν•œλ‹€. + +겉넓이와 λΆ€ν”Όλ₯Ό κ³„μ‚°ν•˜κΈ°μ— μ•žμ„œ λ°‘λ³€μ˜ 넓이λ₯Ό $A$라 ν•˜μž. +κ·Έλ¦Ό \ref{fig:regular_polygon_top}κ³Ό 같이, λ³€μ˜ κ°œμˆ˜κ°€ $n$개인 μ •λ‹€κ°ν˜•μ€ μ€‘μ‹¬μ—μ„œ 각 κΌ­μ§“μ μœΌλ‘œ 선을 그으면 +$n$개의 μ΄λ“±λ³€μ‚Όκ°ν˜•μœΌλ‘œ λΆ„ν• λœλ‹€. + +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{regular_polygon_top.png} + \caption{μ •λ‹€κ°ν˜• κΈ°λ‘₯ λ°‘λ©΄} + \label{fig:regular_polygon_top} +\end{figure} + +각 μ΄λ“±λ³€μ‚Όκ°ν˜•μ˜ 꼭짓각은 $\frac{2 \pi}{n}$이고, λ°‘λ³€μ˜ κΈΈμ΄λŠ” $a$이닀. +μ΄λ•Œ κΌ­μ§“μ μ—μ„œ 밑변에 λ‚΄λ¦° μˆ˜μ„ μ€ μ΄λ“±λ³€μ‚Όκ°ν˜•μ„ μ„œλ‘œ 합동인 두 μ§κ°μ‚Όκ°ν˜•μœΌλ‘œ λ‚˜λˆˆλ‹€. +이 μ§κ°μ‚Όκ°ν˜•μ—μ„œ μ‚Όκ°ν•¨μˆ˜λ₯Ό μ΄μš©ν•˜λ©΄ λ†’μ΄λŠ” +\begin{equation*} + \frac{a}{2 \tan{(\pi / n)}} +\end{equation*} +μž„μ„ ꡬ할 수 μžˆλ‹€. + +λ”°λΌμ„œ 각 μ΄λ“±λ³€μ‚Όκ°ν˜•μ˜ λ„“μ΄λŠ” +\begin{equation*} + \frac{1}{2} \cdot a \cdot \frac{a}{2 \tan{(\pi / n)}} + = \frac{a^2}{4 \tan{(\pi / n)}} +\end{equation*} +이닀. + +μ΄λŸ¬ν•œ μ‚Όκ°ν˜•μ΄ $n$개 μ‘΄μž¬ν•˜λ―€λ‘œ, λ°‘λ©΄μ˜ 전체 넓이 $A$λŠ” +\begin{equation} + A = \frac{na^2}{4 \tan{(\pi / n)}} +\end{equation} +이 λœλ‹€. + +μ΄λ•Œ κ³„μ‚°μ˜ 편의λ₯Ό μœ„ν•΄ +\begin{equation} + t = \frac{n}{4 \tan(\pi / n)} +\end{equation} +둜 두면, μœ„ 식은 λ‹€μŒκ³Ό 같이 κ°„λ‹¨νžˆ ν‘œν˜„ν•  수 μžˆλ‹€. +\begin{equation} + A = ta^2 + \label{eq:regular_polygon_A} +\end{equation} + + +λ‹€μŒμœΌλ‘œ λ°‘λ©΄μ˜ λ‘˜λ ˆλ₯Ό $P$라고 ν•˜μž. $P$λŠ” +\begin{equation} + P = na + \label{eq:regular_polygon_P} +\end{equation} +둜 λ‚˜νƒ€λ‚Ό 수 μžˆλ‹€. + +κΈ°λ‘₯의 κ²‰λ„“μ΄λŠ” μœ—λ©΄μ΄ μ—†κΈ° λ•Œλ¬Έμ— λ°‘λ©΄μ˜ 넓이 $A$와 μ˜†λ©΄μ˜ 넓이$Pb$의 합이닀. +\begin{equation} + S(a, b, n) = A + Pb +\end{equation} +λ”°λΌμ„œ μ œμ•½μ‘°κ±΄μ€ +\begin{equation} + S(a, b, n) = k + \quad \Longleftrightarrow \quad + A + Pb = k + \label{eq:regular_polygon_constraint} +\end{equation} +이닀. ν•œνŽΈ λΆ€ν”ΌλŠ” +\begin{equation} + V(a, b, n) = Ab + \label{eq:regular_polygon_v0} +\end{equation} +이닀. + +\subsection{μ œμ•½μ‘°κ±΄μ„ μ΄μš©ν•œ λ³€μˆ˜ 제거} + +식 \ref{eq:regular_polygon_constraint}을 $b$에 λŒ€ν•΄μ„œ μ •λ¦¬ν•˜λ©΄ +\begin{equation} + b = \frac{k - A}{P} + \label{eq:regular_polygon_b} +\end{equation} +κ°€ λœλ‹€. +이λ₯Ό 식 \ref{eq:regular_polygon_v0}에 λŒ€μž…ν•˜λ©΄ λΆ€ν”ΌλŠ” +\begin{equation} + V(a, n) = Ab = A \cdot \frac{k - A}{P} = \frac{t(ka - ta^3)}{n} + \label{eq:regular_polygon_v1} +\end{equation} +이 λœλ‹€. + +λ³Έ λ¬Έμ œλŠ” μ›λž˜ λ°‘λ©΄μ˜ λ³€ 길이 $a$, λ³€μ˜ 수 $n$, 높이 $b$의 μ„Έ λ³€μˆ˜λ‘œ κΈ°μˆ λœλ‹€. +κ·ΈλŸ¬λ‚˜ μ œμ•½μ‘°κ±΄ $A+Pb=k$λ₯Ό $b$에 λŒ€ν•΄ ν’€μ–΄ $b=\frac{k-A}{P}$둜 λ‚˜νƒ€λ‚Έ λ’€ 이λ₯Ό 뢀피식에 λŒ€μž…ν–ˆμœΌλ―€λ‘œ, +$b$λŠ” 더 이상 λ…λ¦½μ μœΌλ‘œ μ„ νƒλ˜λŠ” λ³€μˆ˜κ°€ μ•„λ‹ˆλ‹€. +결과적으둜 λΆ€ν”ΌλŠ” 식 \ref{eq:regular_polygon_v1}와 같이 $(a,n)$에 μ˜ν•΄ κ²°μ •λœλ‹€. + +λ”°λΌμ„œ μ œμ•½μ‘°κ±΄μ„ λ§Œμ‘±ν•˜λ©΄μ„œ λΆ€ν”Όλ₯Ό μ΅œλŒ€ν™”ν•˜λŠ” $(a, n)$ μŒμ„ μ°Ύμ•„μ•Ό ν•œλ‹€. +κ·ΈλŸ¬λ‚˜ μš°μ„  κ³ μ •λœ $n$μ—μ„œ μ΅œλŒ€ λΆ€ν”Όκ°€ λ˜λŠ” $a$λ₯Ό μ°ΎλŠ” 것뢀터 μ§„ν–‰ν•œλ‹€. +즉, $n$을 $3$μ΄μƒμ˜ μž„μ˜μ˜ μ •μˆ˜ $n_0$라고 ν•  λ•Œ, +λ¨Όμ € $n=n_0$λ₯Ό μƒμˆ˜λ‘œ 두고 $a$에 λŒ€ν•΄ λ‹€μŒμ˜ 1λ³€μˆ˜ ν•¨μˆ˜ +\[ + V_{n_0}(a):=V(a,n_0) +\] +λ₯Ό μ΅œλŒ€ν™”ν•œλ‹€. + +\subsection{\texorpdfstring{$n = n_0$μ—μ„œμ˜ $a$에 λŒ€ν•œ 졜적 쑰건}{n = n0μ—μ„œμ˜ a에 λŒ€ν•œ 졜적 쑰건}} + +$n=n_0$λ₯Ό κ³ μ •ν•˜λ©΄ $t$와 $n_0$λŠ” μƒμˆ˜μ΄λ©°, +\begin{equation} + V_{n_0}(a)=\frac{t}{n_0} \left( ka - ta^3 \right) +\end{equation} +λŠ” $a$에 λŒ€ν•œ λ―ΈλΆ„ κ°€λŠ₯ν•œ ν•¨μˆ˜κ°€ λœλ‹€. +$V_{n_0}(a)$κ°€ μ΅œλŒ€κ°€ λ˜λŠ” μ§€μ μ—μ„œλŠ” ν•΄λ‹Ή ν•¨μˆ˜μ˜ λ„ν•¨μˆ˜μ˜ 값이 $0$이 λ˜μ–΄μ•Ό ν•˜κΈ° λ•Œλ¬Έμ— +\begin{equation} + V_{n_0}'(a)=0 +\end{equation} +이 성립해야 ν•œλ‹€. +이λ₯Ό κ³„μ‚°ν•˜λ©΄, +\begin{equation} + V_{n_0}'(a)=\frac{t}{n_0} \left( k - 3ta^2 \right) +\end{equation} +μ΄λ―€λ‘œ, +\begin{equation} + \frac{t}{n_0} \left( k - 3ta^2 \right) = 0 + \quad \Longrightarrow \quad + k - 3ta^2 = 0 + \quad \Longrightarrow \quad + a = \sqrt{\frac{k}{3t}}. + \label{eq:regular_polygon_a_opt} +\end{equation} +λ₯Ό λ„μΆœν•  수 μžˆλ‹€. + +λ˜ν•œ \ref{eq:regular_polygon_a_opt}μ—μ„œ κ΅¬ν•œ $a$κ°€ $V_{n_0}(a)$κ°€ μ΅œλŒ€κ°€ λ˜λŠ” 지점이 되기 μœ„ν•΄μ„œλŠ” +ν•΄λ‹Ή ν•¨μˆ˜μ˜ μ΄κ³„λ„ν•¨μˆ˜κ°€ ν•΄λ‹Ή μ§€μ μ—μ„œ μŒμˆ˜μ—¬μ•Ό ν•œλ‹€. +\[ + V_{n_0}''(a)=\frac{t}{n_0}(-6ta) < 0 \quad (a>0) +\] +μ΄λ―€λ‘œ 식 \ref{eq:regular_polygon_a_opt}μ—μ„œ 얻은 μž„κ³„μ μ€ μ‹€μ œλ‘œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό μ€€λ‹€λŠ” 것을 μ•Œ 수 μžˆλ‹€. +λ”°λΌμ„œ μ •$n_0$각기λ‘₯μ—μ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό λ§Œλ“€κΈ° μœ„ν•œ λ³€μ˜ κΈΈμ΄λŠ” +\begin{equation} + a = \sqrt{\frac{k}{3t}} + \label{eq:regular_polygon_a} +\end{equation} +둜 κ²°μ •λœλ‹€. + +\subsection{졜적 μ‘°κ±΄μ—μ„œμ˜ A와 b} + +식 \ref{eq:regular_polygon_A}에 κ³„μ‚°ν•œ $a$값을 λŒ€μž…ν•˜λ©΄, +\begin{equation} + A = ta^2 = \frac{k}{3} +\end{equation} +이 λœλ‹€. +$A$κ°€ $k$둜만 λ‚˜νƒ€λ‚΄μ–΄μ§„λ‹€λŠ” 것은 $n$의 값에 상관없이 μ •λ‹€κ°ν˜• κΈ°λ‘₯의 λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ λ•Œ, λ°‘λ©΄μ˜ λ„“μ΄λŠ” μΌμ •ν•˜λ‹€λŠ” 것을 μ˜λ―Έν•œλ‹€. +μ΄λŠ” μœ„μ—μ„œ 직윑면체의 λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ˜λŠ” κ²½μš°μ—μ„œλ„ 두 λ°‘λ©΄μ˜ 길이가 λͺ¨λ‘ $\sqrt{\frac{k}{3}}$κ°€ λ˜μ–΄, +λ°‘λ©΄μ˜ 넓이가 $\frac{k}{3}$이 λ˜μ—ˆκΈ° λ•Œλ¬Έμ— 이 κ²°κ³Όλ₯Ό λ§Œμ‘±ν•œλ‹€κ³  ν•  수 μžˆλ‹€. + +κΈ°λ‘₯의 λ†’μ΄λŠ” 식 \ref{eq:regular_polygon_b}에 κ΅¬ν•œ $A$값을 λŒ€μž…ν•˜λ©΄, +\begin{equation} + b = \frac{k - A}{P} = \frac{2k}{3P} +\end{equation} +κ°€ λœλ‹€. + +\subsection{졜적 μ‘°κ±΄μ—μ„œμ˜ λΆ€ν”Ό} + +κΈ°λ‘₯의 λΆ€ν”ΌλŠ” $A$와 $b$의 곱이기에, μ΅œλŒ€ λΆ€ν”ΌλŠ” +\begin{equation} + V_{max} = Ab = \frac{2k^2}{9P} +\end{equation} +κ°€ λœλ‹€. + +μ—¬κΈ°μ„œ $k$λŠ” κ³ μ •λœ μ–‘μˆ˜μ΄λ©°, $P > 0$이닀. +λ”°λΌμ„œ $V_{\max}$λŠ” $P$에 λŒ€ν•˜μ—¬ +\begin{equation} + V_{\max}(P) = \frac{C}{P} + \quad (C = \frac{2k^2}{9} > 0) +\end{equation} +와 같은 ν˜•νƒœλ₯Ό κ°€μ§„λ‹€. + +ν•¨μˆ˜ $V_{\max}(P)$λŠ” $P > 0$μ—μ„œ 단쑰 κ°μ†Œ ν•¨μˆ˜μ΄λ―€λ‘œ, +$P$κ°€ μž‘μ•„μ§ˆμˆ˜λ‘ $V_{\max}$λŠ” μ¦κ°€ν•œλ‹€. +λ”°λΌμ„œ μ΅œλŒ€ λΆ€ν”Όλ₯Ό μ–»κΈ° μœ„ν•΄μ„œλŠ” $P$κ°€ μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό ν•œλ‹€. + +$P$λŠ” 식 \ref{eq:regular_polygon_P}에 $a$값을 λŒ€μž…ν•˜λ©΄ +\begin{equation} + P(n) = na = n \sqrt{\frac{k}{3t}} = 2 \sqrt{\frac{kn \tan{(\pi / n)}}{3}} +\end{equation} +이 λœλ‹€. +μ΄λ•Œ 식 \ref{eq:regular_polygon_a}μ—μ„œ $a$λ₯Ό $n$으둜 λ‚˜νƒ€λ‚΄μ—ˆκΈ° λ•Œλ¬Έμ—, $P$λŠ” $n$에 λŒ€ν•œ μΌλ³€μˆ˜ ν•¨μˆ˜κ°€ λœλ‹€. + +$P(n)$이 μ΅œμ†Œκ°€ 되기 μœ„ν•΄μ„œλŠ” $n \tan{(\pi / n)}$이 μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό ν•œλ‹€. +\begin{equation} + Q(n) = n \tan{(\pi / n)} \quad (n \geq 3) +\end{equation} +라고 ν•˜λ©΄, $Q(n)$이 μ΅œμ†Œκ°€ λ λ•Œμ˜ $n$이 κΈ°λ‘₯의 λΆ€ν”Όλ₯Ό μ΅œλŒ€ν™” ν•œλ‹€. +μ—¬κΈ°μ„œ $n$은 μ •$n$κ°ν˜•μ˜ λ³€μ˜ 개수λ₯Ό μ˜λ―Έν•˜λ―€λ‘œ 본래 μ •μˆ˜ $n \geq 3$에 ν•œμ •λœλ‹€. +κ·ΈλŸ¬λ‚˜ ν•¨μˆ˜ $Q(n)$의 증가 및 κ°μ†Œλ₯Ό λΆ„μ„ν•˜κΈ° μœ„ν•˜μ—¬, $n$을 μ‹€μˆ˜ $n > 2$둜 ν™•μž₯ν•˜μ—¬ 미뢄을 μˆ˜ν–‰ν•œλ‹€. +μ΄λŸ¬ν•œ 연속적 ν™•μž₯은 ν•¨μˆ˜μ˜ κ²½ν–₯성을 νŒŒμ•…ν•˜κΈ° μœ„ν•œ 것이며, ν‘Έν›„ 해석은 λ‹€μ‹œ μ •μˆ˜ $n$에 λŒ€ν•΄ 이루어진닀. + +$Q(n)$이 μ΅œμ†Œκ°€ 될 수 μžˆλŠ” 후보 지점은 κ·Ήκ°’κ³Ό 경계값이 μžˆλ‹€. + +λ¨Όμ € 극값이 μ΅œμ†Ÿκ°’μ΄ 되기 μœ„ν•΄μ„œλŠ” ν•΄λ‹Ή μ§€μ μ—μ„œ λ„ν•¨μˆ˜κ°€ $0$이어야 ν•˜λ©°, μ΄κ³„λ„ν•¨μˆ˜κ°€ μ–‘μˆ˜μ—¬μ•Ό ν•œλ‹€. +$Q(n)$의 λ„ν•¨μˆ˜λŠ” 곱의 미뢄법에 따라 +\begin{equation} + Q'(n) = \tan\left(\frac{\pi}{n}\right) - \frac{\pi}{n}\sec^2\left(\frac{\pi}{n}\right) +\end{equation} +κ°€ λœλ‹€. +μ΄λ•Œ 계산을 κ°„λ‹¨νžˆ ν•˜κΈ° μœ„ν•˜μ—¬ +\begin{equation} + \theta=\frac{\pi}{n} +\end{equation} +이라 두면, +\begin{equation} + Q'(n) = \tan{\theta} - \theta \sec^2{\theta} +\end{equation} +둜 ν‘œν˜„ν•  수 μžˆκΈ°μ—, +\begin{equation} + \tan{\theta} - \theta \sec^2{\theta} = 0 + \quad \Longrightarrow \quad + \sin{\theta} \cos{\theta} = \theta +\end{equation} + +배각곡식에 λ”°λ₯΄λ©΄ +\begin{equation} + 2 \sin{\theta} \cos{\theta} = \sin{2 \theta} + \quad \Longrightarrow \quad + \sin{\theta} \cos{\theta} = \frac{\sin{2 \theta}}{2} +\end{equation} +κ°€ μ„±λ¦½ν•œλ‹€. + +λ”°λΌμ„œ 극값이 μ‘΄μž¬ν•˜κΈ° μœ„ν•΄μ„œλŠ” +\begin{equation} + \theta = \frac{\sin 2\theta}{2} +\end{equation} +λ₯Ό λ§Œμ‘±ν•΄μ•Ό ν•œλ‹€. +이λ₯Ό λΆ„μ„ν•˜κΈ° μœ„ν•˜μ—¬ λ‹€μŒκ³Ό 같은 ν•¨μˆ˜λ₯Ό μ •μ˜ν•œλ‹€. +\begin{equation} + f(\theta) = \frac{\sin 2\theta}{2} - \theta + \quad (\theta \leq \frac{\pi}{3}) +\end{equation} +그러면 μœ„ 방정식은 $f(\theta)=0$을 ν‘ΈλŠ” λ¬Έμ œμ™€ κ°™λ‹€. +이 ν•¨μˆ˜μ˜ λ„ν•¨μˆ˜λŠ” +\begin{equation} + f'(\theta) = \cos{2 \theta} - 1 +\end{equation} +이닀. +μ‚Όκ°ν•¨μˆ˜μ˜ μ„±μ§ˆμ— μ˜ν•΄ $\cos{2 \theta} \leq 1$이며, +$\theta > 0$일 λ•ŒλŠ” $\cos 2\theta < 1$μ΄λ―€λ‘œ +\begin{equation} + f'(\theta) < 0 \quad (\theta > 0) +\end{equation} +κ°€ μ„±λ¦½ν•œλ‹€. λ”°λΌμ„œ $f(\theta)$λŠ” $\theta>0$μ—μ„œ 단쑰 κ°μ†Œν•œλ‹€. +λ˜ν•œ +\begin{equation} + f(0) = 0 +\end{equation} +μ΄λ―€λ‘œ, $f(\theta) = 0$을 λ§Œμ‘±ν•˜λŠ” ν•΄λŠ” $\theta = 0$ ν•˜λ‚˜λΏμ΄λ‹€. +κ·ΈλŸ¬λ‚˜ $\theta=\frac{\pi}{n}$의 μ •μ˜μ— μ˜ν•΄ +$\theta=0$이 되렀면 $n \to \infty$이어야 ν•˜λ©°, +μœ ν•œν•œ $n$μ—μ„œλŠ” $\theta>0$이닀. λ”°λΌμ„œ $Q(n)$은 μœ ν•œν•œ $n$μ—μ„œ λ‚΄λΆ€ 극값을 κ°–μ§€ μ•ŠλŠ”λ‹€. + +λ‚΄λΆ€μ—μ„œ 극값이 μ‘΄μž¬ν•˜μ§€ μ•ŠμœΌλ―€λ‘œ, μ΅œμ†Ÿκ°’μ€ κ²½κ³„μ—μ„œλ§Œ λ°œμƒν•  수 μžˆλ‹€. +본래 λ¬Έμ œμ—μ„œ $n$은 μ •$n$κ°ν˜•μ˜ λ³€μ˜ 개수λ₯Ό μ˜λ―Έν•˜λ―€λ‘œ μ •μˆ˜ $n \geq 3$에 ν•œμ •λœλ‹€. +λ”°λΌμ„œ $Q(n)$의 μ •μ˜μ—­μ€ +\begin{equation} + \{n \mid n \geq 3,\; n \in \mathbb{Z}\} +\end{equation} +이닀. + +이 μ •μ˜μ—­μ—μ„œ κ°€μž₯ μž‘μ€ 값은 $n=3$μ΄λ―€λ‘œ, 쒌츑 경계값은 +\begin{equation} + Q(3) \simeq 5.196 +\end{equation} +이닀. + +우츑 경계값을 κ΅¬ν•˜κΈ° μœ„ν•΄μ„œλŠ” κ·Ήν•œμ„ μ‚¬μš©ν•œλ‹€. +이 경우 λ¬΄ν•œλŒ€λ‘œμ˜ κ·Ήν•œκ°’μ„ κ΅¬ν•˜λŠ” 것보닀 $0$으둜의 κ·Ήν•œκ°’μ„ κ΅¬ν•˜λŠ”κ²Œ 더 쉽기 λ•Œλ¬Έμ— +\begin{equation} + u = \frac{1}{n} +\end{equation} +으둜 μ •μ˜ν•˜λ©΄, +\begin{equation} + \lim_{n \to \infty}{Q(n)} = \lim_{u \to 0}{Q(u)} +\end{equation} +이 λœλ‹€. 이λ₯Ό κ³„μ‚°ν•˜λ©΄ +\begin{equation} + \lim_{u \to 0}{Q(u)} = \lim_{u \to 0}{\frac{1}{u} \tan{\pi u}} + = \pi \cdot \lim_{u \to 0}{\frac{\tan{(\pi u)}}{\pi u}} + = \pi \cdot 1 + = \pi +\end{equation} +이 λœλ‹€. +$\lim_{u \to 0}{Q(u)} < Q(3)$이기 λ•Œλ¬Έμ— $Q(u)$의 μ΅œμ†Ÿκ°’μ€ $n$이 λ¬΄ν•œλŒ€μΌλ•Œμž„μ„ μ•Œ 수 μžˆλ‹€. +μ΄λŠ” λ°‘λ©΄μ˜ 변이 더 λ§Žμ„μˆ˜λ‘ κΈ°λ‘₯의 λΆ€ν”Όκ°€ 더 컀짐을 μ˜λ―Έν•œλ‹€. + +μ •λ‹€κ°ν˜•μ—μ„œ λ³€μ˜ λŒ€μˆ˜κ°€ λŠμž„μ—†μ΄ λŠ˜μ–΄λ‚˜λ©΄ 원이 되기 λ•Œλ¬Έμ— +λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ λ˜λŠ” 밑면은 원이 되며, μ΄λŠ” 원기λ‘₯이 λœλ‹€. +λ”°λΌμ„œ μ΄λ²ˆμ—λŠ” μ œμ•½μ‘°κ±΄ν•˜μ—μ„œ κ°€μž₯ λΆ€ν”Όκ°€ 큰 원기λ‘₯에 λŒ€ν•΄μ„œ νƒκ΅¬ν•œλ‹€. + +\section{원기λ‘₯의 μ΅œμ ν™”} + +원기λ‘₯ κΈ°λ‘₯(μœ—λ©΄ μ—†μŒ) λΆ„μ„μ—μ„œλŠ” λ°‘λ©΄ μ›μ˜ λ°˜μ§€λ¦„μ„ $a$, κΈ°λ‘₯의 높이λ₯Ό $b$라 ν•œλ‹€. +μ΄λ•Œ, $a$와 $b$λŠ” κΈΈμ΄μ΄λ―€λ‘œ $a > 0$, $b > 0$이닀. +κ·Έλ¦Ό \ref{fig:cylinder}은 ν•΄λ‹Ή κΈ°λ‘₯을 λ‚˜νƒ€λ‚΄λ©°, 길이 λΉ„μœ¨μ€ μ‹€μ œ 해와 λ¬΄κ΄€ν•˜λ‹€. + +\begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{cylinder.png} + \caption{μœ—λ©΄μ΄ μ—†λŠ” 원기λ‘₯} + \label{fig:cylinder} +\end{figure} + +μœ—λ©΄μ΄ μ—†μœΌλ―€λ‘œ κ²‰λ„“μ΄λŠ” λ°‘λ©΄κ³Ό μ˜†λ©΄μ˜ 합이닀 +\begin{equation} + S(a, b) = \pi a^2 + 2 \pi ab +\end{equation} +κ²‰λ„“μ΄λŠ” k둜 κ³ μ •λ˜μ–΄ 있기 λ•Œλ¬Έμ— μ œμ•½μ‘°κ±΄μ€ +\begin{equation} + S(a, b) = k \quad \Longleftrightarrow \pi a^2 + 2 \pi ab = k + \label{eq:cylinder_constraint} +\end{equation} +이닀. +λΆ€ν”ΌλŠ” +\begin{equation} + V(a, b) = \pi a^2 b + \label{eq:cylinder_V} +\end{equation} +이닀. + +식 \ref{eq:cylinder_constraint}λ₯Ό $b$에 λŒ€ν•΄ μ •λ¦¬ν•˜λ©΄ +\begin{equation} + b = \frac{k - \pi a^2}{2 \pi a} + \label{eq:cylinder_b} +\end{equation} +κ°€ λœλ‹€. +이λ₯Ό μ΄μš©ν•΄ 식 \ref{eq:cylinder_V}에 λŒ€μž…ν•˜λ©΄ λΆ€ν”ΌλŠ” +\begin{equation} + V(a) = \frac{ak - \pi a^3}{2} +\end{equation} +이 λœλ‹€. + +λΆ€ν”Όκ°€ μ΅œλŒ€κ°€ 되기 μœ„ν•΄μ„œλŠ” $V(a)$의 λ„ν•¨μˆ˜κ°€ $0$이 되며, μ΄κ³„λ„ν•¨μˆ˜κ°€ 음수인 지점이어야 ν•œλ‹€. +λ„ν•¨μˆ˜λŠ” +\begin{equation} + V'(a) = \frac{k - 3 \pi a^2}{2} +\end{equation} +이기에, +\begin{equation} + V'(a) = 0 + \quad \Longleftrightarrow \quad + \frac{k - 3 \pi a^2}{2} = 0 + \quad \Longleftrightarrow \quad + a = \sqrt{\frac{k}{3 \pi}} +\end{equation} +이닀. +λ˜ν•œ, +\begin{equation} + V''(a) = -3 \pi a +\end{equation} +둜 $a > 0$이기에 $V''(a) < 0$이 항상 μ„±λ¦½ν•˜κΈ°μ— $a = \sqrt{\frac{k}{3 \pi}}$이 $V(a)$의 μ΅œλŒ“κ°μž„μ„ μ•Œ 수 μžˆλ‹€. + +μ΄λ•Œμ˜ $b$λŠ” 식 \ref{eq:cylinder_b}에 κ΅¬ν•œ $a$값을 λŒ€μž…ν•˜λ©΄, +\begin{equation} + b = \frac{k - \pi a^2}{2 \pi a} + = \frac{k - \pi \frac{k}{3 \pi}}{2 \pi \sqrt{\frac{k}{3 \pi}}} + = \sqrt{\frac{k}{3 \pi}} +\end{equation} +κ°€ λœλ‹€. +λ”°λΌμ„œ 졜적 λΉ„μœ¨μ€ +\begin{equation} + \boxed{a = b} +\end{equation} +이닀. + +μ΅œλŒ€ λΆ€ν”ΌλŠ” +\begin{equation} + V_{max} = \pi a^2 b = \pi a^3 = \pi \left( \frac{k}{3 \pi} \right)^{3/2} +\end{equation} +이닀. + +\section{n에 λ”°λ₯Έ λ³€ν™”} + +\begin{figure}[H] + \centering + + \begin{tikzpicture} + \begin{axis}[ + axis lines=middle, + xlabel=$n$, ylabel=$V_{max}$, + legend style={draw=none, fill=none} + ] + + \pgfmathsetmacro{\k}{100} + + \foreach \n in {3,...,50} { + \addplot[ + only marks, + mark=*, + mark size=1.2pt + ] coordinates { + (\n, {(\k^(3/2)) / (3*sqrt(3*\n*tan(deg(pi/\n))))}) + }; + } + \addlegendentry{$V_{max}(n)$} + \end{axis} + \end{tikzpicture} + + \caption{ν•¨μˆ˜ $V_{max}(n)$} + \label{fig:v_max_n} +\end{figure} + +κ·Έλ¦Ό \ref{fig:v_max_n}은 μ •$n$κ°ν˜• κΈ°λ‘₯μ—μ„œ μ΅œλŒ€ λΆ€ν”Ό $V_{\max}(n)$이 +λ³€μ˜ 개수 $n$에 따라 μ–΄λ–»κ²Œ λ³€ν™”ν•˜λŠ”μ§€λ₯Ό λ‚˜νƒ€λ‚Έ 것이닀. +μ΄λ•Œ $k$λŠ” μΌμ •ν•œ κ°’μœΌλ‘œ κ³ μ •λ˜μ–΄ 있으며, +각 $n$에 λŒ€ν•΄ μ•žμ„œ λ„μΆœν•œ 졜적 쑰건을 μ μš©ν•˜μ—¬ κ³„μ‚°ν•œ 값이닀. + +κ·Έλž˜ν”„λ₯Ό 톡해 λ‹€μŒκ³Ό 같은 νŠΉμ§•μ„ κ΄€μ°°ν•  수 μžˆλ‹€. + +\begin{itemize} + \item $n$이 μ¦κ°€ν• μˆ˜λ‘ $V_{\max}(n)$은 단쑰 μ¦κ°€ν•œλ‹€. + \item μ¦κ°€λŸ‰μ€ 점점 κ°μ†Œν•˜λ©°, νŠΉμ • 값에 점근적으둜 μˆ˜λ ΄ν•˜λŠ” ν˜•νƒœλ₯Ό 보인닀. +\end{itemize} + +μ΄λŸ¬ν•œ κ²½ν–₯은 μ•žμ„œ μœ λ„ν•œ 식을 톡해 μ΄λ‘ μ μœΌλ‘œλ„ μ„€λͺ…ν•  수 μžˆλ‹€. +μ΅œλŒ€ λΆ€ν”ΌλŠ” +\begin{equation} + V_{\max}(n) = \frac{2k^2}{9P(n)} +\end{equation} +이며, λ‘˜λ ˆλŠ” +\begin{equation} + P(n) = 2 \sqrt{\frac{kn \tan(\pi / n)}{3}} +\end{equation} +μ΄λ―€λ‘œ, +\begin{equation} + V_{\max}(n) + = \frac{2k^2}{9 \cdot 2 \sqrt{\frac{kn \tan(\pi / n)}{3}}} + = \frac{k^{3/2}}{3 \sqrt{3}} \cdot \frac{1}{\sqrt{n \tan(\pi / n)}} +\end{equation} +둜 λ‚˜νƒ€λ‚Ό 수 μžˆλ‹€. + +λ”°λΌμ„œ $V_{\max}(n)$의 증감은 ν•¨μˆ˜ +\begin{equation} + Q(n) = n \tan\left(\frac{\pi}{n}\right) +\end{equation} +의 변화에 μ˜ν•΄ κ²°μ •λœλ‹€. + +μ•žμ„œ $Q(n)$이 $n$이 증가함에 따라 κ°μ†Œν•˜λ©°, +\begin{equation} + \lim_{n \to \infty} Q(n) = \pi +\end{equation} +μž„μ„ λ³΄μ˜€λ‹€. +λ”°λΌμ„œ +\begin{equation} + \lim_{n \to \infty} V_{\max}(n) + = \frac{k^{3/2}}{3 \sqrt{3\pi}} +\end{equation} +κ°€ λœλ‹€. + +이 값은 이후 원기λ‘₯의 μ΅œλŒ€ λΆ€ν”Όλ‘œ λ„μΆœλœ κ°’κ³Ό μ •ν™•νžˆ μΌμΉ˜ν•œλ‹€. + +λ˜ν•œ $n$이 μ¦κ°€ν• μˆ˜λ‘ 증가폭이 점점 μž‘μ•„μ§€λŠ” μ΄μœ λŠ” +$\tan(\pi/n)$이 $n$이 컀질수둝 $\frac{\pi}{n}$에 κ°€κΉŒμ›Œμ§€κΈ° λ•Œλ¬Έμ΄λ‹€. +즉, +\begin{equation} + \tan\left(\frac{\pi}{n}\right) \approx \frac{\pi}{n} + \quad (n \to \infty) +\end{equation} +μ΄λ―€λ‘œ, +\begin{equation} + Q(n) \approx \pi +\end{equation} +둜 λΉ λ₯΄κ²Œ μˆ˜λ ΄ν•˜κ²Œ 되고, +이에 따라 $V_{\max}(n)$ μ—­μ‹œ 일정 값에 점점 κ°€κΉŒμ›Œμ§€κ²Œ λœλ‹€. + +κΈ°ν•˜ν•™μ μœΌλ‘œ ν•΄μ„ν•˜λ©΄, +μ •λ‹€κ°ν˜•μ€ λ³€μ˜ κ°œμˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ 원에 κ°€κΉŒμ›Œμ§€λ©°, +같은 넓이λ₯Ό κ°€μ§ˆ λ•Œ λ‘˜λ ˆκ°€ 점점 κ°μ†Œν•œλ‹€. +이둜 인해 λ™μΌν•œ 겉넓이 μ‘°κ±΄μ—μ„œ 더 λ§Žμ€ 면적을 밑면에 ν• λ‹Ήν•  수 있게 되고, +결과적으둜 λΆ€ν”Όκ°€ μ¦κ°€ν•˜κ²Œ λœλ‹€. + +κ·ΈλŸ¬λ‚˜ $n$이 μΆ©λΆ„νžˆ 컀진 μ΄ν›„μ—λŠ” +μ •λ‹€κ°ν˜•κ³Ό 원 μ‚¬μ΄μ˜ 차이가 맀우 μž‘μ•„μ§€κΈ° λ•Œλ¬Έμ—, +λΆ€ν”Ό 증가 μ—­μ‹œ 거의 λ°œμƒν•˜μ§€ μ•ŠλŠ”λ‹€. +μ΄λŠ” κ·Έλž˜ν”„μ—μ„œ 증가가 점점 μ™„λ§Œν•΄μ§€λŠ” ν˜•νƒœλ‘œ λ‚˜νƒ€λ‚œλ‹€. + +λ”°λΌμ„œ μ •$n$κ°ν˜• κΈ°λ‘₯의 μ΅œλŒ€ λΆ€ν”ΌλŠ” +$n$이 증가함에 따라 단쑰 μ¦κ°€ν•˜λ©°, +κ·Έ κ·Ήν•œμ€ 원기λ‘₯의 μ΅œλŒ€ 뢀피와 μΌμΉ˜ν•œλ‹€. + +\section{κ²°λ‘ } + +λ³Έ μ—°κ΅¬λŠ” κ³ μ •λœ 겉넓이 $k$λ₯Ό κ°€μ§€λŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯ μ€‘μ—μ„œ +μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ„ νƒκ΅¬ν•˜μ˜€λ‹€. +μ΄λŠ” 3D ν”„λ¦°ν„°λ‘œ μ œν•œλœ ν•„λΌλ©˜νŠΈ 양을 μ‚¬μš©ν•  λ•Œ +κ°€μž₯ 효율적인 μ»¨ν…Œμ΄λ„ˆ ν˜•νƒœλ₯Ό μ°ΎλŠ” λ¬Έμ œλ‘œλΆ€ν„° μΆœλ°œν•˜μ˜€λ‹€. + +λ¨Όμ € 밑면이 μ§μ‚¬κ°ν˜•μΈ 직윑면체의 경우λ₯Ό λΆ„μ„ν•˜μ˜€λ‹€. +겉넓이 $S=ab+2ac+2bc=k$의 μ œμ•½μ‘°κ±΄ ν•˜μ—μ„œ +λΆ€ν”Ό $V=abc$λ₯Ό μ΅œλŒ€ν™”ν•œ κ²°κ³Ό, +\[ + a=b=2c +\] +일 λ•Œ μ΅œλŒ€ λΆ€ν”Όκ°€ 됨을 λ„μΆœν•˜μ˜€λ‹€. +즉, 밑면은 μ •μ‚¬κ°ν˜•μ΄ 되며, +λ†’μ΄λŠ” λ°‘λ³€ 길이의 절반이 λœλ‹€. +λ˜ν•œ μ΄λ•Œ λ°‘λ©΄μ˜ λ„“μ΄λŠ” 항상 +\[ + A=\frac{k}{3} +\] +이 됨을 ν™•μΈν•˜μ˜€λ‹€. + +이λ₯Ό μΌλ°˜ν™”ν•˜μ—¬ 밑면이 μ •$n$κ°ν˜•μΈ κΈ°λ‘₯을 λΆ„μ„ν•˜μ˜€λ‹€. +μ œμ•½μ‘°κ±΄μ„ μ΄μš©ν•˜μ—¬ 높이λ₯Ό μ†Œκ±°ν•˜κ³  +λΆ€ν”Όλ₯Ό $(a,n)$의 ν•¨μˆ˜λ‘œ ν‘œν˜„ν•œ λ’€, +κ³ μ •λœ $n$μ—μ„œ μ΅œλŒ€κ°€ λ˜λŠ” $a$λ₯Ό κ΅¬ν•˜μ˜€λ‹€. +κ·Έ κ²°κ³Ό, +\[ + a=\sqrt{\frac{k}{3t}} +\] +이며, μ΄λ•Œ μ—­μ‹œ λ°‘λ©΄μ˜ λ„“μ΄λŠ” +\[ + A=\frac{k}{3} +\] +이 됨을 ν™•μΈν•˜μ˜€λ‹€. +μ΄λŠ” λ°‘λ©΄μ˜ ν˜•νƒœμ™€ λ¬΄κ΄€ν•˜κ²Œ +μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” κ²½μš°μ—λŠ” +전체 κ²‰λ„“μ΄μ˜ $\frac{1}{3}$이 밑면에, +$\frac{2}{3}$이 μ˜†λ©΄μ— λ°°λΆ„λœλ‹€λŠ” 것을 μ˜λ―Έν•œλ‹€. + +λ‹€μŒμœΌλ‘œ λΆ€ν”Ό μ΅œλŒ€ν™”λ₯Ό μœ„ν•΄μ„œλŠ” +λ‘˜λ ˆ $P$κ°€ μ΅œμ†Œκ°€ λ˜μ–΄μ•Ό 함을 λ³΄μ˜€κ³ , +이λ₯Ό μœ„ν•΄ ν•¨μˆ˜ +\[ + Q(n)=n\tan\left(\frac{\pi}{n}\right) +\] +을 λΆ„μ„ν•˜μ˜€λ‹€. +λ‚΄λΆ€ 극값이 μ‘΄μž¬ν•˜μ§€ μ•ŠμŒμ„ 증λͺ…ν•˜κ³ , +경계 뢄석을 톡해 +\[ + \lim_{n\to\infty}Q(n)=\pi +\] +μž„μ„ λ„μΆœν•˜μ˜€λ‹€. +μ΄λŠ” λ³€μ˜ κ°œμˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ λ‘˜λ ˆκ°€ κ°μ†Œν•˜λ©°, +κ·Έ κ·Ήν•œμ΄ μ›μ˜ κ²½μš°μ™€ μΌμΉ˜ν•¨μ„ μ˜λ―Έν•œλ‹€. + +λ”°λΌμ„œ κ³ μ •λœ 겉넓이 쑰건 ν•˜μ—μ„œ +μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ€ +밑면이 원인 원기λ‘₯μž„μ„ μ•Œ 수 μžˆλ‹€. +μ΄λŠ” μ •λ‹€κ°ν˜•μ΄ λ³€μ˜ 개수λ₯Ό λ¬΄ν•œνžˆ 늘릴 λ•Œ +원이 λœλ‹€λŠ” κΈ°ν•˜ν•™μ  사싀과도 μΌμΉ˜ν•œλ‹€. + +λ³Έ 연ꡬλ₯Ό 톡해 λ„μΆœλœ κ°€μž₯ μ€‘μš”ν•œ 톡찰은 λ‹€μŒκ³Ό κ°™λ‹€. + +\begin{itemize} + \item μ΅œλŒ€ λΆ€ν”Όκ°€ λ˜λŠ” 경우, λ°‘λ©΄ λ„“μ΄λŠ” 항상 $\frac{k}{3}$이닀. + \item μ΅œλŒ€ λΆ€ν”ΌλŠ” μ˜†λ©΄ λ‘˜λ ˆκ°€ μ΅œμ†Œκ°€ λ˜λŠ” λ„ν˜•μ—μ„œ λ°œμƒν•œλ‹€. + \item μ •λ‹€κ°ν˜•μ€ λ³€μ˜ μˆ˜κ°€ μ¦κ°€ν• μˆ˜λ‘ 원에 μˆ˜λ ΄ν•˜λ©°, + 원기λ‘₯이 졜적 ꡬ쑰가 λœλ‹€. +\end{itemize} + +이 κ²°κ³ΌλŠ” κΈ°ν•˜ν•™μ  λŒ€μΉ­μ„±μ΄ +μ΅œμ ν™”λ₯Ό μœ λ„ν•œλ‹€λŠ” 일반적인 원리와도 μ—°κ²°λœλ‹€. +λŒ€μΉ­μ„±μ΄ λ†’μ•„μ§ˆμˆ˜λ‘ λ™μΌν•œ κ²‰λ„“μ΄λ‘œ +더 큰 λ‚΄λΆ€ 곡간을 확보할 수 μžˆλ‹€. + +λ‹€λ§Œ λ³Έ μ—°κ΅¬μ—μ„œλŠ” λ²½ λ‘κ»˜λ₯Ό λ¬΄μ‹œν•˜κ³  +겉넓이와 ν•„λΌλ©˜νŠΈ μ‚¬μš©λŸ‰μ΄ μ™„μ „νžˆ λΉ„λ‘€ν•œλ‹€κ³  κ°€μ •ν•˜μ˜€λ‹€. +μ‹€μ œ 3D ν”„λ¦°νŒ…μ—μ„œλŠ” 재료의 밀도, +적측 ꡬ쑰, λ‚΄λΆ€ 채움λ₯ (infill) λ“±μ˜ μš”μ†Œκ°€ 영ν–₯을 쀄 수 μžˆλ‹€. +ν–₯ν›„ μ—°κ΅¬μ—μ„œλŠ” μ΄λŸ¬ν•œ ν˜„μ‹€μ  μš”μ†Œλ₯Ό ν¬ν•¨ν•˜μ—¬ +보닀 μ‹€μ œμ μΈ λͺ¨λΈλ‘œ ν™•μž₯ν•  수 μžˆμ„ 것이닀. + +결둠적으둜, +κ³ μ •λœ 겉넓이λ₯Ό κ°€μ§€λŠ” μœ—λ©΄μ΄ μ—†λŠ” κΈ°λ‘₯ 쀑 +μ΅œλŒ€ λΆ€ν”Όλ₯Ό κ°€μ§€λŠ” λ„ν˜•μ€ 원기λ‘₯이며, +μ΄λŠ” λŒ€μΉ­μ„±κ³Ό λ‘˜λ ˆ μ΅œμ†Œν™”μ˜ 원리에 μ˜ν•΄ μ„€λͺ…λœλ‹€. + +\end{document} \ No newline at end of file

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