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zlBeKj=va9n3LJENQY3Jxha&vlMRr)*uV2(C2Yj@=P0ug^PbuP$o^y$t3)M+dB=(qD z1(L4zD=URk#NioY_jN_5mb{}I492xudor1aDx9YfFva6{Y(0*ZS>gg2Dpmjs?aAK7Sk6vrkY?t&)C+$g+KDA=?ZPH7Xa?tU`BhYWqJ>tv!M4NM z##o5XIF3thWRgq6C|AtOxN)}(UldBccBupg*E3w2ll)CCFna}W!^E7T2&;W*9bzI= z7!xZOe+ycKo-a|@%Nv$0fgip~OFjblO241SXV2t(| z1XdDxck|IeB8|oTLt+|CYnV#f$KPhctE*MP%bLZJr|3!2MGj0zF^M_2FfjrK{)YER zLSY+U)uaWbgwOz!>>*!aVgZam5y5PpxBD(a70^wLg$*m}AhU*~cNQ3h8^_Qa`XU8u+#Jt+KX)>^d}7c`Xds%PoJG`K9N^aOv>WvSzT-m^91%Blv46-O<8S{*0Kd4^$JJ9ekweLKUVd! zEflvKcvXJE_(^>_p2D)Hwh-#WD!vbT!mDLN`TD{&dx}l}0HI<-vs%}ylm`~{CZ<`5Su!{#?_o+Nk-N&>Fca9+oCbsygi$^1Gb-O)DQA9Qi=k`{<)lf zG~nFf?ZljGyYrIf`fOJJ%(L}=;q?#qQF_W$jU85teks+ypPOpGG>P;z{fw$!cUxaJ z=vS_A)M`9>P`vm%vifJ(5&rD$AZhQgoGx`4FSWR%=v8?2fsX0DegC9wl%-zOPr~S* zojiu*0=SEoZEu5}s|GyzRy==snBJy3hiwCRJiql`MmMrXdE9ZuWb(#9?AtsdkM6>b zpi`HeoMNx-X*!MHwk!$Ok6yM-Uk23sUIiYVnMhrbuNa!Y9s4cfGAiiPb7|dqR7=+T zN&m3(*za&`1WF!YayZ50) + V_{n_0}''(a)=\frac{t}{n_0}(-6ta) < 0 \quad (a>0) \] -이므로 식 \ref{eq:regular_polygon_a_opt}에서 얻은 임계점은 실제로 최대 부피를 준다. +이므로 식 \ref{eq:regular_polygon_a_opt}에서 얻은 임계점은 실제로 최대 부피를 준다는 것을 알 수 있다. 따라서 정$n_0$각기둥에서 최대 부피를 만들기 위한 변의 길이는 \begin{equation} - a=\sqrt{\frac{k}{3t}} + a = \sqrt{\frac{k}{3t}} + \label{eq:regular_polygon_a} \end{equation} 로 결정된다. +\subsection{최적 조건에서의 A와 b} + +식 \ref{eq:regular_polygon_A}에 계산한 $a$값을 대입하면, +\begin{equation} + A = ta^2 = \frac{k}{3} +\end{equation} +이 된다. +$A$가 $k$로만 나타내어진다는 것은 $n$의 값에 상관없이 정다각형 기둥의 부피가 최대가 될때, 밑면의 넓이는 일정하다는 것을 의미한다. +이는 위에서 직육면체의 부피가 최대가 되는 경우에서도 두 밑면의 길이가 모두 $\sqrt{\frac{k}{3}}$가 되어, +밑면의 넓이가 $\frac{k}{3}$이 되었기 때문에 이 결과를 만족한다고 할 수 있다. + +기둥의 높이는 식 \ref{eq:regular_polygon_b}에 구한 $A$값을 대입하면, +\begin{equation} + b = \frac{k - A}{P} = \frac{2k}{3P} +\end{equation} +가 된다. + +\subsection{최적 조건에서의 부피} + +기둥의 부피는 $A$와 $b$의 곱이기에, 최대 부피는 +\begin{equation} + V_{max} = Ab = \frac{2k^2}{9P} +\end{equation} +가 된다. + +여기서 $k$는 고정된 양수이며, $P > 0$이다. +따라서 $V_{\max}$는 $P$에 대하여 +\begin{equation} + V_{\max}(P) = \frac{C}{P} + \quad (C = \frac{2k^2}{9} > 0) +\end{equation} +와 같은 형태를 가진다. + +함수 $V_{\max}(P)$는 $P > 0$에서 단조 감소 함수이므로, +$P$가 작아질수록 $V_{\max}$는 증가한다. +따라서 최대 부피를 얻기 위해서는 $P$가 최소가 되어야 한다. + +$P$는 식 \ref{eq:regular_polygon_P}에 $a$값을 대입하면 +\begin{equation} + P(n) = na = n \sqrt{\frac{k}{3t}} = 2 \sqrt{\frac{kn \tan{(\pi / n)}}{3}} +\end{equation} +이 된다. +이때 식 \ref{eq:regular_polygon_a}에서 $a$를 $n$으로 나타내었기 때문에, $P$는 $n$에 대한 일변수 함수가 된다. + +$P(n)$이 최소가 되기 위해서는 $n \tan{(\pi / n)}$이 최소가 되어야 한다. +\begin{equation} + Q(n) = n \tan{(\pi / n)} \quad (n \geq 3) +\end{equation} +라고 하면, $Q(n)$이 최소가 될때의 $n$이 기둥의 부피를 최대화 한다. +여기서 $n$은 정$n$각형의 변의 개수를 의미하므로 본래 정수 $n \geq 3$에 한정된다. +그러나 함수 $Q(n)$의 증가 및 감소를 분석하기 위하여, $n$을 실수 $n > 2$로 확장하여 미분을 수행한다. +이러한 연속적 확장은 함수의 경향성을 파악하기 위한 것이며, 푸후 해석은 다시 정수 $n$에 대해 이루어진다. + +$Q(n)$이 최소가 될 수 있는 후보 지점은 극값과 경계값이 있다. + +먼저 극값이 최솟값이 되기 위해서는 해당 지점에서 도함수가 $0$이어야 하며, 이계도함수가 양수여야 한다. +$Q(n)$의 도함수는 곱의 미분법에 따라 +\begin{equation} + Q'(n) = \tan\left(\frac{\pi}{n}\right) - \frac{\pi}{n}\sec^2\left(\frac{\pi}{n}\right) +\end{equation} +가 된다. +이때 계산을 간단히 하기 위하여 +\begin{equation} + \theta=\frac{\pi}{n} +\end{equation} +이라 두면, +\begin{equation} + Q'(n) = \tan{\theta} - \theta \sec^2{\theta} +\end{equation} +로 표현할 수 있기에, +\begin{equation} + \tan{\theta} - \theta \sec^2{\theta} = 0 + \quad \Longrightarrow \quad + \sin{\theta} \cos{\theta} = \theta +\end{equation} + +배각공식에 따르면 +\begin{equation} + 2 \sin{\theta} \cos{\theta} = \sin{2 \theta} + \quad \Longrightarrow \quad + \sin{\theta} \cos{\theta} = \frac{\sin{2 \theta}}{2} +\end{equation} +가 성립한다. + +따라서 극값이 존재하기 위해서는 +\begin{equation} + \theta = \frac{\sin 2\theta}{2} +\end{equation} +를 만족해야 한다. +이를 분석하기 위하여 다음과 같은 함수를 정의한다. +\begin{equation} + f(\theta) = \frac{\sin 2\theta}{2} - \theta + \quad (\theta \leq \frac{\pi}{3}) +\end{equation} +그러면 위 방정식은 $f(\theta)=0$을 푸는 문제와 같다. +이 함수의 도함수는 +\begin{equation} + f'(\theta) = \cos{2 \theta} - 1 +\end{equation} +이다. +삼각함수의 성질에 의해 $\cos{2 \theta} \leq 1$이며, +$\theta > 0$일 때는 $\cos 2\theta < 1$이므로 +\begin{equation} + f'(\theta) < 0 \quad (\theta > 0) +\end{equation} +가 성립한다. 따라서 $f(\theta)$는 $\theta>0$에서 단조 감소한다. +또한 +\begin{equation} + f(0) = 0 +\end{equation} +이므로, $f(\theta) = 0$을 만족하는 해는 $\theta = 0$ 하나뿐이다. +그러나 $\theta=\frac{\pi}{n}$의 정의에 의해 +$\theta=0$이 되려면 $n \to \infty$이어야 하며, +유한한 $n$에서는 $\theta>0$이다. 따라서 $Q(n)$은 유한한 $n$에서 내부 극값을 갖지 않는다. + +내부에서 극값이 존재하지 않으므로, 최솟값은 경계에서만 발생할 수 있다. +본래 문제에서 $n$은 정$n$각형의 변의 개수를 의미하므로 정수 $n \geq 3$에 한정된다. +따라서 $Q(n)$의 정의역은 +\begin{equation} + \{n \mid n \geq 3,\; n \in \mathbb{Z}\} +\end{equation} +이다. + +이 정의역에서 가장 작은 값은 $n=3$이므로, 좌측 경계값은 +\begin{equation} + Q(3) \simeq 5.196 +\end{equation} +이다. + +우측 경계값을 구하기 위해서는 극한을 사용한다. +이 경우 무한대로의 극한값을 구하는 것보다 $0$으로의 극한값을 구하는게 더 쉽기 때문에 +\begin{equation} + u = \frac{1}{n} +\end{equation} +으로 정의하면, +\begin{equation} + \lim_{n \to \infty}{Q(n)} = \lim_{u \to 0}{Q(u)} +\end{equation} +이 된다. 이를 계산하면 +\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 된다. +$\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무한대일때임을 알 수 있다. +이는 밑면의 변이 더 많을수록 기둥의 부피가 더 커짐을 의미한다. + +정다각형에서 변의 대수가 끊임없이 늘어나면 원이 되기 때문에 +부피가 최대가 되는 밑면은 원이 되며, 이는 원기둥이 된다. +따라서 이번에는 제약조건하에서 가장 부피가 큰 원기둥에 대해서 탐구한다. + +\section{원기둥의 최적화} + +원기둥 기둥(윗면 없음) 분석에서는 밑면 원의 반지름을 $a$, 기둥의 높이를 $b$라 한다. +이때, $a$와 $b$는 길이이므로 $a > 0$, $b > 0$이다. +그림 \ref{fig:cylinder}은 해당 기둥을 나타내며, 길이 비율은 실제 해와 무관하다. + +\begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{cylinder.png} + \caption{윗면이 없는 원기둥} + \label{fig:cylinder} +\end{figure} + +윗면이 없으므로 겉넓이는 밑면과 옆면의 합이다 +\begin{equation} + S(a, b) = \pi a^2 + 2 \pi ab +\end{equation} +겉넓이는 k로 고정되어 있기 때문에 제약조건은 +\begin{equation} + S(a, b) = k \quad \Longleftrightarrow \pi a^2 + 2 \pi ab = k + \label{eq:cylinder_constraint} +\end{equation} +이다. +부피는 +\begin{equation} + V(a, b) = \pi a^2 b + \label{eq:cylinder_V} +\end{equation} +이다. + +식 \ref{eq:cylinder_constraint}를 $b$에 대해 정리하면 +\begin{equation} + b = \frac{k - \pi a^2}{2 \pi a} + \label{eq:cylinder_b} +\end{equation} +가 된다. +이를 이용해 식 \ref{eq:cylinder_V}에 대입하면 부피는 +\begin{equation} + V(a) = \frac{ak - \pi a^3}{2} +\end{equation} +이 된다. + +부피가 최대가 되기 위해서는 $V(a)$의 도함수가 $0$이 되며, 이계도함수가 음수인 지점이어야 한다. +도함수는 +\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} +이다. +또한, +\begin{equation} + V''(a) = -3 \pi a +\end{equation} +로 $a > 0$이기에 $V''(a) < 0$이 항상 성립하기에 $a = \sqrt{\frac{k}{3 \pi}}$이 $V(a)$의 최댓감임을 알 수 있다. + +이때의 $b$는 식 \ref{eq:cylinder_b}에 구한 $a$값을 대입하면, +\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} +가 된다. +따라서 최적 비율은 +\begin{equation} + \boxed{a = b} +\end{equation} +이다. + +최대 부피는 +\begin{equation} + V_{max} = \pi a^2 b = \pi a^3 = \pi \left( \frac{k}{3 \pi} \right)^{3/2} +\end{equation} +이다. + +\section{n에 따른 변화} + +\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{함수 $V_{max}(n)$} + \label{fig:v_max_n} +\end{figure} + +\section{결론} + +본 연구는 고정된 겉넓이 $k$를 가지는 윗면이 없는 기둥 중에서 +최대 부피를 가지는 도형을 탐구하였다. +이는 3D 프린터로 제한된 필라멘트 양을 사용할 때 +가장 효율적인 컨테이너 형태를 찾는 문제로부터 출발하였다. + +먼저 밑면이 직사각형인 직육면체의 경우를 분석하였다. +겉넓이 $S=ab+2ac+2bc=k$의 제약조건 하에서 +부피 $V=abc$를 최대화한 결과, +\[ +a=b=2c +\] +일 때 최대 부피가 됨을 도출하였다. +즉, 밑면은 정사각형이 되며, +높이는 밑변 길이의 절반이 된다. +또한 이때 밑면의 넓이는 항상 +\[ +A=\frac{k}{3} +\] +이 됨을 확인하였다. + +이를 일반화하여 밑면이 정$n$각형인 기둥을 분석하였다. +제약조건을 이용하여 높이를 소거하고 +부피를 $(a,n)$의 함수로 표현한 뒤, +고정된 $n$에서 최대가 되는 $a$를 구하였다. +그 결과, +\[ +a=\sqrt{\frac{k}{3t}} +\] +이며, 이때 역시 밑면의 넓이는 +\[ +A=\frac{k}{3} +\] +이 됨을 확인하였다. +이는 밑면의 형태와 무관하게 +최대 부피를 가지는 경우에는 +전체 겉넓이의 $\frac{1}{3}$이 밑면에, +$\frac{2}{3}$이 옆면에 배분된다는 것을 의미한다. + +다음으로 부피 최대화를 위해서는 +둘레 $P$가 최소가 되어야 함을 보였고, +이를 위해 함수 +\[ +Q(n)=n\tan\left(\frac{\pi}{n}\right) +\] +을 분석하였다. +내부 극값이 존재하지 않음을 증명하고, +경계 분석을 통해 +\[ +\lim_{n\to\infty}Q(n)=\pi +\] +임을 도출하였다. +이는 변의 개수가 증가할수록 둘레가 감소하며, +그 극한이 원의 경우와 일치함을 의미한다. + +따라서 고정된 겉넓이 조건 하에서 +최대 부피를 가지는 도형은 +밑면이 원인 원기둥임을 알 수 있다. +이는 정다각형이 변의 개수를 무한히 늘릴 때 +원이 된다는 기하학적 사실과도 일치한다. + +본 연구를 통해 도출된 가장 중요한 통찰은 다음과 같다. + +\begin{itemize} +\item 최대 부피가 되는 경우, 밑면 넓이는 항상 $\frac{k}{3}$이다. +\item 최대 부피는 옆면 둘레가 최소가 되는 도형에서 발생한다. +\item 정다각형은 변의 수가 증가할수록 원에 수렴하며, + 원기둥이 최적 구조가 된다. +\end{itemize} + +이 결과는 기하학적 대칭성이 +최적화를 유도한다는 일반적인 원리와도 연결된다. +대칭성이 높아질수록 동일한 겉넓이로 +더 큰 내부 공간을 확보할 수 있다. + +다만 본 연구에서는 벽 두께를 무시하고 +겉넓이와 필라멘트 사용량이 완전히 비례한다고 가정하였다. +실제 3D 프린팅에서는 재료의 밀도, +적층 구조, 내부 채움률(infill) 등의 요소가 영향을 줄 수 있다. +향후 연구에서는 이러한 현실적 요소를 포함하여 +보다 실제적인 모델로 확장할 수 있을 것이다. + +결론적으로, +고정된 겉넓이를 가지는 윗면이 없는 기둥 중 +최대 부피를 가지는 도형은 원기둥이며, +이는 대칭성과 둘레 최소화의 원리에 의해 설명된다. + \end{document} \ No newline at end of file

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