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zYOw8UA8+TO66WD%=3x-+B6#egP>;e{<$w~ruE9@HMR-p|WbkI_k->pKHc|GAZu#9` zIq;(pe@tKTJAdhe{o*iq*z~t@kA8Z9HAe_^1CP9f*2Y4Uua>EXmf9H#$ItjtX6 zzeC)LQ<(mz!io%thf*x{a-E#k_S!?$H<_!{FQZ0^bkr|lhs^bg5AB$WbTltXI*LG| zRNU?UjfyfalcqKr`&QEyPlZJttjrFYbCqw4iBg%f`T{H|Xu>wf@lrd1TZ{VgE_tni z2l8vrYr^vHTX=O0sc)~=iqI1eUz*E{tSSncuv4cMdozbpz37KEa-?o*nWd?_J9j~l z4{#p+$>g(bW5w)L&BwNK=83Mml5$|x$n3hsT?>u(!BVC>^^Tzv1MO8UhTd zq~VtdpFM->^%EWQf#QzoS;r-g+}j0unHshJGd3ALUy~<9b2PcZw7vH*)<|4x=6 kCCQTsoyOh6-v0rBV+XpIE2J_W3@|b|H!=z(B}Gq03WHl5Pyhe` diff --git a/latex/main.tex b/latex/main.tex index f1bdc36..9d82fe6 100644 --- a/latex/main.tex +++ b/latex/main.tex @@ -644,6 +644,84 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 \label{fig:v_max_n} \end{figure} +그림 \ref{fig:v_max_n}은 정$n$각형 기둥에서 최대 부피 $V_{\max}(n)$이 +변의 개수 $n$에 따라 어떻게 변화하는지를 나타낸 것이다. +이때 $k$는 일정한 값으로 고정되어 있으며, +각 $n$에 대해 앞서 도출한 최적 조건을 적용하여 계산한 값이다. + +그래프를 통해 다음과 같은 특징을 관찰할 수 있다. + +\begin{itemize} + \item $n$이 증가할수록 $V_{\max}(n)$은 단조 증가한다. + \item 증가량은 점점 감소하며, 특정 값에 점근적으로 수렴하는 형태를 보인다. +\end{itemize} + +이러한 경향은 앞서 유도한 식을 통해 이론적으로도 설명할 수 있다. +최대 부피는 +\begin{equation} + V_{\max}(n) = \frac{2k^2}{9P(n)} +\end{equation} +이며, 둘레는 +\begin{equation} + P(n) = 2 \sqrt{\frac{kn \tan(\pi / n)}{3}} +\end{equation} +이므로, +\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} +로 나타낼 수 있다. + +따라서 $V_{\max}(n)$의 증감은 함수 +\begin{equation} + Q(n) = n \tan\left(\frac{\pi}{n}\right) +\end{equation} +의 변화에 의해 결정된다. + +앞서 $Q(n)$이 $n$이 증가함에 따라 감소하며, +\begin{equation} + \lim_{n \to \infty} Q(n) = \pi +\end{equation} +임을 보였다. +따라서 +\begin{equation} + \lim_{n \to \infty} V_{\max}(n) + = \frac{k^{3/2}}{3 \sqrt{3\pi}} +\end{equation} +가 된다. + +이 값은 이후 원기둥의 최대 부피로 도출된 값과 정확히 일치한다. + +또한 $n$이 증가할수록 증가폭이 점점 작아지는 이유는 +$\tan(\pi/n)$이 $n$이 커질수록 $\frac{\pi}{n}$에 가까워지기 때문이다. +즉, +\begin{equation} + \tan\left(\frac{\pi}{n}\right) \approx \frac{\pi}{n} + \quad (n \to \infty) +\end{equation} +이므로, +\begin{equation} + Q(n) \approx \pi +\end{equation} +로 빠르게 수렴하게 되고, +이에 따라 $V_{\max}(n)$ 역시 일정 값에 점점 가까워지게 된다. + +기하학적으로 해석하면, +정다각형은 변의 개수가 증가할수록 원에 가까워지며, +같은 넓이를 가질 때 둘레가 점점 감소한다. +이로 인해 동일한 겉넓이 조건에서 더 많은 면적을 밑면에 할당할 수 있게 되고, +결과적으로 부피가 증가하게 된다. + +그러나 $n$이 충분히 커진 이후에는 +정다각형과 원 사이의 차이가 매우 작아지기 때문에, +부피 증가 역시 거의 발생하지 않는다. +이는 그래프에서 증가가 점점 완만해지는 형태로 나타난다. + +따라서 정$n$각형 기둥의 최대 부피는 +$n$이 증가함에 따라 단조 증가하며, +그 극한은 원기둥의 최대 부피와 일치한다. + \section{결론} 본 연구는 고정된 겉넓이 $k$를 가지는 윗면이 없는 기둥 중에서 @@ -655,14 +733,14 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 겉넓이 $S=ab+2ac+2bc=k$의 제약조건 하에서 부피 $V=abc$를 최대화한 결과, \[ -a=b=2c + a=b=2c \] 일 때 최대 부피가 됨을 도출하였다. 즉, 밑면은 정사각형이 되며, 높이는 밑변 길이의 절반이 된다. 또한 이때 밑면의 넓이는 항상 \[ -A=\frac{k}{3} + A=\frac{k}{3} \] 이 됨을 확인하였다. @@ -672,11 +750,11 @@ A=\frac{k}{3} 고정된 $n$에서 최대가 되는 $a$를 구하였다. 그 결과, \[ -a=\sqrt{\frac{k}{3t}} + a=\sqrt{\frac{k}{3t}} \] 이며, 이때 역시 밑면의 넓이는 \[ -A=\frac{k}{3} + A=\frac{k}{3} \] 이 됨을 확인하였다. 이는 밑면의 형태와 무관하게 @@ -688,13 +766,13 @@ $\frac{2}{3}$이 옆면에 배분된다는 것을 의미한다. 둘레 $P$가 최소가 되어야 함을 보였고, 이를 위해 함수 \[ -Q(n)=n\tan\left(\frac{\pi}{n}\right) + Q(n)=n\tan\left(\frac{\pi}{n}\right) \] 을 분석하였다. 내부 극값이 존재하지 않음을 증명하고, 경계 분석을 통해 \[ -\lim_{n\to\infty}Q(n)=\pi + \lim_{n\to\infty}Q(n)=\pi \] 임을 도출하였다. 이는 변의 개수가 증가할수록 둘레가 감소하며, @@ -709,10 +787,10 @@ Q(n)=n\tan\left(\frac{\pi}{n}\right) 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. + \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