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b/latex/main.tex index be46cc2..b59a06b 100644 --- a/latex/main.tex +++ b/latex/main.tex @@ -25,7 +25,7 @@ \newlength{\figwidth} \newlength{\figheight} \AtBeginDocument{% - \setlength{\figwidth}{0.5\textwidth}% + \setlength{\figwidth}{0.5\textwidth}% \setlength{\figheight}{0.4\textwidth}% } @@ -44,7 +44,7 @@ \vspace*{\fill} - {\large 19페이지} + {\large 20페이지} \vspace*{12em} \end{titlepage} @@ -352,7 +352,7 @@ $n=n_0$를 고정하면 $t$와 $n_0$는 상수이며, $V_{n_0}(a)=\frac{t}{n_0}( \end{equation} 이 된다. 또한 $V_{n_0}''(a)=\frac{t}{n_0}(-6ta)<0$ ($a>0$)이므로 이 임계점이 최댓값임을 알 수 있다. -\subsection{최적 조건에서의 A와 b} +\subsection{\texorpdfstring{최적 조건에서의 밑면 넓이 $A$와 높이 $b$}{최적 조건에서의 밑면 넓이 A와 높이 b}} 식 \ref{eq:regular_polygon_A}에 계산한 $a$값을 대입하면, \begin{equation} @@ -385,6 +385,8 @@ $A$가 $k$로만 나타내어진다는 것은 $n$의 값에 상관없이 정다 $P$가 작아질수록 $V_{\max}$는 증가한다. 따라서 최대 부피를 얻기 위해서는 $P$가 최소가 되어야 한다. +\subsection{\texorpdfstring{둘레 $P$의 최소화}{둘레 P의 최소화}} + $P$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\left(\frac{\pi}{n}\right)}{3}} @@ -506,12 +508,14 @@ $x = \pi u > 0$이라 두면 $u \to 0^+$일 때 $x \to 0^+$이므로 $\lim_{x \t $\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 무한대일때임을 알 수 있다. 이는 밑면의 변이 더 많을수록 기둥의 부피가 더 커짐을 의미한다. +\subsection{해석 및 다음 단계} + 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} -앞서 정다각형 분석에서 $Q(n)$이 순감소하며 $V_{\max}(n)$이 단조 증가함을 보였고, 그 극한이 $\frac{k^{\frac{3}{2}}}{3\sqrt{3\pi}}$임을 확인하였다. 이제 극한의 도형인 원기둥의 최대 부피를 직접 계산하여 이 값과 일치하는지 검증한다. +앞서 정다각형 분석에서 $Q(n)$이 순감소하며 변의 개수가 많아질수록 최대 부피가 커짐을 보였다. 이제 극한의 도형인 원기둥의 최대 부피를 직접 계산하고, 이어서 $n \to \infty$에서의 극한값과 일치하는지 비교한다. 원기둥(윗면 없음) 분석에서는 밑면 원의 반지름을 $a$, 기둥의 높이를 $b$라 한다. 이때, $a$와 $b$는 길이이므로 $a > 0$, $b > 0$이다. @@ -538,6 +542,8 @@ $\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 \end{equation} 이다. +\subsection{제약조건을 이용한 변수 제거} + 식 \ref{eq:cylinder_constraint}를 $b$에 대해 정리하면 \begin{equation} b = \frac{k - \pi a^2}{2 \pi a} @@ -550,6 +556,8 @@ $\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 \end{equation*} 이 된다. +\subsection{\texorpdfstring{최대 부피와 $a$, $b$의 비율}{최대 부피와 a, b의 비율}} + 앞서와 동일한 방법으로 $V'(a)=0$을 풀면, \begin{equation*} \frac{k - 3\pi a^2}{2} = 0 @@ -577,7 +585,9 @@ $\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 \end{equation} 이다. -\section{n에 따른 변화} +\section{세 도형의 비교} + +\subsection{\texorpdfstring{$n$에 따른 최대 부피의 변화}{n에 따른 최대 부피의 변화}} \begin{figure}[H] \centering @@ -634,7 +644,7 @@ $\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 \end{equation*} 가 된다. -이 값은 이후 원기둥의 최대 부피로 도출된 값과 정확히 일치한다. +이 값은 앞서 원기둥에서 도출한 최대 부피와 정확히 일치한다. 이러한 일치는 정다각형이 $n \to \infty$일 때 원으로 수렴한다는 기하학적 사실의 수학적 표현이다. 탐구 초반에 세운 예상인 ``변이 많을수록 부피가 크고 원기둥이 최적일 것이다''가 극한 분석과 원기둥 직접 계산 모두에서 검증된 것이다. 기하학적으로 해석하면, 정다각형은 변의 개수가 증가할수록 원에 가까워지며, 같은 넓이를 가질 때 둘레가 점점 감소한다. @@ -645,27 +655,7 @@ $\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 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 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. +\subsection{최적 조건과 최대 부피의 정리} 세 도형에 대한 최적 조건과 최대 부피를 표 \ref{tab:summary}에 정리한다. @@ -686,7 +676,9 @@ $\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 표에서 밑면 넓이가 세 경우 모두 $\frac{k}{3}$로 동일함을 한눈에 확인할 수 있다. -\subsection*{실제 3D 프린팅 결과} +\section{3D 프린팅을 통한 검증} + +\subsection{컨테이너 설계와 출력} 서론에서 밝혔듯이 본 탐구는 로봇 개발용 나사와 너트를 정리할 컨테이너를, 정해진 필라멘트 양으로 최대한 크게 만들고 싶다는 동기에서 출발하였다. 따라서 도출한 결과가 실제 출력물에서도 성립하는지 확인하기 위해, 직육면체, 정육각기둥, 원기둥 컨테이너를 직접 CAD로 설계하고 3D 프린터로 출력한 뒤 물을 이용하여 부피를 측정하였다. @@ -715,7 +707,7 @@ $\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 \caption{3D 프린팅 컨테이너의 설계와 출력물} \end{figure} -\subsubsection*{이론 최대 부피와 벽 두께를 고려한 부피} +\subsection{이론 최대 부피와 벽 두께를 고려한 부피} $k = 90\,\text{cm}^2$를 앞서 도출한 최적 조건에 대입하여 바깥 치수를 정하였으며(그림 \ref{fig:cad_design}), 벽 두께를 무시한 이론 최대 부피 $V_{\max}$는 표 \ref{tab:summary}의 식으로 계산하였다. @@ -732,7 +724,7 @@ $k = 90\,\text{cm}^2$를 앞서 도출한 최적 조건에 대입하여 바깥 이는 벽의 부피가 대략 겉넓이와 두께의 곱인 $kd = 9.0\,\text{cm}^3$이기 때문이며, 모서리에서 벽이 서로 겹치는 부분만큼 조금 더 작게 나타난다. 벽의 부피는 곧 출력에 사용된 필라멘트의 부피이므로, 이 결과는 연구 설계에서 세운 ``필라멘트 양은 겉넓이에 비례한다''는 가정이 타당함을 보여준다. -\subsubsection*{물을 이용한 부피 측정} +\subsection{물을 이용한 부피 측정} 마지막으로 출력한 세 컨테이너에 증류수를 가득 채운 뒤 담긴 물의 질량을 측정하였다. 증류수는 $1\,\text{cm}^3$당 질량이 약 $1\,\text{g}$이므로, 측정한 질량의 수치를 그대로 부피 $V_{\text{측정}}$으로 볼 수 있다. @@ -764,6 +756,28 @@ $k = 90\,\text{cm}^2$를 앞서 도출한 최적 조건에 대입하여 바깥 원기둥은 직육면체보다 이론상 약 $12.8\,\%$, 측정상 약 $\frac{83 - 73}{73} \approx 13.7\,\%$ 더 큰 부피를 가졌다. 벽이 차지하는 부피는 세 도형이 거의 같으므로, 원래 부피가 작은 직육면체일수록 벽으로 잃는 비율이 커져($10.6\,\%$, 원기둥은 $9.4\,\%$) 실제 출력물에서는 원기둥의 이점이 오히려 조금 더 커진다. +\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}