岡山大学
2010年 理系 第4問
4
4
平面上に半径1の円$C$がある.この円に外接し,さらに隣り合う2つが互いに外接するように,同じ大きさの$n$個の円を図(例1)のように配置し,その一つの円の半径を$R_n$とする.また,円$C$に内接し,さらに隣り合う2つが互いに外接するように,同じ大きさの$n$個の円を図(例2)のように配置し,その一つの円の半径を$r_n$とする.ただし,$n \geqq 3$とする.このとき,次の問いに答えよ.
(1) $R_6,\ r_6$を求めよ.
(2) $\displaystyle \lim_{n \to \infty}n^2(R_n-r_n)$を求めよ.ただし,$\displaystyle \lim_{\theta \to 0}\frac{\sin \theta}{\theta}=1$を用いてよい.
\setlength\unitlength{1truecm} \begin{center} \scalebox{1.5}{ \begin{picture}(6,2.6) \put(0.5,1){\circle{3}} \put(0.5,1){\circle*{0.05}} \put(0.5,1.95){\circle{0.48}} \put(0.5,1.95){\circle*{0.05}} \put(0.5,0.05){\circle{0.48}} \put(0.5,0.05){\circle*{0.05}} \put(0.97,1.83){\circle{0.48}} \put(0.97,1.83){\circle*{0.05}} \put(1.32,1.47){\circle{0.48}} \put(1.32,1.47){\circle*{0.05}} \put(1.45,1){\circle{0.48}} \put(1.45,1){\circle*{0.05}} \put(-0.4,-0.6){\tiny 例1:$\ n=12$の場合} \put(1.32,0.53){\circle{0.48}} \put(1.32,0.53){\circle*{0.05}} \put(0.97,0.17){\circle{0.48}} \put(0.97,0.17){\circle*{0.05}} \put(0.03,0.17){\circle{0.48}} \put(0.03,0.17){\circle*{0.05}} \put(-0.32,0.53){\circle{0.48}} \put(-0.32,0.53){\circle*{0.05}} \put(-0.45,1){\circle{0.48}} \put(-0.45,1){\circle*{0.05}} \put(0.03,1.83){\circle{0.48}} \put(0.03,1.83){\circle*{0.05}} \put(-0.32,1.47){\circle{0.48}} \put(-0.32,1.47){\circle*{0.05}} \put(4.5,1){\circle{3}} \put(4.5,1){\circle*{0.05}} \put(4.5,1.41){\circle{0.63}} \put(4.5,1.41){\circle*{0.05}} \put(4.5,0.59){\circle{0.63}} \put(4.5,0.59){\circle*{0.05}} \put(4.09,1){\circle{0.63}} \put(4.09,1){\circle*{0.05}} \put(4.91,1){\circle{0.63}} \put(4.91,1){\circle*{0.05}} \put(3.7,-0.6){\tiny 例2:$\ n=4$の場合} \end{picture} } \end{center}
(1) $R_6,\ r_6$を求めよ.
(2) $\displaystyle \lim_{n \to \infty}n^2(R_n-r_n)$を求めよ.ただし,$\displaystyle \lim_{\theta \to 0}\frac{\sin \theta}{\theta}=1$を用いてよい.
\setlength\unitlength{1truecm} \begin{center} \scalebox{1.5}{ \begin{picture}(6,2.6) \put(0.5,1){\circle{3}} \put(0.5,1){\circle*{0.05}} \put(0.5,1.95){\circle{0.48}} \put(0.5,1.95){\circle*{0.05}} \put(0.5,0.05){\circle{0.48}} \put(0.5,0.05){\circle*{0.05}} \put(0.97,1.83){\circle{0.48}} \put(0.97,1.83){\circle*{0.05}} \put(1.32,1.47){\circle{0.48}} \put(1.32,1.47){\circle*{0.05}} \put(1.45,1){\circle{0.48}} \put(1.45,1){\circle*{0.05}} \put(-0.4,-0.6){\tiny 例1:$\ n=12$の場合} \put(1.32,0.53){\circle{0.48}} \put(1.32,0.53){\circle*{0.05}} \put(0.97,0.17){\circle{0.48}} \put(0.97,0.17){\circle*{0.05}} \put(0.03,0.17){\circle{0.48}} \put(0.03,0.17){\circle*{0.05}} \put(-0.32,0.53){\circle{0.48}} \put(-0.32,0.53){\circle*{0.05}} \put(-0.45,1){\circle{0.48}} \put(-0.45,1){\circle*{0.05}} \put(0.03,1.83){\circle{0.48}} \put(0.03,1.83){\circle*{0.05}} \put(-0.32,1.47){\circle{0.48}} \put(-0.32,1.47){\circle*{0.05}} \put(4.5,1){\circle{3}} \put(4.5,1){\circle*{0.05}} \put(4.5,1.41){\circle{0.63}} \put(4.5,1.41){\circle*{0.05}} \put(4.5,0.59){\circle{0.63}} \put(4.5,0.59){\circle*{0.05}} \put(4.09,1){\circle{0.63}} \put(4.09,1){\circle*{0.05}} \put(4.91,1){\circle{0.63}} \put(4.91,1){\circle*{0.05}} \put(3.7,-0.6){\tiny 例2:$\ n=4$の場合} \end{picture} } \end{center}
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2015-02-17 15:27:07
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