岡山大学
2010年 理系 第4問
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≧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)\lim_{n→∞}n^2(R_n-r_n)を求めよ.ただし,\lim_{θ→0}\frac{sinθ}{θ}=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}](./thumb/612/1191/2010_4.png)
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}
類題(関連度順)
![](./thumb/385/2485/2011_4s.png)
![](./thumb/411/973/2014_3s.png)
![](./thumb/202/88/2014_4s.png)
コメント(1件)
![]() 解答お願いします |
書き込むにはログインが必要です。