こんにちは。一部表現を変えています。それではどうぞ。
下の図1に示した立体は, 1辺の長さが6cmの正四面体である。
辺ACの中点をMとする。点Pは, 頂点Aを出発し, 辺AB, 辺BC上を毎秒1cmの速さで動き, 12秒後に頂点Cに到着する。
点Qは, 点Pが頂点Aを出発するのと同時に頂点Cを出発し, 辺CD, 辺DA上を, 点Pと同じ速さで動き, 12秒後に頂点Aに到着する。
点Mと点P, 点Mと点Qをそれぞれ結ぶ。次の各問いに答えよ。
(1) 図1において, 点Pが辺AB上にあるとき, とする。
の値が最も小さくなるのは, 点Pが頂点Aを出発してから何秒後であるか。
【図1】
(2) 下の図2は, 図1において, 点Pが頂点Aを出発してから8秒後のとき, 頂点Aと点P, 点Pと点Qをそれぞれ結んだ場合を表している。立体
![Rendered by QuickLaTeX.com \text{Q}-\text{APM}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-af775ecb1a1f74e245c2f175280d7e19_l3.png)
【図2】
【東京一部表現改】
【解答】
(1) 秒後
(2)
【解説】
![Rendered by QuickLaTeX.com \bigtriangleup{\text{APM}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e0e7ba92484af889011abe14027e33c9_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{CQM}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-92d15179078ab16dcf30af9f2a3cf161_l3.png)
![Rendered by QuickLaTeX.com 1 : 2 : \sqrt{3}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3f658b8f3aa414c493c5872cfc597183_l3.png)
![Rendered by QuickLaTeX.com \text{AM}=\text{CM}=3\,\text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-92dca1348e7c0cbf6f75d5aede20e38f_l3.png)
![Rendered by QuickLaTeX.com \text{AP}=\text{CQ}=\dfrac32\, \text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-552ece24af1771795a814d208c7ccdca_l3.png)
よって,
![Rendered by QuickLaTeX.com \dfrac32](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e1cb0fd1f033a0bdd7f732d1825ed94f_l3.png)
![Rendered by QuickLaTeX.com \cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f92a08ced98be124fef39e8b49d7144a_l3.png)
(2)
まず, 正四面体
![Rendered by QuickLaTeX.com \text{A}-\text{BCD}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b8fd6f149748a5871e91c53b6b691ed3_l3.png)
![Rendered by QuickLaTeX.com \text{Q}-\text{APM}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-af775ecb1a1f74e245c2f175280d7e19_l3.png)
頂点Aから正三角形BCDに垂線AHを下ろす。このとき, 直角三角形の合同条件より,
![Rendered by QuickLaTeX.com \bigtriangleup{\text{AHB}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8d59eb034944e9bf30181fcc9e603682_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{AHC}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3fc073574deec818222f02154e545466_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{AHD}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-1abb484add9611de339213c75eca3b74_l3.png)
![Rendered by QuickLaTeX.com \text{BH}=\text{CH}=\text{DH}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6ad74c34e3f5309cbc0abbb221c97b6c_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{HBC}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ee755343f4ca4d127a9ac43a10b19b04_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{HCD}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cc93e516aba64d0642d79bf743528fe4_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{HDB}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9a58f667054dd2d2f65afae742f7fbda_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{HUC}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-dd2beb6af1c5a51fc9a34572f89e0e37_l3.png)
![Rendered by QuickLaTeX.com 1 : 2 : \sqrt3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9d31dbf944a09e886d6cae5ee0c86a5c_l3.png)
![Rendered by QuickLaTeX.com \text{CU}=3\,\text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-16bdf4cfacfd17366fffb1db5ec51c05_l3.png)
![Rendered by QuickLaTeX.com \text{CH}=2\sqrt3\, \text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0e26684df902882fcce4ed9144ed0192_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{ACH}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6ec3122d82027163e70803db6270e7d0_l3.png)
![Rendered by QuickLaTeX.com \text{AH}=\sqrt{6^2-(2\sqrt3)^2}=2\sqrt6\,\text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2d3c5783d3efb59fcaee3e87ca158ba9_l3.png)
1辺
![Rendered by QuickLaTeX.com 6\,\text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-214477f7720400787a9f5081485ef90f_l3.png)
![Rendered by QuickLaTeX.com 6\times3\sqrt3\times\dfrac12=9\sqrt3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9b7ff98d684422e110ca4d3fc11568ac_l3.png)
よって, 正四面体
![Rendered by QuickLaTeX.com \text{A}-\text{BCD}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b8fd6f149748a5871e91c53b6b691ed3_l3.png)
![Rendered by QuickLaTeX.com 9\sqrt3\times2\sqrt6\times\dfrac13=18\sqrt2\, \text{cm}^3\cdots\maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-26b5791231488bf2b20fdf74665a2760_l3.png)
次に, 立体
![Rendered by QuickLaTeX.com \text{Q}-\text{APM}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-af775ecb1a1f74e245c2f175280d7e19_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{AMQ}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-282b539e513e9b69b0d39bea6c3d4f86_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{AMQ}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-282b539e513e9b69b0d39bea6c3d4f86_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{ACD}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0c8585f8f6ab29d792cb1cac8db44b56_l3.png)
P, Qが出発して8秒後ということは,
![Rendered by QuickLaTeX.com \text{BP}=\text{DQ}=2\, \text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-575ef6d9e5b09353dc5bb8992ee05c79_l3.png)
![Rendered by QuickLaTeX.com \text{AQ} : \text{QD} = 2 : 1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-90d60fae0d56a8dffae4d0dcd24df8c8_l3.png)
![Rendered by QuickLaTeX.com \text{AM} : \text{MC} = 1 : 1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-adb4184fa22d529c69c082e49d4b4bcf_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{AMQ}}=\dfrac{1\times2}{2\times3}\bigtriangleup{\text{ACD}}=\dfrac{1}{3}\bigtriangleup{\text{ACD}}\cdots\maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f013f12420d7b7040dfcb424e6d3cea3_l3.png)
また, 底面を
![Rendered by QuickLaTeX.com \bigtriangleup{\text{AMQ}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-282b539e513e9b69b0d39bea6c3d4f86_l3.png)
![Rendered by QuickLaTeX.com \text{Q}-\text{APM}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-af775ecb1a1f74e245c2f175280d7e19_l3.png)
![Rendered by QuickLaTeX.com \dfrac{\text{PC}}{\text{BC}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9aaa71f4ea396373bef6ac8d6d053825_l3.png)
![Rendered by QuickLaTeX.com \dfrac{2}{3}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-91b9590d9e32c417dc58fec8ec138aad_l3.png)
![Rendered by QuickLaTeX.com \cdots\maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d8b0c048f7718887e467d512cbce82c3_l3.png)
よって, 求める体積は,
![Rendered by QuickLaTeX.com \maru1, \maru2, \maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a94935c102c23549ddb1822473356c2d_l3.png)
![Rendered by QuickLaTeX.com 18\sqrt2\times\dfrac13\times\dfrac23=4\sqrt2\, (\text{cm}^3)\cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-110697b6b138f70898e37bdd17974f2f_l3.png)