{"id":520,"date":"2012-12-12T20:36:41","date_gmt":"2012-12-12T11:36:41","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=520"},"modified":"2021-09-29T20:12:39","modified_gmt":"2021-09-29T11:12:39","slug":"thr200803","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr200803\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2008\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(\\theta\\) \u3092 \\(0 \\lt \\theta \\lt \\dfrac{2 \\pi}{3}\\) \u306e\u7bc4\u56f2\u306b\u3042\u308b\u5b9f\u6570\u3068\u3057, \u7a7a\u9593\u306e \\(4\\) \u70b9 O , A , B , C \u304c, \\(\\text{OA} = \\text{OB} = \\text{OC} = 1\\) \u304b\u3064 \\(\\angle \\text{AOB} = \\angle \\text{BOC} = \\angle \\text{COA} = \\theta\\) \u3092\u307f\u305f\u3059\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u25b3ABC \u306e\u91cd\u5fc3\u3092 G \u3068\u3059\u308b\u3068\u304d, AG \u3068 OG \u3092\u305d\u308c\u305e\u308c \\(\\theta\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\theta\\) \u3092\u52d5\u304b\u3057\u305f\u3068\u304d, O , A , B , C \u3092\u9802\u70b9\u3068\u3059\u308b\u56db\u9762\u4f53\u306e\u4f53\u7a4d\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

    (1)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\n\\text{AB} & = \\text{BC} =\\text{CA} =\\sqrt{1^2 +1^2 -2 \\cdot 1 \\cdot 1 \\cdot \\cos \\theta} \\\\\r\n& = \\sqrt{2 ( 1 -\\cos \\theta )}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u25b3ABC \u306f\u6b63\u4e09\u89d2\u5f62\u3060\u304b\u3089\r\n\\[\\begin{align}\r\n\\text{AG} & = \\dfrac{2}{3} \\cdot \\dfrac{\\sqrt{3}}{2} \\text{AB} \\\\\r\n& = \\underline{\\sqrt{\\dfrac{2( 1 -\\cos \\theta )}{3}}}\r\n\\end{align}\\]\r\n\u307e\u305f, \u4e09\u5e73\u65b9\u306e\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\n\\text{OG} & = \\sqrt{1^2 -\\dfrac{2( 1 -\\cos \\theta )}{3}} \\\\\r\n& = \\underline{\\sqrt{\\dfrac{1 +2 \\cos \\theta}{3}}}\r\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n

    \u25b3ABC \u306e\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} \\text{AB} \\cdot \\dfrac{\\sqrt{3}}{2} \\text{AB} \\\\\r\n& = \\dfrac{\\sqrt{3} ( 1 -\\cos \\theta )}{2}\r\n\\end{align}\\]\r\n\u56db\u9762\u4f53 OABC \u306e\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = \\dfrac{1}{3} S \\cdot \\text{OG} \\\\\r\n& = \\dfrac{( 1 -\\cos \\theta ) \\sqrt{1 +2 \\cos \\theta}}{6} \\quad ... [1]\r\n\\end{align}\\]\r\n[1] \u306e\u5206\u5b50\u306b\u7740\u76ee\u3057\u3066, \\(f(t) = (1-t)^2 (1+2t) \\ \\left( -\\dfrac{1}{2} \\lt t \\lt 0 \\right)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf'(t) & = 2(t-1) (2t+1) +(t-1)^2 \\cdot 2 \\\\\r\n& = 6t (t-1)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & \\left( -\\frac{1}{2} \\right) \\ & \\cdots & 0 & \\cdots & (1) \\\\ \\hline f'(t) & & + & 0 & - & \\\\ \\hline f(x) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(t)\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\nf(0) = 1\r\n\\]\r\n\u3088\u3063\u3066, \\(V\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{1}{6} f(0) = \\underline{\\dfrac{1}{6}} \\quad \\left( \\theta = \\dfrac{\\pi}{2} \\text{\u306e\u3068\u304d} \\right)\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(\\theta\\) \u3092 \\(0 \\lt \\theta \\lt \\dfrac{2 \\pi}{3}\\) \u306e\u7bc4\u56f2\u306b\u3042\u308b\u5b9f\u6570\u3068\u3057, \u7a7a\u9593\u306e \\(4\\) \u70b9 O , A , B , C \u304c, \\(\\text{OA} = \\tex … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[76],"tags":[148,16],"class_list":["post-520","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2008","tag-tohoku_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/520","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=520"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/520\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=520"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=520"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=520"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}