{"id":91,"date":"2011-11-27T10:16:58","date_gmt":"2011-11-27T01:16:58","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=91"},"modified":"2021-03-15T19:51:36","modified_gmt":"2021-03-15T10:51:36","slug":"tkr201001","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201001\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2010\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(3\\) \u8fba\u306e\u9577\u3055\u304c \\(a\\) \u3068 \\(b\\) \u3068 \\(c\\) \u306e\u76f4\u65b9\u4f53\u3092,\r\n\u9577\u3055\u304c \\(b\\) \u306e \\(1\\) \u8fba\u3092\u56de\u8ee2\u8ef8\u3068\u3057\u3066 \\(90^{\\circ}\\) \u56de\u8ee2\u3055\u305b\u308b\u3068\u304d,\r\n\u76f4\u65b9\u4f53\u304c\u901a\u904e\u3059\u308b\u70b9\u5168\u4f53\u304c\u3064\u304f\u308b\u7acb\u4f53\u3092 \\(V\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(V\\) \u306e\u4f53\u7a4d\u3092 \\(a\\) , \\(b\\) , \\(c\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(a+b+c=1\\) \u306e\u3068\u304d, \\(V\\) \u306e\u4f53\u7a4d\u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\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

    \u76f4\u65b9\u4f53\u304c\u901a\u904e\u3059\u308b\u70b9\u5168\u4f53\u304c\u4f5c\u308b\u7acb\u4f53\u3092 \\(D\\) \u3068\u304a\u304f.
    \r\n\\(D\\) \u3092, \u9577\u3055 \\(b\\) \u306e\u8fba\u306b\u5782\u76f4\u306a\u65b9\u5411\u304b\u3089\u898b\u308b\u3068, \u4e0b\u56f3\u659c\u7dda\u90e8.<\/p>\r\n\"\"\r\n

    \u3053\u306e\u9762\u7a4d \\(S\\) \u306f\r\n\\[\r\nS = \\dfrac{\\pi}{4} \\left( a^2+c^2 \\right) +ac\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nV = bS = \\underline{ b \\left\\{ \\dfrac{\\pi}{4} \\left( a^2+c^2 \\right) + ac \\right\\} }\r\n\\]\r\n

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

    \\(a+b+c=1\\) \u3088\u308a\r\n\\[\r\nb=1-a-c\r\n\\]\r\n\\(p=a+c\\) , \\(q=ac\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n0 \\lt p \\lt 1 , \\quad 0 \\lt q \\leqq \\dfrac{p^2}{4}\r\n\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\nV & = (1-p)\\left\\{ \\dfrac{\\pi}{4} \\left( p^2-2q \\right) +q \\right\\} \\\\\r\n& = (1-p)\\left\\{ \\left( 1 - \\dfrac{\\pi}{2} \\right) q + \\dfrac{\\pi}{4} p^2 \\right\\}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 \\(p\\) \u3092\u5b9a\u6570\u3068\u307f\u306a\u3059\u3068, \\(V\\) \u306f \\(q\\) \u306e \\(1\\) \u6b21\u95a2\u6570\u3067\u3042\u308a, \u50be\u304d \\(( 1-p ) \\left( 1- \\dfrac{\\pi}{2} \\right) \\lt 0\\) \u306a\u306e\u3067\r\n\\[\r\n(1-p) \\left\\{ \\left( 1 - \\dfrac{\\pi}{2} \\right) \\dfrac{p^2}{4} + \\dfrac{\\pi}{4} p^2 \\right\\} \\leqq V \\lt (1-p) \\cdot \\dfrac{\\pi}{4} \\, p^2 \\\\\r\n\\text{\u2234} \\quad \\dfrac{\\pi + 2}{8} \\, p^2 (1-p) \\leqq V \\lt \\dfrac{\\pi}{4} \\, p^2 (1-p) \\quad ... [1]\r\n\\]\r\n\u3064\u304e\u306b \\(p\\) \u3092\u5909\u6570\u3068\u307f\u306a\u3059.
    \r\n\\(f(p) = p^2(1-p) = p^2-p^3\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(p) = 2p -3p^2 = p (2-3p)\r\n\\]\r\n\\(f'(p)=0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\np = 0 , \\, \\dfrac{2}{3}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(0 \\lt p \\lt 1\\) \u306b\u304a\u3051\u308b \\(f(p)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a\r\n\\[\r\n\\begin{array}{c|ccccc} p & 0 & \\cdots & \\dfrac{2}{3} & \\cdots & 1 \\\\ \\hline f'(p) & 0 & + & 0 & - & \\\\ \\hline f(p) & 0 & \\nearrow & \\dfrac{4}{27} & \\searrow & 0 \\end{array}\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n0 \\lt p^2(1-p) \\leqq \\dfrac{4}{27} \\quad ... [2]\r\n\\]\r\n\u3088\u3063\u3066, [1] [2] \u3088\u308a\r\n\\[\r\n\\underline{0 \\lt V \\lt \\dfrac{\\pi}{27}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(3\\) \u8fba\u306e\u9577\u3055\u304c \\(a\\) \u3068 \\(b\\) \u3068 \\(c\\) \u306e\u76f4\u65b9\u4f53\u3092, \u9577\u3055\u304c \\(b\\) \u306e \\(1\\) \u8fba\u3092\u56de\u8ee2\u8ef8\u3068\u3057\u3066 \\(90^{\\circ}\\) \u56de\u8ee2\u3055\u305b\u308b\u3068\u304d, \u76f4\u65b9\u4f53\u304c\u901a\u904e\u3059\u308b\u70b9\u5168\u4f53\u304c\u3064\u304f\u308b\u7acb\u4f53\u3092 … \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":[19],"tags":[139,14],"class_list":["post-91","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2010","tag-tokyo_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/91","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=91"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/91\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=91"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=91"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=91"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}