{"id":188,"date":"2011-12-02T22:58:32","date_gmt":"2011-12-02T13:58:32","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=188"},"modified":"2021-09-10T20:59:08","modified_gmt":"2021-09-10T11:59:08","slug":"tok200904","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok200904\/","title":{"rendered":"\u6771\u5de5\u59272009\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(xyz\\) \u7a7a\u9593\u306e\u539f\u70b9\u3068\u70b9 \\((1, 1, 1)\\) \u3092\u901a\u308b\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\ell\\) \u4e0a\u306e\u70b9 \\(\\left( \\dfrac{t}{3} , \\dfrac{t}{3} , \\dfrac{t}{3} \\right)\\) \u3092\u901a\u308a \\(\\ell\\) \u3068\u5782\u76f4\u306a\u5e73\u9762\u304c, \\(xy\\) \u5e73\u9762\u3068\u4ea4\u308f\u3063\u3066\u3067\u304d\u308b\u76f4\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u4e0d\u7b49\u5f0f \\(0 \\leqq y \\leqq x(1-x)\\) \u306e\u8868\u3059 \\(xy\\) \u5e73\u9762\u5185\u306e\u9818\u57df\u3092 \\(D\\) \u3068\u3059\u308b. \\(\\ell\\) \u3092\u8ef8\u3068\u3057\u3066 \\(D\\) \u3092\u56de\u8ee2\u3055\u305b\u3066\u5f97\u3089\u308c\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

    \\(P \\, \\left( \\dfrac{t}{3} , \\dfrac{t}{3} , \\dfrac{t}{3} \\right)\\) \u3092\u901a\u308a, \\(\\ell\\) \u306b\u5782\u76f4\u306a\u5e73\u9762\u3092 \\(H\\) \u3068\u304a\u304f.
    \r\n\\(H\\) \u4e0a\u306e\u70b9\u3092 \\(( x , y, z )\\) \u3068\u304a\u3051\u3070,\r\n\\[\\begin{gather}\r\n1 \\cdot \\left( x-\\dfrac{t}{3} \\right) +1 \\cdot \\left( y-\\dfrac{t}{3} \\right) +1 \\cdot \\left( z-\\dfrac{t}{3} \\right) = 0 \\\\\r\n\\text{\u2234} \\quad x+y+z = t\r\n\\end{gather}\\]\r\n\\(xy\\) \u5e73\u9762\u306f \\(z = 0\\) \u306a\u306e\u3067, \u4ee3\u5165\u3059\u308c\u3070, \u6c42\u3081\u308b\u65b9\u7a0b\u5f0f\u306f\r\n\\[\r\n\\underline{x+y=t} \\quad ... [1]\r\n\\]\r\n

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

    \\(C : \\ y = x(1-x)\\) \u3088\u308a, \\(y' = 1-2x\\) .
    \r\n\\(x = 1\\) \u306e\u3068\u304d\u306f, \\(y' = -1\\) .
    \r\n\u3057\u305f\u304c\u3063\u3066, \u9818\u57df \\(D\\) \u3068\u76f4\u7dda [1] \u304c\u5171\u6709\u70b9\u3092\u3082\u3064\u306e\u306f, \\(0 \\leqq t \\leqq 1\\) \u306e\u3068\u304d.
    \r\n\u3053\u306e\u3068\u304d, \u5e73\u9762 \\(H\\) \u306b\u3088\u308b\u56de\u8ee2\u4f53\u306e\u65ad\u9762\u7a4d \\(S(t)\\) \u3068\u304a\u304f.
    \r\n[1] \u3068 \\(C\\) \u306e\u4ea4\u70b9\u3092 \\(Q\\) , [1] \u3068 \\(y=0\\) \u3068\u306e\u4ea4\u70b9\u3092 \\(R\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nS(t) = \\pi \\left( PR^2 -PQ^2 \\right)\r\n\\]\r\n\u3053\u3053\u3067, [1] \u3068 \\(C\\) \u306e\u5f0f\u3088\u308a\r\n\\[\\begin{align}\r\nx(1-x) & = t-x \\\\\r\nx^2-2x+t & = 0 \\\\\r\n\\text{\u2234} \\quad x = 1 & -\\sqrt{1-t} \\quad ( \\ \\text{\u2235} \\ 0 \\leqq x \\leqq 1 ) \\\\\r\n\\text{\u2234} \\quad y = t & -\\left( 1 -\\sqrt{1-t} \\right) = t-1 +\\sqrt{1-t}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(Q \\ \\left( 1 -\\sqrt{1-t} , t-1 +\\sqrt{1-t} , 0 \\right)\\) .
    \r\n\u307e\u305f, \\(R \\, ( t , 0 , 0 )\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nPQ^2 & = \\left( 1 -\\dfrac{t}{3} -\\sqrt{1-t} \\right)^2 +\\left( \\dfrac{2t}{3} -1+\\sqrt{1-t} \\right)^2 +\\left( -\\dfrac{t}{3} \\right)^2 \\\\\r\n& = \\left( 1-\\dfrac{t}{3} \\right)^2 -\\left( 2 -\\dfrac{2t}{3} \\right) \\sqrt{1-t} +(1-t) \\\\\r\n& \\qquad +\\left( \\dfrac{2t}{3} -1 \\right)^2 +\\left( \\dfrac{4t}{3} -2 \\right) \\sqrt{1-t} +(1-t) +\\dfrac{t^2}{9} \\\\\r\n& = \\dfrac{2t^2}{3} -4t +4 +(2t-4) \\sqrt{1-t} , \\\\\r\nPR^2 & = \\left( \\dfrac{2t}{3} \\right)^2 +\\left( -\\dfrac{t}{3} \\right)^2 +\\left( -\\dfrac{t}{3} \\right)^2 \\\\\r\n& = \\dfrac{2t^2}{3}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS(t) & = \\pi \\left\\{ -4(1-t) -2(t-2) \\sqrt{1-t} \\right\\} \\\\\r\n& = \\pi \\left\\{ 2 (1-t)^{\\frac{3}{2}} -4(1-t) +2 (1-t)^{\\frac{1}{2}} \\right\\}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(\\ell\\) \u3092\u56de\u8ee2\u8ef8 \\(s\\) \u8ef8\u3068\u3059\u308b\u3068, \u5fae\u5c0f\u90e8\u5206 \\(ds\\) \u306f\r\n\\[\\begin{align}\r\nds^2 & = \\left( \\dfrac{t}{3} \\right)^2 +\\left( \\dfrac{t}{3} \\right)^2 +\\left( \\dfrac{t}{3} \\right)^2 = \\dfrac{dt^2}{3} \\\\\r\n& \\text{\u2234} \\quad ds = \\dfrac{dt}{\\sqrt{3}}\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\\\r\nV & = \\displaystyle\\int _ 0^{\\frac{1}{\\sqrt{3}}} S(t) \\, ds \\\\\r\n& = \\pi \\displaystyle\\int _ 0^1 \\left\\{ 2 (1-t)^{\\frac{3}{2}} -4(1-t) +2 (1-t)^{\\frac{1}{2}} \\right\\} \\dfrac{dt}{\\sqrt{3}} \\\\\r\n& = \\dfrac{\\pi}{\\sqrt{3}} \\left[ -\\dfrac{4}{5} (1-t)^{\\frac{5}{2}} +2(1-t)^2 -\\dfrac{4}{3} (1-t)^{\\frac{3}{2}} \\right] _ 0^1 \\\\\r\n& = \\dfrac{\\pi}{\\sqrt{3}} \\left( \\dfrac{4}{5} -2 +\\dfrac{4}{3} \\right) \\\\\r\n& = \\underline{\\dfrac{2 \\sqrt{3} \\pi}{45}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u7a7a\u9593\u306e\u539f\u70b9\u3068\u70b9 \\((1, 1, 1)\\) \u3092\u901a\u308b\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3059\u308b. (1)\u3000\\(\\ell\\) \u4e0a\u306e\u70b9 \\(\\left( \\dfrac{t}{3} , \\dfrac{t}{3} , \\df … \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":[24],"tags":[141,15],"class_list":["post-188","post","type-post","status-publish","format-standard","hentry","category-toko_2009","tag-toko","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/188","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=188"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/188\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=188"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=188"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}