{"id":390,"date":"2012-05-05T10:30:04","date_gmt":"2012-05-05T01:30:04","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=390"},"modified":"2021-09-25T13:55:08","modified_gmt":"2021-09-25T04:55:08","slug":"osr201203","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201203\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2012\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(xyz\\) \u7a7a\u9593\u306b \\(3\\) \u70b9 O \\((0,0,0)\\) , A \\((1,0,1)\\) , B \\((0,\\sqrt{3},1)\\) \u304c\u3042\u308b. \u5e73\u9762 \\(z=0\\) \u306b\u542b\u307e\u308c, \u4e2d\u5fc3\u304c O , \u534a\u5f84\u304c \\(1\\) \u306e\u5186\u3092 \\(W\\) \u3068\u3059\u308b.\r\n\u70b9 P \u304c\u7dda\u5206 OA \u4e0a\u3092, \u70b9 Q \u304c\u5186 \\(W\\) \u306e\u5468\u304a\u3088\u3073\u5185\u90e8\u3092\u52d5\u304f\u3068\u304d, \\(\\overrightarrow{\\text{OR}} =\\overrightarrow{\\text{OP}} +\\overrightarrow{\\text{OQ}}\\) \u3092\u307f\u305f\u3059\u70b9R\u5168\u4f53\u304c\u3064\u304f\u308b\u7acb\u4f53\u3092 \\(V _ A\\) \u3068\u304a\u304f.\r\n\u540c\u69d8\u306b\u70b9 P \u304c\u7dda\u5206 OB \u4e0a\u3092, \u70b9 Q \u304c\u5186 \\(W\\) \u306e\u5468\u304a\u3088\u3073\u5185\u90e8\u3092\u52d5\u304f\u3068\u304d, \\(\\overrightarrow{\\text{OR}} =\\overrightarrow{\\text{OP}} +\\overrightarrow{\\text{OQ}}\\) \u3092\u307f\u305f\u3059\u70b9 R \u5168\u4f53\u304c\u3064\u304f\u308b\u7acb\u4f53\u3092 \\(V _ B\\) \u3068\u304a\u304f.\r\n\u3055\u3089\u306b \\(V _ A\\) \u3068 \\(V _ B\\) \u306e\u91cd\u306a\u308a\u5408\u3046\u90e8\u5206\u3092 \\(V\\) \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\u5e73\u9762 \\(z =\\cos \\theta \\ \\left( 0 \\leqq \\theta \\leqq \\dfrac{\\pi}{2} \\right)\\) \u306b\u3088\u308b\u7acb\u4f53 \\(V\\) \u306e\u5207\u308a\u53e3\u306e\u9762\u7a4d\u3092 \\(\\theta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u7acb\u4f53 \\(V\\) \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 \u304c OA \u4e0a\u3092\u52d5\u304d, \\(z =\\cos \\theta\\) \u306e\u3068\u304d\u306e\u70b9\u3092 \\(\\text{P} _ A\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\overrightarrow{\\text{OP} _ A} =\\left( \\begin{array}{c} \\cos \\theta \\\\ 0 \\\\ \\cos \\theta \\end{array} \\right)\r\n\\]\r\nP \u304c OB \u4e0a\u3092\u52d5\u304d, \\(z =\\cos \\theta\\) \u306e\u3068\u304d\u306e\u70b9\u3092 \\(\\text{P} _ B\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\overrightarrow{\\text{OP} _ B} =\\left( \\begin{array}{c} 0 \\\\ \\sqrt{3} \\cos \\theta \\\\ \\cos \\theta \\end{array} \\right)\r\n\\]\r\n\\(\\overrightarrow{\\text{OQ}}\\) \u304c\u3064\u304f\u308b\u9818\u57df\u306f \\(W\\) \u3068\u540c\u3058\u5f62, \u3064\u307e\u308a \\(z\\) \u8ef8\u306b\u5782\u76f4\u306a\u534a\u5f84 \\(1\\) \u306e\u5186\u3067\u3042\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066, \\(z =\\cos \\theta\\) \u306b\u3088\u308b \\(V _ A , V _ B\\) \u306e\u5207\u308a\u53e3\u306f, \\(\\text{P} _ A\\) , \\(\\text{P} _ B\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u3068\u306a\u308b.
    \r\n\\(\\text{P} _ A \\text{P} _ B = 2\\cos \\theta \\leqq 2\\) \u306a\u306e\u3067, \\(2\\) \u3064\u306e\u5186\u306f\u5fc5\u305a\u4ea4\u308f\u308a, \\(V\\) \u306e\u5207\u308a\u53e3\u306f\u4e0b\u56f3\u306e\u659c\u7dda\u90e8\uff08\u5883\u754c\u3092\u542b\u3080\uff09\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n

    \\(2\\) \u5186\u306e\u4ea4\u70b9\u306e\u4e00\u65b9\u3092T\u3068\u304a\u304f\u3068, \\(\\angle \\text{TP} _ A\\text{P} _ B =\\theta\\) \u306a\u306e\u3067, \u6c42\u3081\u308b\u9762\u7a4d\u3092 \\(S\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nS & =2 \\left( \\dfrac{1}{2} \\cdot 1^2 \\cdot 2 \\theta -\\dfrac{1}{2} \\cdot 1^2 \\sin 2 \\theta \\right) \\\\\r\n& =\\underline{2 \\theta -\\sin 2 \\theta}\r\n\\end{align}\\]\r\n

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

    \\(z = \\cos \\theta\\) \u3088\u308a\r\n\\[\\begin{gather}\r\n\\dfrac{dz}{d \\theta} = -\\sin \\theta , \\\\\r\n\\begin{array}{c|ccc} z & 0 & \\rightarrow & 1 \\\\ \\hline \\theta & \\dfrac{\\pi}{2} & \\rightarrow & 0 \\end{array}\r\n\\end{gather}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nV & = \\displaystyle\\int _ 0^1 S \\, dz \\\\\r\n& = \\displaystyle\\int _ 0^{\\frac{\\pi}{2}} ( 2 \\theta -\\sin 2 \\theta ) \\sin \\theta \\, d \\theta \\\\\r\n& = -\\displaystyle\\int _ 0^{\\frac{\\pi}{2}} 2 \\sin^2 \\theta ( \\sin \\theta )' \\, d \\theta \\\\\r\n& \\qquad +2 \\left[ \\theta \\cos \\theta \\right] _ 0^{\\frac{\\pi}{2}} -2 \\displaystyle\\int _ 0^{\\frac{\\pi}{2}} \\cos \\theta \\, d \\theta \\\\\r\n& = -\\left[ \\dfrac{2}{3} \\sin^3 \\theta \\right] _ 0^{\\frac{\\pi}{2}} -2 \\left[ \\sin \\theta \\right] _ 0^{\\frac{\\pi}{2}} \\\\\r\n& = -\\dfrac{2}{3} +2 = \\underline{\\dfrac{4}{3}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u7a7a\u9593\u306b \\(3\\) \u70b9 O \\((0,0,0)\\) , A \\((1,0,1)\\) , B \\((0,\\sqrt{3},1)\\) \u304c\u3042\u308b. \u5e73\u9762 \\(z=0\\) \u306b\u542b\u307e\u308c, \u4e2d\u5fc3\u304c O , \u534a\u5f84\u304c \\(1 … \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":[60],"tags":[142,68],"class_list":["post-390","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2012","tag-osaka_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/390","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=390"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/390\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=390"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=390"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=390"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}