{"id":1382,"date":"2017-02-17T22:04:44","date_gmt":"2017-02-17T13:04:44","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1382"},"modified":"2021-03-09T15:49:41","modified_gmt":"2021-03-09T06:49:41","slug":"tkr201606","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201606\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2016\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\u5ea7\u6a19\u7a7a\u9593\u5185\u3092, \u9577\u3055 \\(2\\) \u306e\u7dda\u5206 AB \u304c\u6b21\u306e \\(2\\) \u6761\u4ef6 (a) , (b) \u3092\u307f\u305f\u3057\u306a\u304c\u3089\u52d5\u304f.<\/p>\r\n

    \r\n

  1. (a)\u3000\u70b9 A \u306f\u5e73\u9762 \\(z=0\\) \u4e0a\u306b\u3042\u308b.<\/p><\/li>\r\n

  2. (b)\u3000\u70b9 C \\(( 0 , 0 , 1 )\\) \u304c\u7dda\u5206 AB \u4e0a\u306b\u3042\u308b.<\/p><\/li>\r\n<\/ol>\r\n

    \u3053\u306e\u3068\u304d, \u7dda\u5206 AB \u304c\u901a\u904e\u3059\u308b\u3053\u3068\u306e\u3067\u304d\u308b\u7bc4\u56f2\u3092 \\(K\\) \u3068\u3059\u308b. \\(K\\) \u3068\u4e0d\u7b49\u5f0f \\(z \\geqq 1\\) \u306e\u8868\u3059\u7bc4\u56f2\u3068\u306e\u5171\u901a\u90e8\u5206\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p>\r\n

    \r\n\r\n

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

    \\(K\\) \u306f \\(z\\) \u8ef8\u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u306e\u3067, \\(xz\\) \u5e73\u9762\u4e0a\u3067, \u7dda\u5206 AB \u306e\u901a\u904e\u9818\u57df\u3092\u8003\u3048\u308b.
    \r\n\\(K\\) \u306e\u3046\u3061, \\(z \\geqq 1\\) \u3092\u307f\u305f\u3059\u90e8\u5206\u306f, \u7dda\u5206 BC \u306e\u901a\u904e\u9818\u57df\u3067\u3042\u308b.
    \r\n\u3055\u3089\u306b, \u70b9 B \u306f\u70b9 D \\(( 0, 0, 2 )\\) \u304b\u3089\u70b9 C \u307e\u3067\u52d5\u304f\u304c, \u3053\u306e\u9593, \\(\\angle \\text{BCD}\\) \u306f\u5358\u8abf\u5897\u52a0, \u7dda\u5206 BC \u306e\u9577\u3055\u306f\u5358\u8abf\u6e1b\u5c11\u3059\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066, \u7dda\u5206 BC \u306e\u901a\u904e\u9818\u57df\u306f, \u70b9 B \u306e\u8ecc\u8de1\u3068 \\(z\\) \u8ef8\u306b\u56f2\u307e\u308c\u305f\u90e8\u5206\u3068\u306a\u308a, \u3053\u308c\u306e \\(z\\) \u8ef8\u306b\u3088\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
    \r\n\\(\\angle \\text{OCA} = \\theta\\) \u3068\u304a\u304f\u3068, \\(0 \\leqq \\theta \\leqq \\dfrac{\\pi}{3}\\) \u3067\u3042\u308a\r\n\\[\r\n\\text{AC} = \\dfrac{1}{\\cos \\theta} , \\quad \\text{BC} = 2 -\\dfrac{1}{\\cos \\theta}\r\n\\]\r\n\u70b9 B \u304b\u3089 \\(z\\) \u8ef8\u306b\u4e0b\u308d\u3057\u305f\u5782\u7dda\u306e\u8db3\u3092\u70b9 D \u3068\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\text{CD} & = \\text{BC} \\cos \\theta = 2 \\cos \\theta -1 \\\\\r\n\\text{BD} & = \\text{BC} \\sin \\theta = 2 \\sin \\theta -\\tan \\theta\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u70b9 B \u306e\u5ea7\u6a19\u3092 \\(( X , Z )\\) \u3068\u3059\u308c\u3070\r\n\\[\r\n( X , Z ) = \\left( 2 \\sin \\theta -\\tan \\theta , 2 \\cos \\theta \\right)\r\n\\]\r\n\u3042\u3089\u305f\u3081\u3066, \\(t = \\cos \\theta\\) \u3068\u304a\u304f\u3068, \\(\\dfrac{1}{2} \\leqq t \\leqq 1\\) \u3067\r\n\\[\r\n( X , Z ) = \\left( \\dfrac{( 2t -1 ) \\sqrt{1 -t^2}}{t} , 2t \\right)\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = \\pi \\displaystyle\\int _ {1}^{2} X^2 \\, dZ \\\\\r\n& = \\pi \\displaystyle\\int _ {\\frac{1}{2}}^{1} \\dfrac{( 2t-1 )^2 (1 -t^2)}{t^2} \\cdot 2 \\, dt \\\\\r\n& = 2 \\pi \\displaystyle\\int _ {\\frac{1}{2}}^{1} \\left( -4t^2 +4t +3 -\\dfrac{4}{t} +\\dfrac{1}{t^2} \\right) \\, dt \\\\\r\n& = 2 \\pi \\left[ -\\dfrac{4 t^3}{3} +2t^2 +3t -4 \\log t -\\dfrac{1}{t} \\right] _ {\\frac{1}{2}}^{1} \\\\\r\n& = 2 \\pi \\left\\{ \\dfrac{8}{3} -\\left( -\\dfrac{1}{6} +4 \\log 2 \\right) \\right\\} \\\\\r\n& = \\underline{\\left( \\dfrac{17}{3} -8 \\log 2 \\right) \\pi}\r\n\\end{align}\\]","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u7a7a\u9593\u5185\u3092, \u9577\u3055 \\(2\\) \u306e\u7dda\u5206 AB \u304c\u6b21\u306e \\(2\\) \u6761\u4ef6 (a) , (b) \u3092\u307f\u305f\u3057\u306a\u304c\u3089\u52d5\u304f. (a)\u3000\u70b9 A \u306f\u5e73\u9762 \\(z=0\\) \u4e0a\u306b\u3042\u308b. (b)\u3000\u70b9 C \\(( 0 , 0 , 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":[150],"tags":[139,162],"class_list":["post-1382","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2016","tag-tokyo_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1382","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=1382"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1382\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1382"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1382"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1382"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}