{"id":314,"date":"2012-02-24T01:14:11","date_gmt":"2012-02-23T16:14:11","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=314"},"modified":"2021-10-30T15:24:39","modified_gmt":"2021-10-30T06:24:39","slug":"wsr200805","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr200805\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52008\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(xyz\\) \u5ea7\u6a19\u7a7a\u9593\u306b\u304a\u3044\u3066, \u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u7403\u9762 S \u4e0a\u306b\u70b9 N \\((0,0,1)\\) \u3092\u3068\u308b.\r\n\u307e\u305f \\(0 \\lt \\theta \\lt \\pi\\) \u3092\u307f\u305f\u3059 \\(\\theta\\) \u306b\u5bfe\u3057, \u6b21\u306e \\(2\\) \u3064\u306e\u6761\u4ef6<\/p>\r\n

    \r\n

  1. (a)\u3000\\(\\text{NP} = \\text{NQ}\\)<\/p><\/li>\r\n

  2. (b)\u3000\\(\\angle \\text{PNQ} = \\theta\\)<\/p><\/li>\r\n<\/ol>\r\n

    \u3092\u307f\u305f\u3059 S \u4e0a\u306e\u52d5\u70b9 P , Q \u306b\u3064\u3044\u3066, \u7dda\u5206 PQ \u304c\u901a\u904e\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u56f3\u5f62 T \u3092\u8003\u3048\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\u70b9 P \u3068\u70b9 Q \u306e \\(z\\) \u5ea7\u6a19\u306f\u7b49\u3057\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\u70b9 P \u304c\u5e73\u9762 \\(z=h\\) \u4e0a\u306b\u3042\u308b\u3068\u304d, \u7dda\u5206 PQ \u306e\u9577\u3055\u3092 \\(\\theta\\) \u3068 \\(h\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

    3. (3)<\/strong>\u3000T \u3092\u5e73\u9762 \\(z=h\\) \u3067\u5207\u3063\u305f\u3068\u304d\u306e\u65ad\u9762\u306e\u6982\u5f62\u3092\u63cf\u304d, \u305d\u306e\u9762\u7a4d\u3092 \\(\\theta\\) \u3068 \\(h\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

    4. (4)<\/strong>\u3000\\(h\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306b\u6ce8\u610f\u3057\u3066, T \u306e\u4f53\u7a4d \\(V\\) \u3092 \\(\\theta\\) \u3067\u8868\u3057, \\(\\theta\\) \u3092\u52d5\u304b\u3057\u305f\u3068\u304d\u306e \\(V\\) \u306e\u6700\u5927\u5024\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\"waseda_r_2008_05_01\"\r\n

      \u25b3ONP \u3068 \u25b3ONQ \u306b\u3064\u3044\u3066, OP \u306f\u5171\u901a, OP=OQ , NP=NQ \u306a\u306e\u3067\r\n\\[\r\n\\triangle \\text{ONP} \\equiv \\triangle \\text{ONQ} \\quad ... [1]\r\n\\]\r\nP , Q \u304b\u3089 ON \uff08\u3059\u306a\u308f\u3061 \\(z\\) \u8ef8\uff09\u306b\u4e0b\u308d\u3057\u305f\u5782\u7dda\u306e\u8db3\u3092 \\(\\text{H} _ 1 , \\text{H} _ 2\\) \u3068\u304a\u3051\u3070, [1] \u3088\u308a\r\n\\[\r\n\\text{OH} _ 1 = \\text{OH} _ 2\r\n\\]\r\n\u3088\u3063\u3066, \u70b9 P \u3068 Q \u306e \\(z\\) \u5ea7\u6a19\u306f\u7b49\u3057\u3044.<\/p>\r\n

      (2)<\/strong><\/p>\r\n\"waseda_r_2008_05_02\"\r\n

      H \\((0,0,h)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\text{HN} & = 1-h , \\\\\r\n\\text{HP} & = \\text{HQ} = \\sqrt{1-h^2} , \\\\\r\n\\text{NP} & = \\text{NQ} = \\sqrt{\\text{HN}^2 +\\text{HP}^2} \\\\\r\n& = \\sqrt{(1-h)^2+(1-h^2)} = \\sqrt{2(1-h)}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\text{PQ} = 2 \\text{NP} \\sin \\dfrac{\\theta}{2} = \\underline{2 \\sqrt{2(1-h)} \\sin \\dfrac{\\theta}{2}}\r\n\\]\r\n

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

      \\(\\text{HP} = R\\) , \u70b9 H \u3068\u7dda\u5206 PQ \u306e\u8ddd\u96e2\u3092 \\(r\\) \u3068\u304a\u304f\u3068, \u65ad\u9762\u306e\u6982\u5f62\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"waseda_r_2008_05_03\"\r\n

      \u3053\u3053\u3067, PQ \u306e\u4e2d\u70b9\u3092 M \u3068\u304a\u3051\u3070\r\n\\[\r\nr = \\sqrt{\\text{HP}^2 -\\text{MP}^2}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\pi \\left( R^2 -r^2 \\right) = \\pi \\text{MP}^2 \\\\\r\n& = 2 \\pi (1-h) \\sin^2 \\dfrac{\\theta}{2} \\\\\r\n& = \\underline{\\pi (1-h)( 1-\\cos \\theta )}\r\n\\end{align}\\]\r\n

      (4)<\/strong><\/p>\r\n\"waseda_r_2008_05_04\"\r\n

      \\(h\\) \u304c\u6700\u5c0f\u3068\u306a\u308b\u306e\u306f, \u7dda\u5206 PQ \u304c\u70b9 N \u3092\u901a\u308b\u534a\u5f84 \\(1\\) \u306e\u65ad\u9762\u4e0a\u306b\u3042\u308b\u3068\u304d.
      \r\n\u3053\u306e\u3068\u304d\u306e P \u306e \\(z\\) \u5ea7\u6a19\u306f, \\(- \\cos \\theta\\) \u3067\u3042\u308b\u304b\u3089\r\n\\[\r\n-\\cos \\theta \\leqq h \\leqq 1\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, T\u306e\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = \\displaystyle\\int _ {-\\cos \\theta}^1 \\pi (1-h)( 1-\\cos \\theta ) \\, dh \\\\\r\n& = \\pi ( 1-\\cos \\theta ) \\left[ h -\\dfrac{h^2}{2} \\right] _ {- \\cos \\theta}^1 \\\\\r\n& = \\dfrac{\\pi}{2} ( 1-\\cos \\theta ) \\left( 1 +2\\cos \\theta +\\cos^2 \\theta \\right) \\\\\r\n& = \\dfrac{\\pi}{2} \\left( 1 +\\cos \\theta -\\cos^2 \\theta -\\cos^3 \\theta \\right)\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 \\(c =\\cos \\theta \\ ( -1 \\lt c \\lt 1 )\\) , \\(f(c) = 1+c-c^2-c^3\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nf'(c) = 1-2c-3c^2 = -(3c-1)(c+1)\r\n\\]\r\n\\(f'(c) =0\\) \u3092\u3068\u304f\u3068, \\(c =\\dfrac{1}{3}\\) .
      \r\n\u3057\u305f\u304c\u3063\u3066, \\(f(c)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} c & (-1) & \\cdots & \\dfrac{1}{3} & \\cdots & (1) \\\\ \\hline f'(c) & & + & 0 & - & \\\\ \\hline f(c) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3086\u3048\u306b \\(f(c)\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\nf \\left( \\dfrac{1}{3} \\right) = 1+\\dfrac{1}{3} -\\dfrac{1}{9} -\\dfrac{1}{27} = \\dfrac{32}{27}\r\n\\]\r\n\u3088\u3063\u3066, \\(V\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{\\pi}{2} f \\left( \\dfrac{1}{3} \\right) = \\underline{\\dfrac{16 \\pi}{27}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u5ea7\u6a19\u7a7a\u9593\u306b\u304a\u3044\u3066, \u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u7403\u9762 S \u4e0a\u306b\u70b9 N \\((0,0,1)\\) \u3092\u3068\u308b. \u307e\u305f \\(0 \\lt \\theta \\lt \\pi\\) \u3092\u307f\u305f\u3059 \\(\\theta\\) \u306b\u5bfe … \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":[54],"tags":[147,16],"class_list":["post-314","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2008","tag-waseda_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/314","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=314"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/314\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=314"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=314"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}