{"id":356,"date":"2012-04-07T11:43:20","date_gmt":"2012-04-07T02:43:20","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=356"},"modified":"2021-03-10T17:21:56","modified_gmt":"2021-03-10T08:21:56","slug":"tkr201203","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201203\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2012\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\u5ea7\u6a19\u5e73\u9762\u4e0a\u3067 \\(2\\) \u3064\u306e\u4e0d\u7b49\u5f0f\r\n\\[\r\ny \\geqq \\dfrac{1}{2} x^2 , \\quad \\dfrac{x^2}{4} +4y^2 \\leqq \\dfrac{1}{8}\r\n\\]\r\n\u306b\u3088\u3063\u3066\u5b9a\u307e\u308b\u9818\u57df\u3092 \\(S\\) \u3068\u3059\u308b.\r\n\\(S\\) \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092 \\(V _ 1\\) \u3068\u3057, \\(y\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092 \\(V _ 2\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(V _ 1\\) \u3068 \\(V _ 2\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\dfrac{V _ 2}{V _ 1}\\) \u306e\u5024\u3068 \\(1\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u305b\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\(C : \\ y = \\dfrac{1}{2} x^2\\) , \\(D : \\ \\dfrac{x^2}{4} +4y^2 = \\dfrac{1}{8}\\) \u3068\u304a\u304f.
    \r\n\\(2\\) \u5f0f\u304b\u3089, \\(C\\) \u3068 \\(D\\) \u306e\u4ea4\u70b9\u306e \\(y\\) \u5ea7\u6a19\u306f\r\n\\[\\begin{align}\r\n\\dfrac{y}{2} +4y^2 & = \\dfrac{1}{8} \\\\\r\n32y^2+4y-1 & = 0 \\\\\r\n(4y+1)(8y-1) & =0 \\\\\r\n\\text{\u2234} \\quad y & = \\dfrac{1}{8} \\quad \\left( \\text{\u2235} \\ y>0 \\right)\r\n\\end{align}\\]\r\n\u307e\u305f, \\(x\\) \u5ea7\u6a19\u306f \\(C\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\nx =\\pm \\sqrt{2 \\cdot \\dfrac{1}{8}} =\\pm \\dfrac{1}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u9818\u57df \\(S\\) \u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u3082\u542b\u3080\uff09\u3068\u306a\u308b.<\/p>\r\n\"\"\r\n

    \\(y\\) \u8ef8\u306b\u3064\u3044\u3066\u306e\u5bfe\u79f0\u6027\u304b\u3089\r\n\\[\\begin{align}\r\nV _ 1 & = 2 \\pi \\displaystyle\\int _ 0^{\\frac{1}{2}} \\left\\{ \\left( \\dfrac{1}{32} -\\dfrac{x^2}{16} \\right) -\\left( \\dfrac{1}{2} x^2 \\right)^2 \\right\\} dx \\\\\r\n& = \\dfrac{\\pi}{16} \\displaystyle\\int _ 0^{\\frac{1}{2}} ( 1 -2 x^2 -8 x^4 ) \\, dx \\\\\r\n& = \\dfrac{\\pi}{16} \\left[ x -\\dfrac{2 x^3}{3} -\\dfrac{8 x^5}{5} \\right] _ 0^{\\frac{1}{2}} \\\\\r\n& = \\dfrac{\\pi}{64} \\left( 2 -\\dfrac{1}{3} -\\dfrac{1}{5} \\right) \\\\\r\n& =\\underline{\\dfrac{11 \\pi}{480}}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nV _ 2 & = \\pi \\displaystyle\\int _ 0^{\\frac{1}{8}} 2y \\, dy +\\pi \\displaystyle\\int _ {\\frac{1}{8}}^{\\frac{\\sqrt{2}}{8}} \\left( \\dfrac{1}{2} -16y^2 \\right) \\, dy \\\\\r\n& = \\pi \\left[ y^2 \\right] _ 0^{\\frac{1}{8}} +\\pi \\left[ \\dfrac{y}{2} -\\dfrac{16 y^3}{3} \\right] _ {\\frac{1}{8}}^{\\frac{\\sqrt{2}}{8}} \\\\\r\n& = \\dfrac{\\pi}{64} +\\pi \\left\\{ \\left( \\dfrac{\\sqrt{2}}{16} -\\dfrac{\\sqrt{2}}{48} \\right) -\\left( \\dfrac{1}{16} -\\dfrac{1}{96} \\right) \\right\\} \\\\\r\n& = \\dfrac{1}{64} +\\dfrac{\\sqrt{2} \\pi}{24} -\\dfrac{5 \\pi}{96} \\\\\r\n& =\\underline{\\dfrac{8\\sqrt{2} -7}{192} \\pi}\r\n\\end{align}\\]\r\n

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

    \\(1.4 \\lt \\sqrt{2} \\lt 1.5\\) \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{V _ 2}{V _ 1} -1 & = \\dfrac{40 \\sqrt{2} -35}{22} -1 =\\dfrac{40 \\sqrt{2} -57}{22} \\\\\r\n& \\lt \\dfrac{40 \\cdot 1.4 -57}{22} = -\\dfrac{1}{22} \\lt 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\underline{\\dfrac{V _ 2}{V _ 1} \\lt 1}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u4e0a\u3067 \\(2\\) \u3064\u306e\u4e0d\u7b49\u5f0f \\[ y \\geqq \\dfrac{1}{2} x^2 , \\quad \\dfrac{x^2}{4} +4y^2 \\leqq \\dfrac{1}{8} \\] \u306b\u3088\u3063\u3066\u5b9a\u307e\u308b\u9818\u57df\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":[58],"tags":[139,68],"class_list":["post-356","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2012","tag-tokyo_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/356","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=356"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/356\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=356"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=356"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=356"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}