{"id":7228,"date":"2011-12-02T19:21:08","date_gmt":"2011-12-02T19:21:08","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7228"},"modified":"2022-01-25T23:21:56","modified_gmt":"2022-01-25T23:21:56","slug":"um-poliedro-de-nove-faces","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7228","title":{"rendered":"Um poliedro de nove faces"},"content":{"rendered":"<p><ul id='GTTabs_ul_7228' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7228' class='GTTabs_curr'><a  id=\"7228_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7228' ><a  id=\"7228_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7228'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7229\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7229\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg\" data-orig-size=\"298,360\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Poliedro\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg\" class=\"alignright wp-image-7229\" title=\"Poliedro\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg\" alt=\"\" width=\"240\" height=\"290\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg 298w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60-248x300.jpg 248w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60-124x150.jpg 124w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Considere, agora, o poliedro do problema anterior num referencial o. n. Oxyz, de tal forma que:<\/p>\n<ul>\n<li>o v\u00e9rtice O coincida com a origem do referencial;<\/li>\n<li>o v\u00e9rtice N perten\u00e7a ao semieixo positivo Ox;<\/li>\n<li>o v\u00e9rtice P perten\u00e7a ao semieixo positivo Oy.<\/li>\n<\/ul>\n<ol>\n<li>Escolhendo, ao acaso, tr\u00eas dos nove v\u00e9rtices do s\u00f3lido representado, qual \u00e9 a probabilidade de pertencerem \u00e0 mesma face?<\/li>\n<li>Dois amigos escolheram cada um, em segredo, um dos nove v\u00e9rtices do s\u00f3lido.<br \/>\nQual \u00e9 a probabilidade dos v\u00e9rtices escolhidos pertencerem ambos ao plano $y=x$?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7228' onClick='GTTabs_show(1,7228)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7228'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7229\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7229\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg\" data-orig-size=\"298,360\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Poliedro\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg\" class=\"alignright wp-image-7229\" title=\"Poliedro\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg\" alt=\"\" width=\"240\" height=\"290\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60.jpg 298w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60-248x300.jpg 248w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12P177-60-124x150.jpg 124w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Dos nove v\u00e9rtices, podem ser selecionados tr\u00eas deles de\u00a0$NCP={}^{9}{{C}_{3}}=84$ maneiras diferentes.\n<p>Relativamente \u00e0s faces triangulares do s\u00f3lido, existem ${{N}_{1}}=4\\times {}^{3}{{C}_{3}}=4\\times 1=4$ casos favor\u00e1veis, um por face.<\/p>\n<p>Relativamente \u00e0s faces quadrangulares, existem ${{N}_{2}}=5\\times {}^{4}{{C}_{3}}=5\\times 4=20$ casos favor\u00e1veis, 4 por face.<\/p>\n<p>Logo, a probabilidade pedida \u00e9 $$p=\\frac{4+20}{84}=\\frac{2}{7}$$<\/p>\n<\/li>\n<li>Os v\u00e9rtices que pertencem ao plano de equa\u00e7\u00e3o $y=x$ s\u00e3o: O, Q, U, V e S.\n<p>Nada impede que os amigos possam escolher o mesmo v\u00e9rtice. Portanto, o n\u00famero de casos favor\u00e1veis \u00e9 $NCF={}^{5}A{{&#8216;}_{2}}={{5}^{2}}=25$.<\/p>\n<p>O n\u00famero de casos poss\u00edveis \u00e9 $NCP={}^{9}A{{&#8216;}_{2}}=81$.<\/p>\n<p>Logo, a probabilidade pedida \u00e9 $$p=\\frac{{}^{5}A{{&#8216;}_{2}}}{{}^{9}A{{&#8216;}_{2}}}=\\frac{25}{81}$$<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7228' onClick='GTTabs_show(0,7228)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere, agora, o poliedro do problema anterior num referencial o. n. Oxyz, de tal forma que: o v\u00e9rtice O coincida com a origem do referencial; o v\u00e9rtice N perten\u00e7a ao semieixo&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21050,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,255,215],"series":[],"class_list":["post-7228","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-12-o-ano","tag-analise-combinatoria","tag-probabilidade"],"views":5992,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12V1Pag177-60_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7228","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7228"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7228\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21050"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7228"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7228"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7228"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7228"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}