{"id":7246,"date":"2011-12-04T16:02:04","date_gmt":"2011-12-04T16:02:04","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7246"},"modified":"2021-12-28T01:58:34","modified_gmt":"2021-12-28T01:58:34","slug":"uma-superficie-esferica","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7246","title":{"rendered":"Uma superf\u00edcie esf\u00e9rica"},"content":{"rendered":"<p><ul id='GTTabs_ul_7246' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7246' class='GTTabs_curr'><a  id=\"7246_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7246' ><a  id=\"7246_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_7246'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere, num referencial ortonormado Oxyz, a superf\u00edcie esf\u00e9rica de equa\u00e7\u00e3o ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}=25$.<\/p>\n<p>Considere todos os tri\u00e2ngulos cujos v\u00e9rtices s\u00e3o pontos de interse\u00e7\u00e3o desta superf\u00edcie esf\u00e9rica com os eixos do referencial.<\/p>\n<p>Escolhendo um desses tri\u00e2ngulos ao acaso, determine a probabilidade de estar contido no plano definido por $z=0$.<br \/>\nIndique o resultado em percentagem.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7246' onClick='GTTabs_show(1,7246)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7246'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>\u00a0<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Supesferica.gif\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7247\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7247\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Supesferica.gif\" data-orig-size=\"590,265\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&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=\"Superf\u00edcie esf\u00e9rica\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Supesferica.gif\" class=\"aligncenter size-full wp-image-7247\" title=\"Superf\u00edcie esf\u00e9rica\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Supesferica.gif\" alt=\"\" width=\"590\" height=\"265\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Supesferica.gif 590w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Supesferica-300x134.gif 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Supesferica-150x67.gif 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Supesferica-400x179.gif 400w\" sizes=\"auto, (max-width: 590px) 100vw, 590px\" \/><\/a><\/p>\n<p>Os eixos coordenados intersetam a superf\u00edcie esf\u00e9rica em seis pontos: $A(5,0,0)$, $B(0,5,0)$, $C(-5,0,0)$, $D(0,-5,0)$, $E(0,0,-5)$ e $F(0,0,5)$.<\/p>\n<p>O n\u00famero de tri\u00e2ngulos distintos que se podem obter considerando tr\u00eas desses seis pontos como v\u00e9rtices \u00e9 $NCP={}^{6}{{C}_{3}}=20$.<\/p>\n<p>Apenas $NCF={}^{4}{{C}_{3}}=4$ desses tri\u00e2ngulos est\u00e3o contidos no plano definido por $z=0$: os que possuem v\u00e9rtices pertencentes ao conjunto $\\left\\{ A,B,C,D \\right\\}$.<\/p>\n<p>Logo, a probabilidade pedida \u00e9 $p=\\frac{{}^{4}{{C}_{3}}}{{}^{6}{{C}_{3}}}=\\frac{4}{20}=\\frac{1}{5}$, ou seja, 20%.<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7246' onClick='GTTabs_show(0,7246)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere, num referencial ortonormado Oxyz, a superf\u00edcie esf\u00e9rica de equa\u00e7\u00e3o ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}=25$. Considere todos os tri\u00e2ngulos cujos v\u00e9rtices s\u00e3o pontos de interse\u00e7\u00e3o desta superf\u00edcie esf\u00e9rica com os eixos do referencial. Escolhendo um&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":13847,"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-7246","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":2487,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag030-3.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7246","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=7246"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7246\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/13847"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7246"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7246"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7246"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7246"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}