{"id":9794,"date":"2012-10-02T16:09:20","date_gmt":"2012-10-02T15:09:20","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9794"},"modified":"2022-01-20T18:35:28","modified_gmt":"2022-01-20T18:35:28","slug":"relacao-entre-os-volumes-de-dois-cubos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9794","title":{"rendered":"Rela\u00e7\u00e3o entre os volumes de dois cubos"},"content":{"rendered":"<p><ul id='GTTabs_ul_9794' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9794' class='GTTabs_curr'><a  id=\"9794_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9794' ><a  id=\"9794_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_9794'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Qual \u00e9 a rela\u00e7\u00e3o entre os volumes de dois cubos, sabendo que a aresta de um \u00e9 o triplo da do outro?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9794' onClick='GTTabs_show(1,9794)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9794'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9797\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9797\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo1.jpg\" data-orig-size=\"72,83\" 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=\"Cubo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo1.jpg\" class=\"aligncenter size-full wp-image-9797\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo1.jpg\" alt=\"\" width=\"72\" height=\"83\" \/><\/a><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo3.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9798\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9798\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo3.jpg\" data-orig-size=\"219,253\" 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=\"Cubo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo3.jpg\" class=\"aligncenter size-full wp-image-9798\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo3.jpg\" alt=\"\" width=\"219\" height=\"253\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo3.jpg 219w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/cubo3-129x150.jpg 129w\" sizes=\"auto, (max-width: 219px) 100vw, 219px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Cubo<\/strong><\/td>\n<td style=\"text-align: center;\">A<\/td>\n<td style=\"text-align: center;\">B<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Volume<\/strong><\/td>\n<td style=\"text-align: center;\">${V_1}$<\/td>\n<td style=\"text-align: center;\">${V_2}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Comprimento da aresta<\/strong><\/td>\n<td style=\"text-align: center;\">${a_1} = a$<\/td>\n<td style=\"text-align: center;\">${a_2} = 3a$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Os cubos A e B s\u00e3o semelhantes (o cubo B uma amplia\u00e7\u00e3o do cubo A), sendo a\u00a0raz\u00e3o de semelhan\u00e7a:<\/p>\n<p>$$r = \\frac{{{a_2}}}{{a{}_1}} = \\frac{{3a}}{a} = 3$$<\/p>\n<p>Quanto \u00e0 raz\u00e3o entre os seus volumes, temos:<\/p>\n<p>$$\\frac{{{V_2}}}{{{V_1}}} = \\frac{{{{\\left( {{a_2}} \\right)}^3}}}{{{{\\left( {a{}_1} \\right)}^3}}} = \\frac{{{{\\left( {3a} \\right)}^3}}}{{{{\\left( a \\right)}^3}}} = {\\left( {\\frac{{3a}}{a}} \\right)^3} = {3^3} = 27$$<\/p>\n<\/p>\n<p>De um modo geral [extensivo a outros s\u00f3lidos e independentemente de ser uma amplia\u00e7\u00e3o ($r &gt; 1$)\u00a0ou redu\u00e7\u00e3o ($0 &lt; r &lt; 1$)], tem-se:<\/p>\n<blockquote>\n<p style=\"text-align: center;\">Rela\u00e7\u00e3o entre os seus volumes:<\/p>\n<p>$$\\frac{{{V_2}}}{{{V_1}}} = {r^3}$$<\/p>\n<p style=\"text-align: center;\">Rela\u00e7\u00e3o entre as \u00e1reas das suas faces:<\/p>\n<p>$$\\frac{{{A_2}}}{{{A_1}}} = {r^2}$$<\/p>\n<p style=\"text-align: center;\">Rela\u00e7\u00e3o entre o comprimento das suas arestas:<\/p>\n<p>$$\\frac{{{a_2}}}{{a{}_1}} = r$$<\/p>\n<\/blockquote>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9794' onClick='GTTabs_show(0,9794)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Qual \u00e9 a rela\u00e7\u00e3o entre os volumes de dois cubos, sabendo que a aresta de um \u00e9 o triplo da do outro? Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":20760,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,430,323,109],"series":[],"class_list":["post-9794","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-modulo-inicial","tag-10-o-ano","tag-modulo-inicial","tag-razao-de-semelhanca","tag-volume"],"views":2788,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag033-2_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9794","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=9794"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9794\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20760"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9794"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9794"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9794"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}