{"id":4976,"date":"2010-10-31T18:51:28","date_gmt":"2010-10-31T18:51:28","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=4976"},"modified":"2022-01-19T14:39:09","modified_gmt":"2022-01-19T14:39:09","slug":"dois-cones-com-a-mesma-base","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=4976","title":{"rendered":"Dois cones com a mesma base"},"content":{"rendered":"<p><ul id='GTTabs_ul_4976' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_4976' class='GTTabs_curr'><a  id=\"4976_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_4976' ><a  id=\"4976_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_4976'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4977\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4977\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone.jpg\" data-orig-size=\"287,465\" 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=\"Duplo cone\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone.jpg\" class=\"alignright wp-image-4977\" title=\"Duplo cone\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone-185x300.jpg\" alt=\"\" width=\"200\" height=\"324\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone-185x300.jpg 185w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone-92x150.jpg 92w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone.jpg 287w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/><\/a>Na figura est\u00e3o representados dois cones com a mesma base.<\/p>\n<p>[ABCD] \u00e9 um losango de 36 cm<sup>2<\/sup> de \u00e1rea e cuja diagonal menor mede metade da diagonal maior.<\/p>\n<ol>\n<li>Determina a medida das diagonais do losango.<\/li>\n<li>Determina o volume do s\u00f3lido.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_4976' onClick='GTTabs_show(1,4976)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_4976'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"float: right;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":204,\r\n\"height\":480,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/p>\n<p>Consideremos o losango [ABCD] inscrito num ret\u00e2ngulo, cujos pontos m\u00e9dios dos lados s\u00e3o os v\u00e9rtices do losango.<\/p>\n<p>As diagonais e os lados do losango dividem o ret\u00e2ngulo em 8 tri\u00e2ngulos ret\u00e2ngulos geometricamente iguais (Porqu\u00ea).<\/p>\n<p>Conclui-se, por isso, que a \u00e1rea do losango \u00e9 metade da \u00e1rea do ret\u00e2ngulo, isto \u00e9:<\/p>\n<p style=\"text-align: center;\">${{A}_{L}}=\\frac{1}{2}\\times {{A}_{R}}=\\frac{D\\times d}{2}$.<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"4977\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=4977\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone.jpg\" data-orig-size=\"287,465\" 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=\"Duplo cone\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone.jpg\" class=\"alignright wp-image-4977\" title=\"Duplo cone\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone-185x300.jpg\" alt=\"\" width=\"200\" height=\"324\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone-185x300.jpg 185w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone-92x150.jpg 92w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/duplocone.jpg 287w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/>Designando o comprimento da diagonal menor por d, ent\u00e3o o comprimento da diagonal maior pode ser expressa por 2d.\n<p>Substituindo na igualdade acima, obt\u00e9m-se: $36=\\frac{2d\\times d}{2}$.<\/p>\n<p>Resolvendo a equa\u00e7\u00e3o, vem: $36=\\frac{2d\\times d}{2}\\Leftrightarrow {{d}^{2}}=36\\Leftrightarrow d=6$.<\/p>\n<p>Logo, $\\overline{AC}=6\\,cm$ e $\\overline{BD}=12\\,cm$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O volume de um dos cones \u00e9 ${{V}_{C}}=\\frac{1}{3}\\times \\pi \\times {{3}^{2}}\\times 6=18\\pi \\,c{{m}^{3}}$.\n<\/p>\n<p>Logo, o volume do s\u00f3lido \u00e9 ${{V}_{S}}=2\\times {{V}_{C}}=2\\times 18\\pi =36\\pi \\,c{{m}^{3}}$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_4976' onClick='GTTabs_show(0,4976)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e3o representados dois cones com a mesma base. [ABCD] \u00e9 um losango de 36 cm2 de \u00e1rea e cuja diagonal menor mede metade da diagonal maior. Determina a medida&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20638,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,102],"tags":[424,67,109],"series":[],"class_list":["post-4976","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-do-espaco-ao-plano","tag-8-o-ano","tag-geometria","tag-volume"],"views":3153,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/10\/7V2Pag117-5_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4976","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=4976"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/4976\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20638"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4976"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4976"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4976"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=4976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}