{"id":13796,"date":"2018-03-05T01:02:19","date_gmt":"2018-03-05T01:02:19","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13796"},"modified":"2022-01-06T15:01:03","modified_gmt":"2022-01-06T15:01:03","slug":"um-cone-reto","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13796","title":{"rendered":"Um cone reto"},"content":{"rendered":"<p><ul id='GTTabs_ul_13796' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13796' class='GTTabs_curr'><a  id=\"13796_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13796' ><a  id=\"13796_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_13796'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/cone.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13797\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13797\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/cone.png\" data-orig-size=\"520,245\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Cone\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/cone.png\" class=\"alignright size-medium wp-image-13797\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/cone-300x141.png\" alt=\"\" width=\"300\" height=\"141\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/cone-300x141.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/cone-720x340.png 720w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/cone.png 520w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Um cone reto tem 1256 cm<sup>2<\/sup> de \u00e1rea de superf\u00edcie e a sua geratriz \u00e9 tripla do raio da base.<\/p>\n<p>Qual \u00e9 a medida do comprimento, arredondado \u00e0s unidades:<\/p>\n<ol>\n<li>o raio da base do cone?<\/li>\n<li>da geratriz do cone?<\/li>\n<li>da altura do cone?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13796' onClick='GTTabs_show(1,13796)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13796'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag28-3.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13798\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13798\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag28-3.png\" data-orig-size=\"1013,1269\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Planifica\u00e7\u00e3o do cone\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag28-3-817x1024.png\" class=\"alignright wp-image-13798\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag28-3-239x300.png\" alt=\"\" width=\"320\" height=\"401\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag28-3-239x300.png 239w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag28-3-768x962.png 768w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag28-3-817x1024.png 817w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/2017-18-MF9P2-pag28-3.png 1013w\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" \/><\/a>Comecemos por determinar uma express\u00e3o da \u00e1rea da superf\u00edcie lateral do cone em fun\u00e7\u00e3o da raio da base.<br \/>\nOra, a superf\u00edcie lateral do cone \u00e9 um setor circular de raio [BV], cuja \u00e1rea \u00e9 diretamente proporcional ao comprimento do arco BAB&#8217;:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\frac{{{A_L}}}{{2\\pi \\times r}} = \\frac{{\\pi \\times {g^2}}}{{2\\pi \\times g}}}&amp; \\Leftrightarrow &amp;{\\frac{{{A_L}}}{{2\\pi \\times r}} = \\frac{{\\pi \\times {{\\left( {3r} \\right)}^2}}}{{2\\pi \\times 3r}}}\\\\{}&amp; \\Leftrightarrow &amp;{\\frac{{{A_L}}}{{2\\pi \\times r}} = \\frac{{3r}}{2}}\\\\{}&amp; \\Leftrightarrow &amp;{{A_L} = 3\\pi \\times {r^2}}\\end{array}\\]<\/p>\n<p>A \u00e1rea da superf\u00edcie total do cone pode ser expressa, em fun\u00e7\u00e3o de r, por:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{A_T}}&amp; = &amp;{{A_B} + {A_L}}\\\\{}&amp; = &amp;{\\pi \\times {r^2} + 3\\pi \\times {r^2}}\\\\{}&amp; = &amp;{4\\pi \\times {r^2}}\\end{array}\\]<\/p>\n<ol>\n<li>O comprimento do raio da base do cone \u00e9, aproximadamente, 10 cm:<br \/>\n\\[\\begin{array}{*{20}{l}}{{A_T} = 1256}&amp; \\Leftrightarrow &amp;{4\\pi \\times {r^2} = 1256}\\\\{}&amp; \\Leftrightarrow &amp;{r = \\sqrt {\\frac{{314}}{\\pi }} }\\\\{}&amp;{}&amp;{r \\approx 10}\\end{array}\\]<\/li>\n<li>O comprimento da geratriz do cone \u00e9, aproximadamente, 30 cm:<br \/>\n\\[g = 3\\sqrt {\\frac{{314}}{\\pi }} \\approx 30\\]<\/li>\n<li>A altura do cone \u00e9, aproximadamente, 28 cm:<br \/>\nPor aplica\u00e7\u00e3o do Teorema de Pit\u00e1goras, temos:<br \/>\n\\[h = \\sqrt {{{\\left( {3r} \\right)}^2} &#8211; {r^2}} = \\sqrt {8{r^2}} = 2\\sqrt 2 \\,r\\]<br \/>\nPortanto, vem:<br \/>\n\\[h = 2\\sqrt 2 \\times \\sqrt {\\frac{{314}}{\\pi }} \\approx 28\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13796' onClick='GTTabs_show(0,13796)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um cone reto tem 1256 cm2 de \u00e1rea de superf\u00edcie e a sua geratriz \u00e9 tripla do raio da base. Qual \u00e9 a medida do comprimento, arredondado \u00e0s unidades: o raio&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":13797,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,466],"tags":[426,108,470,109],"series":[],"class_list":["post-13796","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-distancias-areas-e-volumes-de-solidos","tag-9-o-ano","tag-area","tag-cone","tag-volume"],"views":2976,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/cone.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13796","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=13796"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13796\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/13797"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13796"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13796"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13796"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}