{"id":9891,"date":"2012-10-03T11:01:56","date_gmt":"2012-10-03T10:01:56","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9891"},"modified":"2022-01-20T19:53:23","modified_gmt":"2022-01-20T19:53:23","slug":"a-altura-do-cone","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9891","title":{"rendered":"A altura do cone"},"content":{"rendered":"<p><ul id='GTTabs_ul_9891' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9891' class='GTTabs_curr'><a  id=\"9891_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9891' ><a  id=\"9891_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_9891'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<div id=\"attachment_9895\" style=\"width: 381px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9895\" data-attachment-id=\"9895\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9895\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg\" data-orig-size=\"371,169\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349259019&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=\"Cone\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Sector circular e cone sem base&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg\" class=\"   wp-image-9895 size-full\" title=\"Cone\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg\" alt=\"\" width=\"371\" height=\"169\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg 371w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone-300x136.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone-150x68.jpg 150w\" sizes=\"auto, (max-width: 371px) 100vw, 371px\" \/><\/a><p id=\"caption-attachment-9895\" class=\"wp-caption-text\">Setor circular e cone sem base<\/p><\/div>\n<p>O setor circular $ARC$ tem raio $9$ cm e o \u00e2ngulo mede $80^\\circ $.<\/p>\n<p>Quando se corta o setor circular e se junta, com fita adesiva, os segmentos $AR$ e $RC$ forma-se o cone, sem base, representado na figura.<\/p>\n<p>Determine a altura do cone.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9891' onClick='GTTabs_show(1,9891)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9891'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<div id=\"attachment_9895\" style=\"width: 381px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9895\" data-attachment-id=\"9895\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9895\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg\" data-orig-size=\"371,169\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349259019&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=\"Cone\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Sector circular e cone sem base&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg\" class=\"   wp-image-9895 size-full\" title=\"Cone\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg\" alt=\"\" width=\"371\" height=\"169\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone.jpg 371w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone-300x136.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/planifcone-150x68.jpg 150w\" sizes=\"auto, (max-width: 371px) 100vw, 371px\" \/><\/a><p id=\"caption-attachment-9895\" class=\"wp-caption-text\">Setor circular e cone sem base<\/p><\/div>\n<p>Um olhar atento permite concluir que o per\u00edmetro da base do cone \u00e9 igual ao comprimento do arco $AC$ do setor circular.<\/p>\n<p>Designando por $m$ o comprimento do arco $AC$ (em cent\u00edmetros), vem:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{\\frac{{80^\\circ }}{{360^\\circ }} = \\frac{m}{{2\\pi\u00a0 \\times 9}}}&amp; \\Leftrightarrow &amp;{m = \\frac{{2\\pi\u00a0 \\times 9 \\times 80^\\circ }}{{360^\\circ }}}\\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{m = 4\\pi }<br \/>\n\\end{array}$$<\/p>\n<p>Assim, o comprimento (em cent\u00edmetros) do raio da base do cone \u00e9: $$r = \\frac{{4\\pi }}{{2\\pi }} = 2$$<\/p>\n<p>Por aplica\u00e7\u00e3o Teorema de Pit\u00e1goras e dado que o comprimento da geratriz do cone \u00e9 igual ao raio do setor circular, resulta:<\/p>\n<p>$$h = \\sqrt {{9^2} &#8211; {2^2}}\u00a0 = \\sqrt {77} $$<\/p>\n<p>Portanto, o cone tem $\\sqrt {77} $ cent\u00edmetros de altura.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9891' onClick='GTTabs_show(0,9891)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado O setor circular $ARC$ tem raio $9$ cm e o \u00e2ngulo mede $80^\\circ $. Quando se corta o setor circular e se junta, com fita adesiva, os segmentos $AR$ e $RC$&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20766,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,108,430,109],"series":[],"class_list":["post-9891","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-area","tag-modulo-inicial","tag-volume"],"views":2603,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag033-9_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9891","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=9891"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9891\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20766"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9891"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9891"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9891"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9891"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}