{"id":13803,"date":"2018-03-05T12:11:36","date_gmt":"2018-03-05T12:11:36","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13803"},"modified":"2022-01-16T18:43:14","modified_gmt":"2022-01-16T18:43:14","slug":"13803","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13803","title":{"rendered":"Outro cone reto"},"content":{"rendered":"<p><ul id='GTTabs_ul_13803' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13803' class='GTTabs_curr'><a  id=\"13803_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13803' ><a  id=\"13803_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_13803'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13805\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13805\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4.png\" data-orig-size=\"620,405\" 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\/9V2Pag28-4.png\" class=\"alignright wp-image-13805\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4-300x196.png\" alt=\"\" width=\"480\" height=\"314\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4-300x196.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4.png 620w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/a>Um cone reto com 28,5 cm de altura tem 13718\u03c0 cm<sup>3<\/sup> de volume.<\/p>\n<p>Calcula:<\/p>\n<ol>\n<li>o valor exato da \u00e1rea da superf\u00edcie do cone;<\/li>\n<li>a medida da amplitude do setor circular que se obt\u00e9m quando se planifica o cone.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13803' onClick='GTTabs_show(1,13803)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13803'>\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\/9V2Pag28-4.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13805\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13805\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4.png\" data-orig-size=\"620,405\" 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\/9V2Pag28-4.png\" class=\"alignright wp-image-13805\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4-300x196.png\" alt=\"\" width=\"480\" height=\"314\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4-300x196.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-4.png 620w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/a>Comecemos por determinar uma express\u00e3o da \u00e1rea da superf\u00edcie do cone em fun\u00e7\u00e3o de <em>r<\/em> e de <em>g<\/em>.<\/p>\n<p>Ora, a superf\u00edcie lateral do cone \u00e9 um setor circular de raio [VA], cuja \u00e1rea \u00e9 diretamente proporcional ao comprimento do arco correspondente a esse setor circular:<\/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{g}{2}}\\\\{}&amp; \\Leftrightarrow &amp;{{A_L} = \\pi \\times r \\times g}\\end{array}\\]<\/p>\n<p>A \u00e1rea da superf\u00edcie total do cone pode ser expressa, em fun\u00e7\u00e3o de <em>r<\/em> e de <em>g<\/em>, por:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{A_T}}&amp; = &amp;{{A_B} + {A_L}}\\\\{}&amp; = &amp;{\\pi \\times {r^2} + \\pi \\times r \\times g}\\\\{}&amp; = &amp;{\\pi \\times r\\left( {r + g} \\right)}\\end{array}\\]<\/p>\n<ol>\n<li>Comecemos por determinar o valor de <em>r<\/em>:<br \/>\n\\[\\begin{array}{*{20}{l}}{V = 13718\\pi }&amp; \\Leftrightarrow &amp;{\\frac{1}{3}\\pi {r^2} \\times 28,5 = 13718\\pi }\\\\{}&amp; \\Leftrightarrow &amp;{r = \\sqrt {\\frac{{3 \\times 13718}}{{28,5}}} }\\\\{}&amp; \\Leftrightarrow &amp;{r = \\sqrt {1444} }\\\\{}&amp; \\Leftrightarrow &amp;{r = 38}\\end{array}\\]<br \/>\nPor aplica\u00e7\u00e3o do Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [OAV], temos:<br \/>\n\\[g = \\sqrt {{r^2} + {h^2}} = \\sqrt {{{38}^2} + {{28,5}^2}} = 47,5\\]<br \/>\nAssim, o valor da\u00a0\u00e1rea da superf\u00edcie do cone, em cm<sup>2<\/sup>, \u00e9:<br \/>\n\\[\\begin{array}{*{20}{l}}{{A_T}}&amp; = &amp;{\\pi \\times 38\\left( {38 + 47,5} \\right)}\\\\{}&amp; = &amp;{3249\\pi }\\end{array}\\]<\/li>\n<li>Como a amplitude do setor circular \u00e9 diretamente proporcional ao comprimento do arco correspondente, vem:<br \/>\n\\[\\begin{array}{*{20}{l}}{\\frac{{360^\\circ }}{{2\\pi \\times g}} = \\frac{\\alpha }{{2\\pi \\times r}}}&amp; \\Leftrightarrow &amp;{\\alpha = \\frac{{2\\pi \\times r}}{{2\\pi \\times g}} \\times 360^\\circ }\\\\{}&amp; \\Leftrightarrow &amp;{\\alpha = \\frac{r}{g} \\times 360^\\circ }\\end{array}\\]<br \/>\nAssim, \u00e9 de \\(288^\\circ \\) a medida da amplitude do setor circular que se obt\u00e9m quando se planifica o cone:<br \/>\n\\[\\alpha = \\frac{{38}}{{47,5}} \\times 360^\\circ = 288^\\circ \\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13803' onClick='GTTabs_show(0,13803)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um cone reto com 28,5 cm de altura tem 13718\u03c0 cm3 de volume. Calcula: o valor exato da \u00e1rea da superf\u00edcie do cone; a medida da amplitude do setor circular que&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20400,"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],"series":[],"class_list":["post-13803","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"],"views":2806,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag028-4_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13803","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=13803"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13803\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20400"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13803"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13803"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13803"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}