{"id":14138,"date":"2018-03-12T03:31:03","date_gmt":"2018-03-12T03:31:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14138"},"modified":"2022-01-11T10:50:31","modified_gmt":"2022-01-11T10:50:31","slug":"a-geratriz-de-um-cone-reto-mede-40-cm","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14138","title":{"rendered":"A geratriz de um cone reto mede 40 cm"},"content":{"rendered":"<p><ul id='GTTabs_ul_14138' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14138' class='GTTabs_curr'><a  id=\"14138_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14138' ><a  id=\"14138_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_14138'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14139\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14139\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8.png\" data-orig-size=\"218,443\" 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\/9V2Pag056-8.png\" class=\"alignright wp-image-14139\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8.png\" alt=\"\" width=\"180\" height=\"366\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8.png 218w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8-148x300.png 148w\" sizes=\"auto, (max-width: 180px) 100vw, 180px\" \/><\/a>A geratriz do cone reto da figura mede 40 cm e faz um \u00e2ngulo de 80 graus com o di\u00e2metro da base.<br \/>\nEm cada al\u00ednea, apresenta os valores arredondados \u00e0s d\u00e9cimas.<\/p>\n<ol>\n<li>Calcula a altura do cone.<\/li>\n<li>Determina o volume do cone.<\/li>\n<li>Qual \u00e9 a \u00e1rea da superf\u00edcie deste cone?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14138' onClick='GTTabs_show(1,14138)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14138'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14139\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14139\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8.png\" data-orig-size=\"218,443\" 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\/9V2Pag056-8.png\" class=\"alignright wp-image-14139\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8-148x300.png\" alt=\"\" width=\"180\" height=\"366\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8-148x300.png 148w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag056-8.png 218w\" sizes=\"auto, (max-width: 180px) 100vw, 180px\" \/><\/a>No tri\u00e2ngulo ret\u00e2ngulo [OBV], temos:<br \/>\n\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm sen}\\nolimits} 80^\\circ = \\frac{{\\overline {OV} }}{{\\overline {VB} }}}&amp; \\Leftrightarrow &amp;{{\\mathop{\\rm sen}\\nolimits} 80^\\circ = \\frac{h}{{40}}}\\\\{}&amp; \\Leftrightarrow &amp;{h = 40 \\times {\\mathop{\\rm sen}\\nolimits} 80^\\circ }\\\\{}&amp;{}&amp;{h \\approx 39,4}\\end{array}\\]<br \/>\nPortanto, o cone tem, aproximadamente, 39,4 cm de altura.<br \/>\n\u00ad<\/li>\n<li>Comecemos por determinar o comprimento do raio da base do cone:<br \/>\n\\[\\begin{array}{*{20}{l}}{\\cos 80^\\circ = \\frac{{\\overline {OB} }}{{\\overline {VB} }}}&amp; \\Leftrightarrow &amp;{\\cos 80^\\circ = \\frac{r}{{40}}}\\\\{}&amp; \\Leftrightarrow &amp;{r = 40 \\times \\cos 80^\\circ }\\\\{}&amp;{}&amp;{r \\approx 6,9}\\end{array}\\]<br \/>\nO cone tem, aproximadamente, 1990,2 cm<sup>3<\/sup> de volume:<br \/>\n\\[V = \\frac{1}{3} \\times \\pi \\times {\\left( {40 \\times \\cos 80^\\circ } \\right)^2} \\times 40 \\times {\\mathop{\\rm sen}\\nolimits} 80^\\circ \\approx 1990,2\\]<br \/>\n\u00ad<\/li>\n<li>Determinemos agora uma express\u00e3o para a \u00e1rea da superf\u00edcie lateral do cone:<br \/>\n\\[\\begin{array}{*{20}{l}}{\\frac{{{A_L}}}{{2\\pi r}} = \\frac{{\\pi {g^2}}}{{2\\pi g}}}&amp; \\Leftrightarrow &amp;{\\frac{{{A_L}}}{{2\\pi r}} = \\frac{g}{2}}\\\\{}&amp; \\Leftrightarrow &amp;{{A_L} = \\pi rg}\\end{array}\\]<br \/>\nA superf\u00edcie do cone, tem aproximadamente, 1024,4 cm<sup>2<\/sup> de \u00e1rea:<br \/>\n\\[A = \\pi \\times {\\left( {40 \\times \\cos 80^\\circ } \\right)^2} + \\pi \\times 40 \\times 40 \\times \\cos 80^\\circ \\approx 1024,4\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14138' onClick='GTTabs_show(0,14138)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A geratriz do cone reto da figura mede 40 cm e faz um \u00e2ngulo de 80 graus com o di\u00e2metro da base. Em cada al\u00ednea, apresenta os valores arredondados \u00e0s d\u00e9cimas.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14113,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,108,470,493,490,109],"series":[],"class_list":["post-14138","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-trigonometria-9--ano","tag-9-o-ano","tag-area","tag-cone","tag-razao-trigonometrica","tag-seno","tag-volume"],"views":2752,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat55.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14138","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=14138"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14138\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14113"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14138"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}