{"id":13821,"date":"2018-03-05T22:27:21","date_gmt":"2018-03-05T22:27:21","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13821"},"modified":"2022-01-16T18:48:14","modified_gmt":"2022-01-16T18:48:14","slug":"uma-cavidade-num-cilindro","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13821","title":{"rendered":"Uma cavidade num cilindro"},"content":{"rendered":"<p><ul id='GTTabs_ul_13821' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13821' class='GTTabs_curr'><a  id=\"13821_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13821' ><a  id=\"13821_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_13821'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-6b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13822\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13822\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-6b.png\" data-orig-size=\"150,275\" 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=\"Cilindro\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-6b.png\" class=\"alignright size-full wp-image-13822\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-6b.png\" alt=\"\" width=\"150\" height=\"275\" \/><\/a>Um cilindro reto com 40 cm de altura apresenta uma cavidade com a forma de um cone reto cuja base \u00e9 conc\u00eantrica com a base do cilindro e com metade da altura deste.<\/p>\n<p>Sabendo que o raio da base do cilindro mede 25 cm e que supera em 10 cm o raio da base do cone, calcula a \u00e1rea da superf\u00edcie e o volume do s\u00f3lido, arredondados \u00e0s d\u00e9cimas.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13821' onClick='GTTabs_show(1,13821)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13821'>\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\/9V2Pag28-6b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13822\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13822\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-6b.png\" data-orig-size=\"150,275\" 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=\"Cilindro\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-6b.png\" class=\"alignright size-full wp-image-13822\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-6b.png\" alt=\"\" width=\"150\" height=\"275\" \/><\/a>Recapitulemos os dados:<\/p>\n<ul style=\"list-style-type: square;\">\n<li>altura do cilindro: \\(h = 40\\) cm<\/li>\n<li>altura do cone:\u00a0\\(h&#8217; = 20\\) cm<\/li>\n<li>raio da base do cilindro:\u00a0\\(r = 25\\) cm<\/li>\n<li>raio da base do cone:\u00a0\\(r&#8217; = 15\\) cm<\/li>\n<\/ul>\n<p>Comecemos por determinar o comprimento <em>g<\/em>, em cm, da geratriz do cone, por aplica\u00e7\u00e3o do Teorema de Pit\u00e1goras:<\/p>\n<p>\\[g = \\sqrt {{{\\left( {h&#8217;} \\right)}^2} + {{\\left( {r&#8217;} \\right)}^2}} = \\sqrt {{{20}^2} + {{15}^2}} = 25\\]<\/p>\n<p>A \u00e1rea da superf\u00edcie do s\u00f3lido \u00e9, aproximadamente, 10681,4 cm<sup>2<\/sup>:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{A_T}}&amp; = &amp;{{A_{Cilindro}} &#8211; {A_{BCone}} + {A_{LCone}}}\\\\{}&amp; = &amp;{\\left( {2 \\times \\pi {r^2} + 2\\pi r \\times h} \\right) &#8211; \\pi \\times {{\\left( {r&#8217;} \\right)}^2} + \\pi rg}\\\\{}&amp; = &amp;{\\left( {2 \\times \\pi \\times {{25}^2} + 2\\pi \\times 25 \\times 40} \\right) &#8211; \\pi \\times {{15}^2} + \\pi \\times 15 \\times 25}\\\\{}&amp; = &amp;{3400\\pi }\\\\{}&amp; \\approx &amp;{10681,4}\\end{array}\\]<\/p>\n<p>O volume do s\u00f3lido \u00e9, aproximadamente, 73827,4 cm<sup>3<\/sup>:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{V_T}}&amp; = &amp;{{V_{Cilindro}} &#8211; {V_{Cone}}}\\\\{}&amp; = &amp;{\\pi {r^2} \\times h &#8211; \\frac{1}{3} \\times \\pi \\times {{\\left( {r&#8217;} \\right)}^2} \\times h&#8217;}\\\\{}&amp; = &amp;{\\pi \\times {{25}^2} \\times 40 &#8211; \\frac{1}{3} \\times \\pi \\times {{15}^2} \\times 20}\\\\{}&amp; = &amp;{23500\\pi }\\\\{}&amp; \\approx &amp;{73827,4}\\end{array}\\]<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13821' onClick='GTTabs_show(0,13821)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um cilindro reto com 40 cm de altura apresenta uma cavidade com a forma de um cone reto cuja base \u00e9 conc\u00eantrica com a base do cilindro e com metade da&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20401,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,466],"tags":[426,108,469,470,109],"series":[],"class_list":["post-13821","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-cilindro","tag-cone","tag-volume"],"views":1982,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag028-6_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13821","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=13821"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13821\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20401"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13821"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13821"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13821"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13821"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}