{"id":13811,"date":"2018-03-05T20:26:34","date_gmt":"2018-03-05T20:26:34","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13811"},"modified":"2022-01-06T15:04:44","modified_gmt":"2022-01-06T15:04:44","slug":"um-solido-constituido-por-um-cilindro-e-dois-cones","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13811","title":{"rendered":"Um s\u00f3lido constitu\u00eddo por um cilindro e dois cones"},"content":{"rendered":"<p><ul id='GTTabs_ul_13811' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13811' class='GTTabs_curr'><a  id=\"13811_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13811' ><a  id=\"13811_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_13811'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13812\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13812\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b.png\" data-orig-size=\"440,120\" 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 e cones\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b.png\" class=\"alignright size-medium wp-image-13812\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b-300x82.png\" alt=\"\" width=\"300\" height=\"82\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b-300x82.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b.png 440w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Um s\u00f3lido \u00e9 formado por um cilindro e por dois cones retos com a mesma altura e cuja base \u00e9 a base do cilindro.<\/p>\n<p>O cilindro tem 18 cm de altura e 1152\u03c0 cm<sup>3<\/sup> de volume. A \u00e1rea da superf\u00edcie do s\u00f3lido \u00e9 560\u03c0 cm<sup>2<\/sup>.<\/p>\n<p>Qual \u00e9 o volume do s\u00f3lido?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13811' onClick='GTTabs_show(1,13811)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13811'>\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\/9V2Pag28-5b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13812\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13812\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b.png\" data-orig-size=\"440,120\" 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 e cones\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b.png\" class=\"alignright size-medium wp-image-13812\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b-300x82.png\" alt=\"\" width=\"300\" height=\"82\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b-300x82.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5b.png 440w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Seja <em>r<\/em> o comprimento do raio da base do cilindro, <em>g<\/em> o comprimento da geratriz do cone e <em>h<\/em> o comprimento da altura do cone, todos eles em cent\u00edmetros.<\/p>\n<p>Comecemos por determinar <em>r<\/em>:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{V_{Cilindro}} = 1152\\pi }&amp; \\Leftrightarrow &amp;{\\pi {r^2} \\times 18 = 1152\\pi }\\\\{}&amp; \\Leftrightarrow &amp;{r = \\sqrt {\\frac{{1152}}{{18}}} }\\\\{}&amp; \\Leftrightarrow &amp;{r = \\sqrt {64} }\\\\{}&amp; \\Leftrightarrow &amp;{r = 8}\\end{array}\\]<\/p>\n<p>Calculemos agora a \u00e1rea da superf\u00edcie lateral do cilindro:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{A_{LCilindro}}}&amp; = &amp;{\\left( {2\\pi \\times 8} \\right) \\times 18}\\\\{}&amp; = &amp;{288\\pi }\\end{array}\\]<\/p>\n<p>Calculemos seguidamente a \u00e1rea da superf\u00edcie lateral de um dos dois cones iguais:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{A_{LCone}}}&amp; = &amp;{\\frac{{{A_T} &#8211; {A_{LCilindro}}}}{2}}\\\\{}&amp; = &amp;{\\frac{{560\\pi &#8211; 288\\pi }}{2}}\\\\{}&amp; = &amp;{136\\pi }\\end{array}\\]<\/p>\n<p>Determinemos agora <em>g<\/em>:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\frac{{{A_{LCone}}}}{{2\\pi \\times r}} = \\frac{{\\pi \\times {g^2}}}{{2\\pi \\times g}}}&amp; \\Leftrightarrow &amp;{\\frac{{136\\pi }}{{16\\pi }} = \\frac{g}{2}}\\\\{}&amp; \\Leftrightarrow &amp;{g = \\frac{{2 \\times 136}}{{16}}}\\\\{}&amp; \\Leftrightarrow &amp;{g = 17}\\end{array}\\]<\/p>\n<p>Por aplica\u00e7\u00e3o do Teorema de Pit\u00e1goras, determinemos <em>h<\/em>:<\/p>\n<p>\\[h = \\sqrt {{g^2} &#8211; {r^2}} = \\sqrt {{{17}^2} &#8211; {8^2}} = 15\\]<\/p>\n<p>Finalmente, conclui-se que o s\u00f3lido tem 1792\u03c0 cm<sup>3<\/sup> de volume:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}V&amp; = &amp;{{V_{Cilindro}} + 2 \\times {V_{Cone}}}\\\\{}&amp; = &amp;{1152\\pi + 2 \\times \\frac{1}{3} \\times \\pi \\times {8^2} \\times 15}\\\\{}&amp; = &amp;{1792\\pi }\\end{array}\\]<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13811' onClick='GTTabs_show(0,13811)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um s\u00f3lido \u00e9 formado por um cilindro e por dois cones retos com a mesma altura e cuja base \u00e9 a base do cilindro. O cilindro tem 18 cm de altura&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":13813,"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-13811","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":1969,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag28-5.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13811","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=13811"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13811\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/13813"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13811"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13811"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13811"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13811"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}