{"id":13902,"date":"2018-03-07T23:13:54","date_gmt":"2018-03-07T23:13:54","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13902"},"modified":"2022-01-06T18:20:55","modified_gmt":"2022-01-06T18:20:55","slug":"uma-demonstracao-de-arquimedes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13902","title":{"rendered":"Uma demonstra\u00e7\u00e3o de Arquimedes"},"content":{"rendered":"<p><ul id='GTTabs_ul_13902' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13902' class='GTTabs_curr'><a  id=\"13902_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13902' ><a  id=\"13902_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_13902'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13904\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13904\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16b.png\" data-orig-size=\"210,245\" 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=\"Sempre em p\u00e9\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16b.png\" class=\"alignright size-full wp-image-13904\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16b.png\" alt=\"\" width=\"210\" height=\"245\" \/><\/a>Arquimedes demonstrou que o volume de um cilindro, em que a altura coincide com o raio da base, \u00e9 igual \u00e0 soma do volume do cone, de base e altura iguais \u00e0 do cilindro, com o volume de semiesfera, de base igual \u00e0 do cone.<\/p>\n<p>Na figura, o cone e a semiesfera t\u00eam a mesma base, cuja \u00e1rea \u00e9 de 100\u03c0 cm2.<\/p>\n<p>Calcula:<\/p>\n<ol>\n<li>o raio da base;<\/li>\n<li>a altura do cone;<\/li>\n<li>o volume do s\u00f3lido formado pelo cone e pela semiesfera;<\/li>\n<li>o volume do cilindro cuja base e a altura s\u00e3o iguais \u00e0s do cone.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13902' onClick='GTTabs_show(1,13902)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13902'>\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\/9V2Pag033-16_Arquimedes.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13907\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13907\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16_Arquimedes.png\" data-orig-size=\"640,300\" 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=\"Arquimedes\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16_Arquimedes.png\" class=\"aligncenter wp-image-13907\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16_Arquimedes.png\" alt=\"\" width=\"480\" height=\"225\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16_Arquimedes.png 640w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16_Arquimedes-300x141.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16_Arquimedes-520x245.png 520w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><\/a><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13904\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13904\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16b.png\" data-orig-size=\"210,245\" 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=\"Sempre em p\u00e9\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16b.png\" class=\"alignright size-full wp-image-13904\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16b.png\" alt=\"\" width=\"210\" height=\"245\" \/><\/a>O raio da base \u00e9 10 cm:<br \/>\n\\[\\begin{array}{*{20}{l}}{{A_b} = 100\\pi }&amp; \\Leftrightarrow &amp;{\\pi {r^2} = 100\\pi }\\\\{}&amp; \\Leftrightarrow &amp;{r = 10}\\end{array}\\]<\/li>\n<li>A altura do cone \u00e9 10 cm, pois \\(h = r\\), visto que o tri\u00e2ngulo assinalado na figura \u00e9 ret\u00e2ngulo is\u00f3sceles.<br \/>\n\u00ad<\/li>\n<li>O\u00a0volume do s\u00f3lido formado pelo cone e pela semiesfera \u00e9 1000\u03c0 cm<sup>3<\/sup>:<br \/>\n\\[\\begin{array}{*{20}{l}}{{V_{Cone + SE}}}&amp; = &amp;{{V_{Cone}} + {V_{SE}}}\\\\{}&amp; = &amp;{\\frac{1}{3} \\times \\pi \\times {{10}^2} \\times 10 + \\frac{1}{2} \\times \\frac{4}{3} \\times \\pi \\times {{10}^3}}\\\\{}&amp; = &amp;{\\frac{{1000\\pi }}{3} + \\frac{{2000\\pi }}{3}}\\\\{}&amp; = &amp;{1000\\pi }\\end{array}\\]<\/li>\n<li>O\u00a0volume do cilndro cuja base e a altura s\u00e3o iguais \u00e0s do cone \u00e9 tamb\u00e9m 1000\u03c0 cm<sup>3<\/sup>:<br \/>\n\\[\\begin{array}{*{20}{l}}{{V_{Cilindro}}}&amp; = &amp;{\\pi \\times {{10}^2} \\times 10}\\\\{}&amp; = &amp;{1000\\pi }\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13902' onClick='GTTabs_show(0,13902)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Arquimedes demonstrou que o volume de um cilindro, em que a altura coincide com o raio da base, \u00e9 igual \u00e0 soma do volume do cone, de base e altura iguais&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":13905,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,466],"tags":[426,469,470,482,109],"series":[],"class_list":["post-13902","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-cilindro","tag-cone","tag-esfera","tag-volume"],"views":2039,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-16.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13902","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=13902"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13902\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/13905"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13902"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13902"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13902"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13902"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}