{"id":13924,"date":"2018-03-08T22:50:43","date_gmt":"2018-03-08T22:50:43","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=13924"},"modified":"2022-01-16T19:18:13","modified_gmt":"2022-01-16T19:18:13","slug":"sobre-uma-esfera","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=13924","title":{"rendered":"Sobre uma esfera"},"content":{"rendered":"<p><ul id='GTTabs_ul_13924' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_13924' class='GTTabs_curr'><a  id=\"13924_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_13924' ><a  id=\"13924_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_13924'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-19b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13925\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13925\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-19b.png\" data-orig-size=\"240,245\" 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=\"Esfera\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-19b.png\" class=\"alignright size-full wp-image-13925\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-19b.png\" alt=\"\" width=\"240\" height=\"245\" \/><\/a>Uma esfera \u00e9 seccionada por um plano a 8 cm do centro.<br \/>\nA sec\u00e7\u00e3o obtida \u00e9 um c\u00edrculo com 36 cm<sup>2<\/sup> de \u00e1rea.<\/p>\n<p>Determina a \u00e1rea da superf\u00edcie da esfera e o seu volume, arredondado \u00e0s d\u00e9cimas.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_13924' onClick='GTTabs_show(1,13924)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_13924'>\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\/9V2Pag033-19b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"13925\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=13925\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-19b.png\" data-orig-size=\"240,245\" 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=\"Esfera\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-19b.png\" class=\"alignright size-full wp-image-13925\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-19b.png\" alt=\"\" width=\"240\" height=\"245\" \/><\/a>Comecemos por determinar, em cm, o comprimento do raio da sec\u00e7\u00e3o:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{A = 36}&amp; \\Leftrightarrow &amp;{\\pi \\times {s^2} = 36}\\\\{}&amp; \\Leftrightarrow &amp;{s = \\sqrt {\\frac{{36}}{\\pi }} }\\\\{}&amp; \\Leftrightarrow &amp;{s = \\frac{6}{{\\sqrt \\pi }}}\\end{array}\\]<\/p>\n<p>Determinemos agora, em cm, o comprimento do raio da esfera, aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [<em>MNO<\/em>]:<\/p>\n<p>\\[r = \\sqrt {{{\\overline {OM} }^2} + {{\\overline {MN} }^2}} = \\sqrt {{8^2} + {{\\left( {\\frac{6}{{\\sqrt \\pi }}} \\right)}^2}} = \\sqrt {64 + \\frac{{36}}{\\pi }} \\]<\/p>\n<p>A \u00e1rea da superf\u00edcie da esfera \u00e9, aproximadamente, 948,2 cm<sup>2<\/sup>:<\/p>\n<p>\\[A = 4 \\times \\pi \\times {\\left( {\\sqrt {64 + \\frac{{36}}{\\pi }} } \\right)^2} = 4 \\times \\pi \\times \\left( {64 + \\frac{{36}}{\\pi }} \\right) = 256\\pi + 144 \\approx 948,2\\]<\/p>\n<p>A esfera tem, aproximadamente, 2745,7 cm<sup>3<\/sup> de volume:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}V&amp; = &amp;{\\frac{4}{3} \\times \\pi \\times {{\\left( {\\sqrt {64 + \\frac{{36}}{\\pi }} } \\right)}^3}}\\\\{}&amp; = &amp;{\\frac{4}{3} \\times \\pi \\times \\left( {64 + \\frac{{36}}{\\pi }} \\right) \\times \\sqrt {64 + \\frac{{36}}{\\pi }} }\\\\{}&amp; = &amp;{\\left( {\\frac{{256\\pi }}{3} + 48} \\right) \\times \\sqrt {64 + \\frac{{36}}{\\pi }} }\\\\{}&amp; \\approx &amp;{2745,7}\\end{array}\\]<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_13924' onClick='GTTabs_show(0,13924)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma esfera \u00e9 seccionada por um plano a 8 cm do centro. A sec\u00e7\u00e3o obtida \u00e9 um c\u00edrculo com 36 cm2 de \u00e1rea. Determina a \u00e1rea da superf\u00edcie da esfera e&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20407,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,466],"tags":[426,108,482,109],"series":[],"class_list":["post-13924","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-esfera","tag-volume"],"views":3035,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag033-19_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13924","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=13924"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/13924\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20407"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13924"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13924"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13924"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=13924"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}