{"id":10073,"date":"2012-10-05T12:59:55","date_gmt":"2012-10-05T11:59:55","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10073"},"modified":"2022-01-14T17:57:54","modified_gmt":"2022-01-14T17:57:54","slug":"duas-esferas-e-um-cilindro","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10073","title":{"rendered":"Duas esferas e um cilindro"},"content":{"rendered":"<p><ul id='GTTabs_ul_10073' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10073' class='GTTabs_curr'><a  id=\"10073_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10073' ><a  id=\"10073_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_10073'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Duas esferas, de raios $2$ e $3$, est\u00e3o encaixadas num recipiente cil\u00edndrico de di\u00e2metro $9$ (unidades em cent\u00edmetro).<\/p>\n<ol>\n<li>Fa\u00e7a um desenho e explique a sua constru\u00e7\u00e3o. \u00a0<\/li>\n<li>Qual \u00e9 o volume de l\u00edquido necess\u00e1rio para cobrir totalmente as duas esferas. \u00a0<\/li>\n<li>Se o l\u00edquido cobrir exatamente a esfera maior, que se encontra no fundo, que parte da esfera menor fica fora?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10073' onClick='GTTabs_show(1,10073)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10073'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2esferascilindro.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10104\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10104\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2esferascilindro.png\" data-orig-size=\"591,493\" 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;}\" data-image-title=\"Duas esferas e um cilindro\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2esferascilindro.png\" class=\"alignright  wp-image-10104\" title=\"Duas esferas e um cilindro\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2esferascilindro.png\" alt=\"\" width=\"355\" height=\"296\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2esferascilindro.png 591w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2esferascilindro-300x250.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2esferascilindro-150x125.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2esferascilindro-400x333.png 400w\" sizes=\"auto, (max-width: 355px) 100vw, 355px\" \/><\/a>Na figura est\u00e1 representado o corte efetuado no conjunto cilindro e esferas por um plano que cont\u00e9m o eixo do cilindro e os centros das duas esferas. As sec\u00e7\u00f5es obtidas s\u00e3o, respetivamente, um ret\u00e2ngulo e dois c\u00edrculos, tangentes entre si, conforme est\u00e1 ilustrado no desenho. \u00a0<\/p>\n<\/li>\n<li>Para cobrir totalmente as duas esferas, o l\u00edquido tem de atingir a altura (em cent\u00edmetros): ${h_1} = \\overline {AE}\u00a0 + \\overline {BC}\u00a0 + \\overline {BR} $.\n<p>Como o cateto $\\left[ {AC} \\right]$ do tri\u00e2ngulo ret\u00e2ngulo $\\left[ {ABC} \\right]$ tem $4$ cm de comprimento, ent\u00e3o $\\overline {BC}\u00a0 = 3$ cm.<\/p>\n<p>Logo, ${h_1} = \\overline {AE}\u00a0 + \\overline {BC}\u00a0 + \\overline {BR}\u00a0 = 3 + 3 + 2 = 8$ cm.<\/p>\n<p>Assim, o volume de l\u00edquido (em cm<sup>3<\/sup>) necess\u00e1rio para cobrir totalmente as duas esferas \u00e9:<br \/>$$\\begin{array}{*{20}{l}} {{V_{L\u00edquido}}}&amp; = &amp;{\\pi\u00a0 \\times {{\\left( {\\frac{9}{2}} \\right)}^2} \\times 8 &#8211; {V_{Esfera &#8211; G}} &#8211; {V_{Esfera &#8211; P}}}\\\\ {}&amp; = &amp;{\\pi\u00a0 \\times {{\\left( {\\frac{9}{2}} \\right)}^2} \\times 8 &#8211; \\frac{4}{3}\\pi\u00a0 \\times {3^3} &#8211; \\frac{4}{3}\\pi\u00a0 \\times {2^3}}\\\\ {}&amp; = &amp;{\\pi\u00a0 \\times \\left( {81 \\times 2 &#8211; 36 &#8211; \\frac{{32}}{3}} \\right)}\\\\ {}&amp; = &amp;{\\pi\u00a0 \\times \\left( {126 &#8211; \\frac{{32}}{3}} \\right)}\\\\ {}&amp; = &amp;{\\frac{{378 &#8211; 32}}{3}\\pi }\\\\ {}&amp; = &amp;{\\frac{{346}}{3}\\pi } \\end{array}$$ \u00a0<\/p>\n<\/li>\n<li>Uma an\u00e1lise atenta do desenho permite concluir que, se o l\u00edquido cobrir exatamente a esfera maior, fica de fora a metade superior da esfera menor (repare que $\\overline {AE}\u00a0 + \\overline {BC}\u00a0 = 3 + 3 = 6$ cm \u00e9 igual ao comprimento do di\u00e2metro da esfera maior).<\/li>\n<\/ol>\n<p>Explore, agora, a aplica\u00e7\u00e3o seguinte:<\/p>\n<p>\u00a0<\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":517,\r\n\"height\":532,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ \"material_id\":12345,\r\n\"ggbBase64\":\"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\"};\r\n\/\/ is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/p>\n<p style=\"text-align: left;\">\u00a0<\/p>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10073' onClick='GTTabs_show(0,10073)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Duas esferas, de raios $2$ e $3$, est\u00e3o encaixadas num recipiente cil\u00edndrico de di\u00e2metro $9$ (unidades em cent\u00edmetro). Fa\u00e7a um desenho e explique a sua constru\u00e7\u00e3o. \u00a0 Qual \u00e9 o volume&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19771,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,67,430,109],"series":[],"class_list":["post-10073","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-modulo-inicial","tag-10-o-ano","tag-geometria","tag-modulo-inicial","tag-volume"],"views":2216,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/2esferascilindro-520x245-1.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10073","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=10073"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10073\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19771"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10073"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10073"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10073"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10073"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}