{"id":9910,"date":"2012-10-03T18:13:04","date_gmt":"2012-10-03T17:13:04","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9910"},"modified":"2022-01-20T20:26:26","modified_gmt":"2022-01-20T20:26:26","slug":"caixa-de-forma-cilindrica-com-bolas-de-tenis","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9910","title":{"rendered":"Caixa de forma cil\u00edndrica com bolas de t\u00e9nis"},"content":{"rendered":"<p><ul id='GTTabs_ul_9910' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9910' class='GTTabs_curr'><a  id=\"9910_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9910' ><a  id=\"9910_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_9910'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Uma caixa de forma cil\u00edndrica com bolas de t\u00e9nis tem $7$ cm de di\u00e2metro e uma altura de $20$ cm.<\/p>\n<p>Se o di\u00e2metro de cada bola de t\u00e9nis for $6$ cm, qual a percentagem do volume ocupado pelas $3$ bolas?<\/p>\n<div id=\"attachment_9911\" style=\"width: 343px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9911\" data-attachment-id=\"9911\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9911\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg\" data-orig-size=\"333,126\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349284221&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=\"Caixa\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Caixa com bolas de t\u00e9nis&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg\" class=\"  wp-image-9911 size-full\" title=\"Caixa\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg\" alt=\"\" width=\"333\" height=\"126\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg 333w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis-300x113.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis-150x56.jpg 150w\" sizes=\"auto, (max-width: 333px) 100vw, 333px\" \/><\/a><p id=\"caption-attachment-9911\" class=\"wp-caption-text\">Caixa com bolas de t\u00e9nis<\/p><\/div>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9910' onClick='GTTabs_show(1,9910)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9910'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Uma caixa de forma cil\u00edndrica com bolas de t\u00e9nis tem $7$ cm de di\u00e2metro e uma altura de $20$ cm.<\/p>\n<p>Se o di\u00e2metro de cada bola de t\u00e9nis for $6$ cm, qual a percentagem do volume ocupado pelas $3$ bolas?<\/p>\n<div id=\"attachment_9911\" style=\"width: 343px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-9911\" data-attachment-id=\"9911\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=9911\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg\" data-orig-size=\"333,126\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349284221&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=\"Caixa\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Caixa com bolas de t\u00e9nis&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg\" class=\"size-full wp-image-9911\" title=\"Caixa\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg\" alt=\"\" width=\"333\" height=\"126\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis.jpg 333w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis-300x113.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/caixabolastenis-150x56.jpg 150w\" sizes=\"auto, (max-width: 333px) 100vw, 333px\" \/><\/a><p id=\"caption-attachment-9911\" class=\"wp-caption-text\">Caixa com bolas de t\u00e9nis<\/p><\/div><\/blockquote>\n<p>Comecemos por calcular o volume da caixa, em cent\u00edmetros c\u00fabicos:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{V_C}}&amp; = &amp;{\\pi\u00a0 \\times {{\\left( {\\frac{7}{2}} \\right)}^2} \\times 20}\\\\<br \/>\n{}&amp; = &amp;{\\pi\u00a0 \\times \\frac{{49}}{4} \\times 20}\\\\<br \/>\n{}&amp; = &amp;{245\\pi }<br \/>\n\\end{array}$$<\/p>\n<p>Calculemos, agora, o volume (em cent\u00edmetros c\u00fabicos) de uma bola de t\u00e9nis:<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{{V_B}}&amp; = &amp;{\\frac{4}{3}\\pi\u00a0 \\times {3^3}}\\\\<br \/>\n{}&amp; = &amp;{36\\pi }<br \/>\n\\end{array}$$<\/p>\n<p>A fra\u00e7\u00e3o de volume da caixa ocupada pelas tr\u00eas bolas \u00e9:<\/p>\n<p>$$\\frac{{3 \\times {V_B}}}{{{V_C}}} = \\frac{{3 \\times 36\\pi }}{{245\\pi }} = \\frac{{108}}{{245}}$$<\/p>\n<p>Portanto, as tr\u00eas bolas de t\u00e9nis ocupam, aproximadamente, $44\\% $ do volume da caixa.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9910' onClick='GTTabs_show(0,9910)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma caixa de forma cil\u00edndrica com bolas de t\u00e9nis tem $7$ cm de di\u00e2metro e uma altura de $20$ cm. Se o di\u00e2metro de cada bola de t\u00e9nis for $6$ cm,&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20769,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,430,109],"series":[],"class_list":["post-9910","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-modulo-inicial","tag-volume"],"views":3106,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag033-11-b_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9910","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=9910"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9910\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20769"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9910"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9910"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9910"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9910"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}