{"id":14762,"date":"2018-04-22T14:55:45","date_gmt":"2018-04-22T13:55:45","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14762"},"modified":"2024-08-04T23:16:52","modified_gmt":"2024-08-04T22:16:52","slug":"o-tempo-que-demora-a-encher-um-tanque","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14762","title":{"rendered":"O tempo que demora a encher um tanque"},"content":{"rendered":"<p><ul id='GTTabs_ul_14762' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14762' class='GTTabs_curr'><a  id=\"14762_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14762' ><a  id=\"14762_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_14762'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>O tempo, em horas, que demora a encher um tanque \u00e9 inversamente proporcional ao n\u00famero de m<sup>3<\/sup> de \u00e1gua que uma torneira debita por hora\u00a0(caudal da torneira). O tanque fica cheio com 60 m<sup>3<\/sup> de \u00e1gua.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14763\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14763\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10.png\" data-orig-size=\"588,108\" 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=\"Tabela\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10.png\" class=\"aligncenter wp-image-14763\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10-300x55.png\" alt=\"\" width=\"340\" height=\"62\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10-300x55.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10.png 588w\" sizes=\"auto, (max-width: 340px) 100vw, 340px\" \/><\/a><\/p>\n<ol>\n<li>A tabela anterior relaciona o caudal da torneira com o tempo necess\u00e1rio para encher o tanque.<br \/>\nQual \u00e9 o valor de <em>a<\/em>?<\/li>\n<li>Qual dos gr\u00e1ficos seguintes pode representar a rela\u00e7\u00e3o entre o caudal, em m<sup>3<\/sup> por hora, da torneira que enche o tanque e o tempo, em horas, que \u00e9 necess\u00e1rio para encher o tanque?<br \/>\n<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14765\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14765\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png\" data-orig-size=\"720,615\" 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=\"Gr\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png\" class=\"aligncenter wp-image-14765\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png\" alt=\"\" width=\"520\" height=\"444\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png 720w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b-300x256.png 300w\" sizes=\"auto, (max-width: 520px) 100vw, 520px\" \/><\/a><\/li>\n<li>Para um determinado caudal da torneira que enche o tanque, a altura, <em>h<\/em>, que a \u00e1gua atinge no tanque, <em>t<\/em> horas depois de se iniciar o enchimento, \u00e9 dada, em dec\u00edmetros, por \\(h = 1,5\\,t\\). Se o enchimento do tanque se iniciar hoje \u00e0s 15 horas, a que horas a \u00e1gua atingir\u00e1, no tanque, 3,75 dm de altura?<br \/>\nApresenta a resposta em horas e minutos.<br \/>\nApresenta todos os c\u00e1lculos que efetuares.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14762' onClick='GTTabs_show(1,14762)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14762'>\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\/04\/9V2Pag115-10.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14763\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14763\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10.png\" data-orig-size=\"588,108\" 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=\"Tabela\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10.png\" class=\"aligncenter wp-image-14763\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10-300x55.png\" alt=\"\" width=\"340\" height=\"62\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10-300x55.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10.png 588w\" sizes=\"auto, (max-width: 340px) 100vw, 340px\" \/><\/a><\/p>\n<ol>\n<li>Como as grandezas s\u00e3o inversamente proporcionais, ent\u00e3o o produto das medidas correspondentes dessas grandezas \u00e9 constante.<br \/>\nAssim, temos:\\[\\begin{array}{*{20}{l}}{5 \\times 12 = a \\times 8}&amp; \\Leftrightarrow &amp;{a = \\frac{{5 \\times 12}}{8}}\\\\{}&amp; \\Leftrightarrow &amp;{a = 7,5}\\end{array}\\]Portanto, \\({a = 7,5}\\) (m<sup>3<\/sup>\/h).<br \/>\n\u00ad<\/li>\n<li>Considerando o caudal (c), em m<sup>3<\/sup> por hora, da torneira que enche o tanque e o tempo (t), em horas, que \u00e9 necess\u00e1rio para encher o tanque, sabe-se que\u00a0\\(c \\times t = 60\\), que \u00e9 equivalente a\u00a0\\(t = \\frac{{60}}{t}\\) e cuja representa\u00e7\u00e3o gr\u00e1fica corresponde \u00e0 apresentada na alternativa [A].<br \/>\n<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14765\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14765\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png\" data-orig-size=\"720,615\" 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=\"Gr\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png\" class=\"aligncenter wp-image-14765\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png\" alt=\"\" width=\"520\" height=\"444\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b.png 720w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10b-300x256.png 300w\" sizes=\"auto, (max-width: 520px) 100vw, 520px\" \/><\/a><\/li>\n<li>Para <em>h<\/em> igual a 3,75 dm, temos:\\[\\begin{array}{*{20}{l}}{3,75 = 1,5\\,t}&amp; \\Leftrightarrow &amp;{t = \\frac{{3,75}}{{1,5}}}\\\\{}&amp; \\Leftrightarrow &amp;{t = 2,5}\\end{array}\\]Isto \u00e9, a \u00e1gua demorar\u00e1 2,5 horas a atingir, no tanque, 3,75 dm de altura. Por isso, se o enchimento do tanque se iniciar hoje \u00e0s 15 horas, essa altura da \u00e1gua, no tanque, ocorrer\u00e1 hoje \u00e0s 17 horas e 30 minutos.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14762' onClick='GTTabs_show(0,14762)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado O tempo, em horas, que demora a encher um tanque \u00e9 inversamente proporcional ao n\u00famero de m3 de \u00e1gua que uma torneira debita por hora\u00a0(caudal da torneira). O tanque fica cheio&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14767,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[97,249],"tags":[426,497,250],"series":[],"class_list":["post-14762","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-aplicando","category-proporcionalidade-inversa-e-funcoes-algebricas","tag-9-o-ano","tag-proporcionalidade-direta","tag-proporcionalidade-inversa-2"],"views":4747,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag115-10a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14762","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=14762"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14762\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14767"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14762"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14762"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14762"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14762"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}