{"id":7174,"date":"2011-11-15T21:43:24","date_gmt":"2011-11-15T21:43:24","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7174"},"modified":"2022-01-15T21:31:31","modified_gmt":"2022-01-15T21:31:31","slug":"um-comboio","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7174","title":{"rendered":"Um comboio"},"content":{"rendered":"<p><ul id='GTTabs_ul_7174' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7174' class='GTTabs_curr'><a  id=\"7174_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7174' ><a  id=\"7174_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_7174'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Um comboio, que viaja \u00e0 velocidade m\u00e9dia de 75 km\/h, faz um certo percurso em 36 minutos.<\/p>\n<p>Quanto tempo demoraria a fazer o mesmo percurso, se andasse a uma velocidade m\u00e9dia de 45 km\/h?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7174' onClick='GTTabs_show(1,7174)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7174'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>As grandezas velocidade m\u00e9dia (${{v}_{m}}$) e tempo ($t$) gasto no percurso s\u00e3o grandezas inversamente proporcionais, sabendo-se: \\[{{v}_{m}}=\\frac{d}{t}\\]<\/p>\n<p>Come\u00e7ando por tabelar os dados, temos:<\/p>\n<table class=\"aligncenter\" style=\"width: 40%; border-color: #000000;\" border=\"1\" cellspacing=\"1px\" cellpadding=\"1px\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"border-color: #000000; background-color: #dfeded;\">$t$ &#8211; tempo (horas)<\/td>\n<td style=\"border-color: #000000; background-color: #dfeded;\">0,6<\/td>\n<td style=\"border-color: #000000; background-color: #dfeded;\">t<\/td>\n<\/tr>\n<tr>\n<td style=\"border-color: #000000; background-color: #dfeded;\">${{v}_{m}}$ &#8211; velocidade m\u00e9dia (km\/h)<\/td>\n<td style=\"border-color: #000000; background-color: #dfeded;\">75<\/td>\n<td style=\"border-color: #000000; background-color: #dfeded;\">45<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00a0Nota que: \\[36\\,\\,\\min =\\frac{36}{60}\\,\\,h=\\frac{6}{10}\\,\\,h=0,6\\,\\,h\\]<\/p>\n<p>Como as grandezas da tabela s\u00e3o inversamente proporcionais, ent\u00e3o \u00e9 constante o produto dos valores correspondentes: \\[0,6\\times 75=t\\times 45\\]<\/p>\n<p>Resolvendo esta equa\u00e7\u00e3o, obtemos: \\[\\begin{array}{*{35}{l}}<br \/>\n0,6\\times 75=t\\times 45 &amp; \\Leftrightarrow\u00a0 &amp; t=\\frac{0,6\\times 75}{45}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; t=1\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, se andasse a uma velocidade m\u00e9dia de 45 km\/h, o comboio demoraria uma hora.<br \/>\n\u00ad<\/p>\n<p><strong>Alternativa<\/strong>:<\/p>\n<p>Usando a equa\u00e7\u00e3o escrita no in\u00edcio da p\u00e1gina, podemos come\u00e7ar por determinar o comprimento do percurso efetuado pelo comboio: \\[\\begin{array}{*{35}{l}}<br \/>\n75=\\frac{d}{0,6} &amp; \\Leftrightarrow\u00a0 &amp; d=0,6\\times 75\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; d=45\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Sabendo que o percurso \u00e9 de 45 km, podemos agora determinar o tempo que o comboio demorar\u00e1 a efetu\u00e1-lo \u00e0 velocidade m\u00e9dia de 45 km\/h. Naturalmente, n\u00e3o \u00e9 necess\u00e1rio utilizar novamente a f\u00f3rmula: basta interpretar os dados, para concluir que o tempo necess\u00e1rio \u00e9 uma hora.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7174' onClick='GTTabs_show(0,7174)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um comboio, que viaja \u00e0 velocidade m\u00e9dia de 75 km\/h, faz um certo percurso em 36 minutos. Quanto tempo demoraria a fazer o mesmo percurso, se andasse a uma velocidade m\u00e9dia&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20290,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,249],"tags":[426,250],"series":[],"class_list":["post-7174","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-proporcionalidade-inversa-e-funcoes-algebricas","tag-9-o-ano","tag-proporcionalidade-inversa-2"],"views":3139,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/9V1Pag069-4_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7174","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=7174"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7174\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20290"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7174"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}