{"id":10911,"date":"2012-10-25T12:28:29","date_gmt":"2012-10-25T11:28:29","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10911"},"modified":"2022-01-02T00:54:17","modified_gmt":"2022-01-02T00:54:17","slug":"a-visita-de-estudo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10911","title":{"rendered":"A visita de estudo"},"content":{"rendered":"<p><ul id='GTTabs_ul_10911' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10911' class='GTTabs_curr'><a  id=\"10911_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10911' ><a  id=\"10911_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_10911'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>\u00c0s 12:00 horas, o Josu\u00e9 disse \u00e0s amigas Rute e In\u00eas: &#8220;O 7.\u00ba ano vai fazer uma visita de estudo.&#8221;<\/p>\n<p>\u00c0s 12:05 horas, a Rute e a In\u00eas tinham contado a novidade ao Joel, ao Pedro, \u00e0 T\u00e2nia e \u00e0 Sara. E assim se espalhou a not\u00edcia: cada alno encarregou-se de avisar outros dois colegas no espa\u00e7o de cinco minutos.<\/p>\n<ol>\n<li>Quantos alunos souberam a not\u00edcia \u00e0s 12:10 horas? \u00c0s 12:15 horas? E \u00e0s 12:30 horas?<\/li>\n<li>Ao fim de meia hora, quantas pessoas sabiam a not\u00edcia da visita de estudo?<br \/>\nExplica como chegaste \u00e0 tua resposta.<\/li>\n<li>Ao fim de 35 minutos todos os alunos do 7.\u00ba ano sabiam da not\u00edcia.<br \/>\nQuantos eram os alunos do 7.\u00ba ano?<br \/>\nExplica como chegaste \u00e0 tua resposta.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10911' onClick='GTTabs_show(1,10911)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10911'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Comecemos por construir uma tabela com os dados conhecidos e outros calculados segundo a regra estabelecida:<\/p>\n<table border=\"0\" align=\"center\">\n<caption><strong>Como se espalhou a not\u00edcia da visita de estudo<\/strong><\/caption>\n<tbody>\n<tr>\n<td>Hora<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">12:00<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">12:05<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">12:10<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">12:15<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">&#8230;<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">Termo geral<\/td>\n<\/tr>\n<tr>\n<td>N\u00famero de minutos decorridos<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$0 \\times 5 = 0$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$1 \\times 5 = 5$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$2 \\times 5 = 10$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$3 \\times 5 = 15$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">&#8230;<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$n \\times 5 = 5n$<\/td>\n<\/tr>\n<tr>\n<td>N\u00famero de pessoas que souberam a not\u00edcia<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$2$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$2 \\times 2 = 4$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$2 \\times 2 \\times 2 = 8$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$2 \\times 2 \\times 2 \\times 2 = 16$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">&#8230;<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$\\underbrace {2 \\times 2 \\times &#8230; \\times 2}_{\\left( {n + 1} \\right)\\,\\,fatores} = {2^{n + 1}}$<\/td>\n<\/tr>\n<tr>\n<td>N\u00famero de pessoas que sabiam a not\u00edcia<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$1 + 2 = 3$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$1 + 2 + 4 = 7$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$1 + 2 + 4 + 8 = 15$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$1 + 2 + 4 + 8 + 16 = 31$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">&#8230;<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">\u00a0${2^{n + 2}} &#8211; 1$<\/td>\n<\/tr>\n<tr>\n<td>$n$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$0$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$1$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$2$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$3$<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">&#8230;<\/td>\n<td style=\"border: 1px solid #b0e0e6; text-align: center;\">$n$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00ad<\/p>\n<ol>\n<li>\u00c0s 12:10 horas, souberam a not\u00edcia $8$ alunos.\n<p>\u00c0s 12:15 horas, souberam a not\u00edcia $16$ alunos.<\/p>\n<p>At\u00e9 \u00e0s 12:30 horas decorreram $6$ vezes $5$ minutos ap\u00f3s as 12:00\u00a0horas. Logo, \u00e0s 12:30 horas, souberam a not\u00edcia $\\underbrace {2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2}_{7\\,\\,fatores} = 128$ alunos.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\u00c0s 12:00 horas, duas pessoas souberam a novidade. Passados $5$ minutos, souberam a novidade o dobro das pessoas que tomaram conhecimento \u00e0s 12:00, e assim sucessivamente.\n<p>O Josu\u00e9 era o \u00fanico aluno do 7.\u00ba ano que sabia a not\u00edcia antes das 12:00 horas.<\/p>\n<p>Como at\u00e9 \u00e0s 12:30 decorreram $6$ vezes $5$ minutos ap\u00f3s as 12:00 horas, o n\u00famero de alunos do 7.\u00ba ao que sabiam da visita de estudo \u00e9:<br \/>\n$$1 + 2 + \\underbrace {4 + 8 + 16 + 32 + 64 + 128}_{6\\,\\,parcelas} = 255$$\u00ad<\/p>\n<\/li>\n<li>\u00c0s 12:35 horas, souberam a not\u00edcia os restantes $2 \\times 128 = 256$ alunos do 7.\u00ba ano.\n<p>Logo, o 7.\u00ba ano tem $255 + 256 = 511$ alunos.<br \/>\n\u00ad<\/p>\n<\/li>\n<\/ol>\n<p><strong>Nota<\/strong>:<\/p>\n<p>Em alternativa, podem ser utilizadas as express\u00f5es do termo geral das sequ\u00eancias apresentadas na \u00faltima coluna da tabela. Experimenta!<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10911' onClick='GTTabs_show(0,10911)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado \u00c0s 12:00 horas, o Josu\u00e9 disse \u00e0s amigas Rute e In\u00eas: &#8220;O 7.\u00ba ano vai fazer uma visita de estudo.&#8221; \u00c0s 12:05 horas, a Rute e a In\u00eas tinham contado a&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19188,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[317,97,318],"tags":[428,320,319],"series":[],"class_list":["post-10911","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-7-o-ano","category-aplicando","category-numeros-inteiros","tag-7-o-ano","tag-multiplicacao","tag-numeros-inteiros-2"],"views":2243,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat74.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10911","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=10911"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10911\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10911"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10911"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10911"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}