{"id":7338,"date":"2012-01-19T01:38:34","date_gmt":"2012-01-19T01:38:34","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7338"},"modified":"2021-12-30T21:01:06","modified_gmt":"2021-12-30T21:01:06","slug":"a-populacao-de-uma-cidade","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7338","title":{"rendered":"A popula\u00e7\u00e3o de uma cidade"},"content":{"rendered":"<p><ul id='GTTabs_ul_7338' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7338' class='GTTabs_curr'><a  id=\"7338_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7338' ><a  id=\"7338_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_7338'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>A popula\u00e7\u00e3o de uma cidade aumenta 5% por ano.<\/p>\n<p>Sup\u00f5e-se que no in\u00edcio de 1990 a popula\u00e7\u00e3o era de 100.000 habitantes.<\/p>\n<ol>\n<li>Designe por $P(n)$ o n\u00famero de habitantes no in\u00edcio do ano $1989+n$ (com $n\\in \\mathbb{N}$).<br \/>\nQual o valor de $P(1)$?<br \/>\nEstabele\u00e7a uma rela\u00e7\u00e3o entre $P(n)$ e $P(n+1)$ e, em seguida, deduza a express\u00e3o de $P(n)$ em fun\u00e7\u00e3o de $n$.<\/li>\n<li>Qual ser\u00e1 o n\u00famero de habitantes da referida cidade no in\u00edcio do ano 2010?<\/li>\n<li>A partir de que ano ser\u00e1 o n\u00famero de habitantes da cidade superior a 200.000?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7338' onClick='GTTabs_show(1,7338)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7338'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $P(1)={{P}_{1990}}=100000={{10}^{5}}$.<br \/>\nComo a popula\u00e7\u00e3o da cidade aumenta 5% por ano, tem-se: $P(n+1)=P(n)+0,05\\times P(n)=1,05\\times P(n)$.<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\nP(2) &amp; = &amp; 1,05\\times P(1) &amp; {} &amp; {}\u00a0 \\\\<br \/>\nP(3) &amp; = &amp; 1,05\\times P(2) &amp; = &amp; {{1,05}^{2}}\\times P(1)\u00a0 \\\\<br \/>\nP(4) &amp; = &amp; 1,05\\times P(3) &amp; = &amp; {{1,05}^{3}}\\times P(1)\u00a0 \\\\<br \/>\n&#8230; &amp; {} &amp; &#8230; &amp; {} &amp; &#8230;\u00a0 \\\\<br \/>\nP(n) &amp; = &amp; 1,05\\times P(n-1) &amp; = &amp; {{1,05}^{n-1}}\\times P(1)\u00a0 \\\\<br \/>\n\\end{array}$$<br \/>\nComo $P(1)={{10}^{5}}$, ent\u00e3o $P(n)={{10}^{5}}\\times {{1,05}^{n-1}}$.<br \/>\n\u00ad<\/li>\n<li>No in\u00edcio de 2010, o n\u00famero de habitantes da referida cidade \u00e9 ${{P}_{2010}}=P(21)={{10}^{5}}\\times {{1,05}^{20}}\\approx 265330$, aproximadamente.<br \/>\n\u00ad<\/li>\n<li>Ora, $$\\begin{matrix}<br \/>\n\\begin{array}{*{35}{l}}<br \/>\nP(n)&gt;2\\times {{10}^{5}} &amp; \\Leftrightarrow\u00a0 &amp; {{10}^{5}}\\times {{1,05}^{n-1}}&gt;2\\times {{10}^{5}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{1,05}^{n-1}}&gt;2\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; n-1&gt;{{\\log }_{1,05}}2\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; n&gt;1+{{\\log }_{1,05}}2\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; n&gt;1+\\frac{\\ln 2}{\\ln 1,05}\u00a0 \\\\<br \/>\n\\end{array}\u00a0 \\\\<br \/>\n{}\u00a0 \\\\<br \/>\n{}\u00a0 \\\\<br \/>\n\\end{matrix}$$<br \/>\nComo $1+\\frac{\\ln 2}{\\ln 1,05}\\approx 15,2$, o n\u00famero de habitantes da cidade ser\u00e1 superior a 200.000 a partir de 2005 (1989+16).<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7338' onClick='GTTabs_show(0,7338)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A popula\u00e7\u00e3o de uma cidade aumenta 5% por ano. Sup\u00f5e-se que no in\u00edcio de 1990 a popula\u00e7\u00e3o era de 100.000 habitantes. Designe por $P(n)$ o n\u00famero de habitantes no in\u00edcio do&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19699,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,267],"tags":[427,268],"series":[],"class_list":["post-7338","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-exponenciais-e-logaritmicas","tag-12-o-ano","tag-funcao-exponencial"],"views":2735,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat260.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7338","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=7338"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7338\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19699"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7338"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7338"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7338"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7338"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}