{"id":11756,"date":"2014-02-05T01:44:58","date_gmt":"2014-02-05T01:44:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11756"},"modified":"2022-01-22T01:34:27","modified_gmt":"2022-01-22T01:34:27","slug":"uma-especie-rara-de-insetos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11756","title":{"rendered":"Uma esp\u00e9cie rara de insetos"},"content":{"rendered":"<p><ul id='GTTabs_ul_11756' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11756' class='GTTabs_curr'><a  id=\"11756_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11756' ><a  id=\"11756_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_11756'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Uma esp\u00e9cie rara de insetos foi descoberta na floresta tropical do Brasil.<\/p>\n<p>Ambientalistas colocaram os insetos numa \u00e1rea protegida.<\/p>\n<p>A popula\u00e7\u00e3o de insetos no m\u00eas $t$, ap\u00f3s terem sido colocados na \u00e1rea protegida, \u00e9 dado pela fun\u00e7\u00e3o: \\[P\\left( t \\right) = \\frac{{45\\left( {1 + 0,6t} \\right)}}{{3 + 0,02t}}\\]<\/p>\n<ol>\n<li>Qual era a popula\u00e7\u00e3o quando $t = 0$?<\/li>\n<li>Qual ser\u00e1 a popula\u00e7\u00e3o de insetos 10 anos depois de terem sido colocados na \u00e1rea protegida?<\/li>\n<li>Quando \u00e9 que a popula\u00e7\u00e3o atinge os $549$ insetos?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11756' onClick='GTTabs_show(1,11756)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11756'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Quando $t = 0$, a popula\u00e7\u00e3o era de $P\\left( 0 \\right) = \\frac{{45\\left( {1 + 0,6 \\times 0} \\right)}}{{3 + 0,02 \\times 0}} = \\frac{{45}}{3} = 15$ insetos.<br \/>\n\u00ad<\/li>\n<li>Como $P\\left( {120} \\right) = \\frac{{45\\left( {1 + 0,6 \\times 120} \\right)}}{{3 + 0,02 \\times 120}} = \\frac{{45 \\times 73}}{{5,4}} \\approx 608$, \u00e9 esperada uma popula\u00e7\u00e3o de $608$ insetos 10 anos depois de terem sido colocados na \u00e1rea protegida.<br \/>\n\u00ad<\/li>\n<li>Ora, tem-se sucessivamente:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{P\\left( t \\right) = 549}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\frac{{45\\left( {1 + 0,6t} \\right)}}{{3 + 0,02t}} = 549}&amp; \\wedge &amp;{t \\geqslant 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{45\\left( {1 + 0,6t} \\right) = 549\\left( {3 + 0,02t} \\right)}&amp; \\wedge &amp;{t \\geqslant 0}&amp;{\\left( {Porqu\u00ea?} \\right)}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{45 + 27t = 1647 + 10,98t}&amp; \\wedge &amp;{t \\geqslant 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{16,02t = 1602}&amp; \\wedge &amp;{t \\geqslant 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{t = \\frac{{1602}}{{16,02}}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{t = 100}<br \/>\n\\end{array}\\]<\/p>\n<p>A popula\u00e7\u00e3o atinge os $549$ insetos $100$ meses, ou seja, $8$ anos e $4$ meses ap\u00f3s terem sido colocados na \u00e1rea protegida.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11756' onClick='GTTabs_show(0,11756)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma esp\u00e9cie rara de insetos foi descoberta na floresta tropical do Brasil. Ambientalistas colocaram os insetos numa \u00e1rea protegida. A popula\u00e7\u00e3o de insetos no m\u00eas $t$, ap\u00f3s terem sido colocados na&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20870,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,125],"tags":[422,160,126],"series":[],"class_list":["post-11756","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-racionais","tag-11-o-ano","tag-equacao","tag-funcao-racional"],"views":2549,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11V2Pag050-4_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11756","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=11756"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11756\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20870"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11756"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11756"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11756"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11756"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}