{"id":11839,"date":"2014-03-17T22:15:17","date_gmt":"2014-03-17T22:15:17","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11839"},"modified":"2022-01-25T01:02:15","modified_gmt":"2022-01-25T01:02:15","slug":"uma-colonia-de-bacterias","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11839","title":{"rendered":"Uma col\u00f3nia de bact\u00e9rias"},"content":{"rendered":"<p><ul id='GTTabs_ul_11839' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11839' class='GTTabs_curr'><a  id=\"11839_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11839' ><a  id=\"11839_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_11839'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Volvox.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11840\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11840\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Volvox.jpg\" data-orig-size=\"360,340\" 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;}\" data-image-title=\"Volvox\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Volvox.jpg\" class=\"alignright  wp-image-11840\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Volvox.jpg\" alt=\"Volvox\" width=\"173\" height=\"163\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Volvox.jpg 360w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Volvox-300x283.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Volvox-150x141.jpg 150w\" sizes=\"auto, (max-width: 173px) 100vw, 173px\" \/><\/a>A popula\u00e7\u00e3o inicial de uma col\u00f3nia de bact\u00e9rias \u00e9 $100 000$ unidades.<\/p>\n<p>Depois de $t$ horas, a col\u00f3nia tem uma popula\u00e7\u00e3o $P\\left( t \\right)$, que obedece \u00e0 lei polinomial seguinte:<\/p>\n<p>\\[P\\left( t \\right) = 10000\\,{t^3}\\]<\/p>\n<ol>\n<li>Qual \u00e9 o n\u00famero de bact\u00e9rias ap\u00f3s $10$ horas?<\/li>\n<li>Encontre a lei que indica a taxa de varia\u00e7\u00e3o da popula\u00e7\u00e3o $P\\left( t \\right)$ em rela\u00e7\u00e3o ao tempo $t$.<\/li>\n<li>Determine essa taxa de varia\u00e7\u00e3o ap\u00f3s $10$ horas.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11839' onClick='GTTabs_show(1,11839)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11839'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>\u00c9 $P\\left( {10} \\right) = 10000 \\times {10^3} = {10^7}$ o n\u00famero de bact\u00e9rias ap\u00f3s $10$ horas.<br \/>\n\u00ad<\/li>\n<li>Seja ${t_0} \\in \\mathbb{R}_0^ + $.<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{P&#8217;\\left( {{t_0}} \\right)}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{P\\left( {{t_0} + h} \\right) &#8211; P\\left( {{t_0}} \\right)}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{10000\\,{{\\left( {{t_0} + h} \\right)}^3} &#8211; 10000\\,{t_0}^3}}{h}} \\\\<br \/>\n{}&amp; = &amp;{10000 \\times \\mathop {\\lim }\\limits_{h \\to 0} \\frac{{\\,\\left( {{t_0}^2 + 2{t_0}h + {h^2}} \\right)\\left( {{t_0} + h} \\right) &#8211; \\,{t_0}^3}}{h}} \\\\<br \/>\n{}&amp; = &amp;{10000 \\times \\mathop {\\lim }\\limits_{h \\to 0} \\frac{{\\,{t_0}^3 + {t_0}^2h + 2{t_0}^2h + 2{t_0}{h^2} + {h^2}{t_0} + {h^3} &#8211; \\,{t_0}^3}}{h}} \\\\<br \/>\n{}&amp; = &amp;{10000 \\times \\mathop {\\lim }\\limits_{h \\to 0} \\frac{{3{t_0}^2h + 3{t_0}{h^2} + {h^3}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{10000 \\times \\mathop {\\lim }\\limits_{h \\to 0} \\left( {3{t_0}^2 + 3{t_0}h + {h^3}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{10000 \\times 3{t_0}^2} \\\\<br \/>\n{}&amp; = &amp;{30000 \\times {t_0}^2}<br \/>\n\\end{array}\\]<br \/>\nPortanto, $P\\left( t \\right) = 30000 \\times {t^2}$, com $t \\in \\mathbb{R}_0^ + $.<br \/>\n\u00ad<\/li>\n<li>\u00a0Ap\u00f3s $10$ horas, a taxa de varia\u00e7\u00e3o \u00e9 $P\\left( {10} \\right) = 30000 \\times {10^2} = 3 \\times {10^6}$ bact\u00e9rias por hora.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11839' onClick='GTTabs_show(0,11839)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A popula\u00e7\u00e3o inicial de uma col\u00f3nia de bact\u00e9rias \u00e9 $100 000$ unidades. Depois de $t$ horas, a col\u00f3nia tem uma popula\u00e7\u00e3o $P\\left( t \\right)$, que obedece \u00e0 lei polinomial seguinte: \\[P\\left(&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20946,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,134],"tags":[422,145],"series":[],"class_list":["post-11839","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-derivadas","tag-11-o-ano","tag-derivadas-2"],"views":2489,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/11V2Pag089-4_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11839","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=11839"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11839\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20946"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11839"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11839"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11839"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11839"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}