{"id":8240,"date":"2012-04-11T18:41:53","date_gmt":"2012-04-11T17:41:53","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8240"},"modified":"2022-01-30T22:03:12","modified_gmt":"2022-01-30T22:03:12","slug":"um-novo-analgesico-o-antidor","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8240","title":{"rendered":"Um novo analg\u00e9sico: o AntiDor"},"content":{"rendered":"<p><ul id='GTTabs_ul_8240' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8240' class='GTTabs_curr'><a  id=\"8240_0\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<li id='GTTabs_li_1_8240' ><a  id=\"8240_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_8240'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><\/p>\n<p>Um laborat\u00f3rio farmac\u00eautico lan\u00e7ou no mercado um novo analg\u00e9sico: o <em>AntiDor<\/em>.<\/p>\n<p>A concentra\u00e7\u00e3o desse medicamento, em decigramas por litro de sangue, $t$ horas ap\u00f3s ser administrado a uma pessoa, \u00e9 dada por $$C(t) = {t^2}{e^{ &#8211; 0,6t}}\\,\\,\\,\\left( {t \\geqslant 0} \\right)$$<\/p>\n<ol>\n<li>Recorrendo exclusivamente a processos anal\u00edticos, determine o valor de $t$ para o qual \u00e9 m\u00e1xima a concentra\u00e7\u00e3o de <em>AntiDor<\/em> no sangue de uma pessoa que o tenha tomado.<br \/>\nCalcule o valor dessa concentra\u00e7\u00e3o m\u00e1xima, apresentando o resultado na unidade considerada, com aproxima\u00e7\u00e3o \u00e0s d\u00e9cimas.<\/li>\n<li>O mesmo laborat\u00f3rio realizou uma campanha de promo\u00e7\u00e3o deste medicamento, baseada no <em>slogan<\/em>: \u00ab<em>AntiDor &#8211; A\u00e7\u00e3o r\u00e1pida e prolongada!<\/em>\u00bb<br \/>\nNuma breve composi\u00e7\u00e3o, comente o <em>slogan<\/em>, tendo em conta que:<\/p>\n<p>&#8211; para a maioria das dores, o <em>AntiDor<\/em> s\u00f3 produz efeito se a sua concentra\u00e7\u00e3o for superior a 1 decigrama por litro de sangue;<\/p>\n<p>&#8211; de acordo com uma associa\u00e7\u00e3o de defesa do consumidor, um bom analg\u00e9sico deve come\u00e7ar a produzir efeito, no m\u00e1ximo, meia hora ap\u00f3s ter sido tomado, e a sua a\u00e7\u00e3o deve permanecer durante, pelo menos, cinco horas (ap\u00f3s ter come\u00e7ado a produzir efeito).<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8240' onClick='GTTabs_show(1,8240)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8240'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Um laborat\u00f3rio farmac\u00eautico lan\u00e7ou no mercado um novo analg\u00e9sico: o <em>AntiDor<\/em>.<\/p>\n<p>A concentra\u00e7\u00e3o desse medicamento, em decigramas por litro de sangue, $t$ horas ap\u00f3s ser administrado a uma pessoa, \u00e9 dada por $$C(t) = {t^2}{e^{ &#8211; 0,6t}}\\,\\,\\,\\left( {t \\geqslant 0} \\right)$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Ora,\u00a0\\[\\begin{array}{*{20}{l}}<br \/>\n{C'(t)}&amp; = &amp;{{{\\left( {{t^2}{e^{ &#8211; 0,6t}}} \\right)}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{2t \\times {e^{ &#8211; 0,6t}} &#8211; 0,6{t^2}{e^{ &#8211; 0,6t}}} \\\\<br \/>\n{}&amp; = &amp;{\\left( {2t &#8211; 0,6{t^2}} \\right){e^{ &#8211; 0,6t}}}<br \/>\n\\end{array}\\]<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{C'(t) = 0}&amp; \\Leftrightarrow &amp;{\\left( {2t &#8211; 0,6{t^2}} \\right){e^{ &#8211; 0,6t}} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{2t &#8211; 0,6{t^2} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{2t\\left( {1 &#8211; 0,3t} \\right) = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{t = 0}&amp; \\vee &amp;{t = \\frac{{10}}{3}}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$t$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${\\frac{{10}}{3}}$<\/td>\n<td style=\"text-align: right; border: #00008b 1px solid;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 $ + \\infty $<\/td>\n<\/tr>\n<tr>\n<td style=\"border: #00008b 1px solid;\">$C'(t) = \\left( {2t &#8211; 0,6{t^2}} \\right){e^{ &#8211; 0,6t}}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"border: #00008b 1px solid;\">$-$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${C(t)}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$\\frac{{100}}{{9{e^2}}}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\searrow $<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>$$C\\left( {\\frac{{10}}{3}} \\right) = {\\left( {\\frac{{10}}{3}} \\right)^2}{e^{ &#8211; 2}} = \\frac{{100}}{{9{e^2}}} \\approx 1,5$$<br \/>\nA concentra\u00e7\u00e3o m\u00e1xima \u00e9, aproximadamente, 1,5 decigramas por litro de sangue. Este valor de concentra\u00e7\u00e3o de <em>AntiDor<\/em> ocorre 3 horas e 20 minutos ap\u00f3s a tomada do medicamento.<br \/>\n\u00ad<\/li>\n<li>\u00ad<br \/>\n<script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":903,\r\n\"height\":353,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/p>\n<p>O slogan \u00e9 enganador, pois a a\u00e7\u00e3o do medicamento n\u00e3o \u00e9 r\u00e1pida e prolongada. O <em>AntiDor<\/em> apenas produz efeito ao fim de mais de uma hora e meia, depois de ter sido tomado, o que ultrapassa em muito a meia hora recomendada pela associa\u00e7\u00e3o de defesa do consumidor. Por outro lado, esse efeito dura menos de quatro horas e meia, ficando aqu\u00e9m das cinco horas preconizadas pela referida associa\u00e7\u00e3o.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8240' onClick='GTTabs_show(0,8240)'>&lt;&lt; Resolu\u00e7\u00e3o<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Resolu\u00e7\u00e3o Resolu\u00e7\u00e3o Resolu\u00e7\u00e3o Um laborat\u00f3rio farmac\u00eautico lan\u00e7ou no mercado um novo analg\u00e9sico: o AntiDor. A concentra\u00e7\u00e3o desse medicamento, em decigramas por litro de sangue, $t$ horas ap\u00f3s ser administrado a uma pessoa, \u00e9 dada&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21157,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,292],"tags":[427,136,144],"series":[],"class_list":["post-8240","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-calculo-diferencial","tag-12-o-ano","tag-derivada","tag-extremos-relativos"],"views":2849,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V2Pag229-94_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8240","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=8240"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8240\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21157"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8240"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8240"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8240"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8240"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}