{"id":8300,"date":"2012-04-13T00:07:37","date_gmt":"2012-04-12T23:07:37","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8300"},"modified":"2022-01-30T22:13:49","modified_gmt":"2022-01-30T22:13:49","slug":"uma-determinada-substancia-e-injetada-na-corrente-sanguinea","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8300","title":{"rendered":"Uma determinada subst\u00e2ncia \u00e9 injetada na corrente sangu\u00ednea"},"content":{"rendered":"<p><ul id='GTTabs_ul_8300' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8300' class='GTTabs_curr'><a  id=\"8300_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8300' ><a  id=\"8300_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_8300'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Quando uma determinada subst\u00e2ncia \u00e9 injetada na corrente sangu\u00ednea, a sua concentra\u00e7\u00e3o $C$, $t$ minutos depois, \u00e9 dada por $$C(t) = \\frac{1}{2}\\left( {{e^{ &#8211; 2t}} &#8211; {e^{ &#8211; 4t}}} \\right)$$<\/p>\n<ol>\n<li>Em que instante ocorre a concentra\u00e7\u00e3o m\u00e1xima e qual o seu valor?<\/li>\n<li>O que se pode dizer sobre a concentra\u00e7\u00e3o, ap\u00f3s um longo per\u00edodo de tempo?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8300' onClick='GTTabs_show(1,8300)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8300'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Quando uma determinada subst\u00e2ncia \u00e9 injetada na corrente sangu\u00ednea, a sua concentra\u00e7\u00e3o $C$, $t$ minutos depois, \u00e9 dada por $$C(t) = \\frac{1}{2}\\left( {{e^{ &#8211; 2t}} &#8211; {e^{ &#8211; 4t}}} \\right)$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Ora,\u00a0\\[\\begin{array}{*{20}{l}}<br \/>\n{C&#8217;\\left( t \\right)}&amp; = &amp;{{{\\left[ {\\frac{1}{2}\\left( {{e^{ &#8211; 2t}} &#8211; {e^{ &#8211; 4t}}} \\right)} \\right]}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{\\frac{1}{2}\\left( { &#8211; 2{e^{ &#8211; 2t}} + 4{e^{ &#8211; 4t}}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{2{e^{ &#8211; 4t}} &#8211; {e^{ &#8211; 2t}}} \\\\<br \/>\n{}&amp; = &amp;{{e^{ &#8211; 2t}}\\left( {2{e^{ &#8211; 2t}} &#8211; 1} \\right)}<br \/>\n\\end{array}\\]<br \/>\nComo $$\\begin{array}{*{20}{l}}<br \/>\n{C'(t) = 0}&amp; \\Leftrightarrow &amp;{{e^{ &#8211; 2t}}\\left( {2{e^{ &#8211; 2t}} &#8211; 1} \\right) = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{2{e^{ &#8211; 2t}} &#8211; 1 = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{e^{ &#8211; 2t}} = \\frac{1}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{ &#8211; 2t = \\ln \\frac{1}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{t =\u00a0 &#8211; \\frac{1}{2}\\ln \\frac{1}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\ln {{\\left( {\\frac{1}{2}} \\right)}^{ &#8211; \\frac{1}{2}}}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\ln \\sqrt 2 }<br \/>\n\\end{array}$$<br \/>\ntemos:<\/p>\n<table class=\" aligncenter\" style=\"width: 60%;\" 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;\">${\\ln \\sqrt 2 }$<\/td>\n<td style=\"border: 1px solid #00008b; text-align: right;\" colspan=\"2\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 $ + \\infty $<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$C'(t)$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\" colspan=\"2\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\" colspan=\"2\">$-$<\/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;\" colspan=\"2\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$\\frac{1}{8}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\" colspan=\"2\">$ \\searrow $<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>A concentra\u00e7\u00e3o m\u00e1xima \u00e9 0,125 e ocorre aproximadamente 0,55 ($\\ln \\sqrt 2\u00a0 \\approx 0,35$) minutos ap\u00f3s a subst\u00e2ncia ter sido injetada na corrente sangu\u00ednea.<br \/>\n\u00ad<\/li>\n<li>Ap\u00f3s um longo per\u00edodo de tempo, a concentra\u00e7\u00e3o anula-se, pois\u00a0\\[\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{t \\to\u00a0 + \\infty } C(t)}&amp; = &amp;{\\mathop {\\lim }\\limits_{t \\to\u00a0 + \\infty } \\left[ {\\frac{1}{2}\\left( {{e^{ &#8211; 2t}} &#8211; {e^{ &#8211; 4t}}} \\right)} \\right]} \\\\<br \/>\n{}&amp; = &amp;{\\frac{1}{2}\\left( {\\mathop {\\lim }\\limits_{t \\to\u00a0 + \\infty } {e^{ &#8211; 2t}} &#8211; \\mathop {\\lim }\\limits_{t \\to\u00a0 + \\infty } {e^{ &#8211; 4t}}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{1}{2}\\left( {0 &#8211; 0} \\right)} \\\\<br \/>\n{}&amp; = &amp;0<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<p style=\"text-align: center;\">\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\": 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