{"id":8204,"date":"2012-04-10T21:35:31","date_gmt":"2012-04-10T20:35:31","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8204"},"modified":"2022-01-30T21:51:17","modified_gmt":"2022-01-30T21:51:17","slug":"capacidade-pulmonar-de-um-ser-humano","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8204","title":{"rendered":"Capacidade pulmonar de um ser humano"},"content":{"rendered":"<p><ul id='GTTabs_ul_8204' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8204' class='GTTabs_curr'><a  id=\"8204_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8204' ><a  id=\"8204_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_8204'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Seja $f$ a fun\u00e7\u00e3o definida em $\\left[ {10,100} \\right]$ por $$f(x) = \\frac{{\\ln x &#8211; 2}}{x}$$<\/p>\n<ol>\n<li>Caraterize a fun\u00e7\u00e3o derivada $f&#8217;$.<\/li>\n<li>Representando por $x$ a idade, em anos, e por $g(x)$ a capacidade\u00a0pulmonar de um ser humano, em litros, admite-se que\u00a0$g(x) = 100 \\times f(x)$.<br \/>\nCalcule para que idade \u00e9 m\u00e1xima a capacidade pulmonar e qual \u00e9 o valor dessa capacidade pulmonar. Apresente os resultados nas unidades consideradas, com aproxima\u00e7\u00e3o \u00e0s d\u00e9cimas.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8204' onClick='GTTabs_show(1,8204)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8204'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Seja $f$ a fun\u00e7\u00e3o definida em $\\left[ {10,100} \\right]$ por $$f(x) = \\frac{{\\ln x &#8211; 2}}{x}$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Como\u00a0\\[\\begin{array}{*{20}{l}}<br \/>\n{f&#8217;\\left( x \\right)}&amp; = &amp;{{{\\left( {\\frac{{\\ln x &#8211; 2}}{x}} \\right)}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\frac{1}{x} \\times x &#8211; 1 \\times \\left( {\\ln x &#8211; 2} \\right)}}{{{x^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{3 &#8211; \\ln x}}{{{x^2}}}}<br \/>\n\\end{array}\\] ent\u00e3o $$\\begin{array}{*{20}{l}}<br \/>\n{f&#8217;:}&amp;{\\left[ {10,100} \\right] \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{{3 &#8211; \\ln x}}{{{x^2}}}}<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>Temos, portanto, $$g(x) = 100 \\times \\frac{{\\ln x &#8211; 2}}{x}$$ e $$g'(x) = 100 \\times \\frac{{3 &#8211; \\ln x}}{{{x^2}}}$$ com $x \\in \\left[ {10,100} \\right]$.<br \/>\n\u00ad<\/p>\n<table class=\" aligncenter\" style=\"width: 90%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$x$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$10$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${{e^3}}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$100$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Sinal de $g'(x) = 100 \\times \\frac{{3 &#8211; \\ln x}}{{{x^2}}}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/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=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">\u00a0Varia\u00e7\u00e3o de $g$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ &#8211; 20 + 10\\ln 10$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$100{e^{ &#8211; 3}}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\searrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ &#8211; 2 + \\ln 100$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>$$g(10) = 100 \\times \\frac{{\\ln 10 &#8211; 2}}{{10}} =\u00a0 &#8211; 20 + 10\\ln 10 \\approx 3,0$$<br \/>\n$$g({e^3}) = 100 \\times \\frac{{\\ln {e^3} &#8211; 2}}{{{e^3}}} = 100 \\times \\frac{1}{{{e^3}}} = 100{e^{ &#8211; 3}} \\approx 5,0$$<br \/>\n$$g(100) = 100 \\times \\frac{{\\ln 100 &#8211; 2}}{{100}} =\u00a0 &#8211; 2 + \\ln 100 \\approx 2,6$$<br \/>\n$${e^3} \\approx 20,1$$<\/p>\n<p>A m\u00e1xima capacidade pulmonar \u00e9, aproximadamente, 5,0 litros, atingindo-se cerca dos 20 anos de idade.<br \/>\n\u00ad<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><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\":779,\r\n\"height\":426,\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 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