{"id":11869,"date":"2014-05-02T01:41:38","date_gmt":"2014-05-02T00:41:38","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11869"},"modified":"2022-01-22T16:48:16","modified_gmt":"2022-01-22T16:48:16","slug":"a-distancia-entre-os-automoveis","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11869","title":{"rendered":"A dist\u00e2ncia entre os autom\u00f3veis"},"content":{"rendered":"<p><ul id='GTTabs_ul_11869' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11869' class='GTTabs_curr'><a  id=\"11869_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11869' ><a  id=\"11869_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_11869'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Dois autom\u00f3veis circulam \u00e0 mesma velocidade, em estradas perpendiculares, em dire\u00e7\u00e3o a um cruzamento.<br \/>\nUm deles encontra-se a $5$ km do cruzamento e o outro a $6$ km.<\/p>\n<p>Representa graficamente a fun\u00e7\u00e3o que d\u00e1 a dist\u00e2ncia entre os dois autom\u00f3veis \u00e0 medida que se aproximam do cruzamento.<\/p>\n<p>Utilizando a calculadora gr\u00e1fica, determina quando \u00e9 que a dist\u00e2ncia entre os autom\u00f3veis \u00e9 m\u00ednima.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11869' onClick='GTTabs_show(1,11869)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11869'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Seja $x$ a dist\u00e2ncia, em km,\u00a0percorrida por cada um dos autom\u00f3veis em fun\u00e7\u00e3o do tempo.<\/p>\n<p>Em cada instante (com exce\u00e7\u00e3o de dois [quais?]), os dois autom\u00f3veis e o cruzamento\u00a0definem os v\u00e9rtices de um tri\u00e2ngulo ret\u00e2ngulo de catetos com comprimentos $\\left| {6 &#8211; x} \\right|$ e $\\left| {5 &#8211; x} \\right|$, em km.<\/p>\n<p>Ent\u00e3o, em cada instante ap\u00f3s o\u00a0momento inicial (inclusive nos dois acima referidos [porqu\u00ea?]), a dist\u00e2ncia, em km, entre os autom\u00f3veis pode ser expressa por: \\[d\\left( x \\right) = \\sqrt {{{\\left( {6 &#8211; x} \\right)}^2} + {{\\left( {5 &#8211; x} \\right)}^2}} \\]<\/p>\n<p>Recorrendo \u00e0 calculadora gr\u00e1fica obt\u00e9m-se:<\/p>\n<table class=\" aligncenter\">\n<tbody>\n<tr>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Janela2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11870\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11870\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Janela2.jpg\" data-orig-size=\"264,136\" 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=\"Janela\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Janela2.jpg\" class=\"alignright size-full wp-image-11870\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Janela2.jpg\" alt=\"Janela\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Janela2.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Janela2-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11871\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11871\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2a.jpg\" data-orig-size=\"264,136\" 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=\"Gr\u00e1fico 1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2a.jpg\" class=\"alignright size-full wp-image-11871\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2a.jpg\" alt=\"Gr\u00e1fico 1\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2a.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2a-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2b.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11872\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11872\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2b.jpg\" data-orig-size=\"264,136\" 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=\"Gr\u00e1fico 2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2b.jpg\" class=\"alignright size-full wp-image-11872\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2b.jpg\" alt=\"Gr\u00e1fico 2\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2b.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/05\/Graf2b-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>A dist\u00e2ncia entre os autom\u00f3veis \u00e9 m\u00ednima quando eles se encontram a $0,5$ km do cruzamento, tendo um deles passado o cruzamento e o outro ainda n\u00e3o o alcan\u00e7ou.<\/p>\n<ul style=\"list-style-type: square;\">\n<li>\n<div style=\"text-align: left;\"><strong>Extens\u00e3o<\/strong>: Confirme\u00a0que a dist\u00e2ncia entre os autom\u00f3veis \u00e9 m\u00ednima quando eles se encontram a $0,5$ km do cruzamento, considerando que a derivada da fun\u00e7\u00e3o \u00e9 definida por: \\[d&#8217;\\left( x \\right) = \\frac{{2x &#8211; 11}}{{\\sqrt {{{\\left( {6 &#8211; x} \\right)}^2} + {{\\left( {5 &#8211; x} \\right)}^2}} }}\\]<\/div>\n<\/li>\n<\/ul>\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\": 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Um deles encontra-se a $5$ km do cruzamento e o outro a $6$ km. 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