{"id":6433,"date":"2011-01-07T21:14:39","date_gmt":"2011-01-07T21:14:39","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6433"},"modified":"2022-01-21T02:58:51","modified_gmt":"2022-01-21T02:58:51","slug":"uma-ponte-aerea","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6433","title":{"rendered":"Uma ponte a\u00e9rea"},"content":{"rendered":"<p><ul id='GTTabs_ul_6433' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6433' class='GTTabs_curr'><a  id=\"6433_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6433' ><a  id=\"6433_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_6433'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Pretende-se organizar uma ponte a\u00e9rea para transportar 1600 pessoas e 90 toneladas de bagagem.<\/p>\n<p>Os avi\u00f5es dispon\u00edveis s\u00e3o de dois tipos: 12 do tipo A e 9 do tipo B.<\/p>\n<p>Com carga completa, um avi\u00e3o do tipo A pode transportar 200 pessoas e 6 toneladas de bagagem. Um avi\u00e3o do tipo B pode transportar 100 pessoas e 15 toneladas.<\/p>\n<p>Quantos avi\u00f5es de cada tipo devem ser contratados para que o custo do aluguer seja m\u00ednimo, sabendo que o aluguer do avi\u00e3o do tipo A custa 4 milh\u00f5es de euros e o do tipo B custa 1 milh\u00e3o de euros.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6433' onClick='GTTabs_show(1,6433)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6433'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><\/p>\n<p>Para facilitar a an\u00e1lise do problema, comecemos por organizar parte dos dados numa tabela:<\/p>\n<table border=\"1\" cellspacing=\"2\" cellpadding=\"1\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\" rowspan=\"2\"><strong>Tipo de avi\u00e3o<\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\" rowspan=\"2\"><strong>Pre\u00e7o de aluguer (M \u20ac)<\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\" rowspan=\"2\"><strong>N.\u00ba de avi\u00f5es dispon\u00edveis<\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\" colspan=\"2\"><strong>Carga completa<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\"><strong>N.\u00ba de pessoas <\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\"><strong>Bagagem (t)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: 1px solid;\"><strong>Tipo A<\/strong><\/td>\n<td style=\"text-align: center; border: 1px solid;\">4<\/td>\n<td style=\"text-align: center; border: 1px solid;\">12<\/td>\n<td style=\"text-align: center; border: 1px solid;\">200<\/td>\n<td style=\"text-align: center; border: 1px solid;\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: 1px solid;\"><strong>Tipo B<\/strong><\/td>\n<td style=\"text-align: center; border: 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: 1px solid;\">9<\/td>\n<td style=\"text-align: center; border: 1px solid;\">100<\/td>\n<td style=\"text-align: center; border: 1px solid;\">15<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As <strong>inc\u00f3gnitas<\/strong> do problema s\u00e3o:<\/p>\n<ul>\n<li><em>x<\/em>, o n\u00famero de avi\u00f5es alugados do tipo A<\/li>\n<li><em>y<\/em>, o n\u00famero de avi\u00f5es alugados do tipo B<\/li>\n<\/ul>\n<p>Organizemos os dados numa tabela para podermos relacion\u00e1-los mais facilmente.<\/p>\n<table border=\"1\" cellspacing=\"2\" cellpadding=\"1\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\" rowspan=\"2\"><strong>Tipo de avi\u00e3o<\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\" rowspan=\"2\"><strong>N.\u00ba de avi\u00f5es alugados<\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\" colspan=\"2\"><strong>Carga transportada<\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\" rowspan=\"2\"><strong>Custo do aluguer (M \u20ac)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\"><strong>N.\u00ba de pessoas<\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\"><strong>Bagagem (t)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: 1px solid;\"><strong>Tipo A<\/strong><\/td>\n<td style=\"text-align: center; border: 1px solid;\">$x$<\/td>\n<td style=\"text-align: center; border: 1px solid;\">$200x$<\/td>\n<td style=\"text-align: center; border: 1px solid;\">$6x$<\/td>\n<td style=\"text-align: center; border: 1px solid;\">$4x$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: 1px solid;\"><strong>Tipo B<\/strong><\/td>\n<td style=\"text-align: center; border: 1px solid;\">$y$<\/td>\n<td style=\"text-align: center; border: 1px solid;\">$100y$<\/td>\n<td style=\"text-align: center; border: 1px solid;\">$15y$<\/td>\n<td style=\"text-align: center; border: 1px solid;\">$y$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\"><strong>Total<\/strong><\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\">$x+y$<\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\">$200x+100y$<\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\">$6x+15y$<\/td>\n<td style=\"text-align: center; background-color: #efefef; border: 1px solid;\">$4x+y$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\">\n<p style=\"text-align: left;\">O problema tem as seguintes <strong>restri\u00e7\u00f5es<\/strong>:<\/p>\n<ol>\n<li>\n<div style=\"text-align: left;\">$x\\in {{\\mathbb{N}}_{0}}$ e $y\\in {{\\mathbb{N}}_{0}}$<br \/>\nO n\u00famero de avi\u00f5es a alugar de cada tipo \u00e9 inteiro n\u00e3o negativo;<\/div>\n<\/li>\n<li>\n<div style=\"text-align: left;\">$x\\le 12$<br \/>\nS\u00f3 h\u00e1\u00a012 avi\u00f5es dispon\u00edveis do tipo A;<\/div>\n<\/li>\n<li>$y\\le 9$<br \/>\nS\u00f3 h\u00e1\u00a09 avi\u00f5es dispon\u00edveis do tipo B;<\/li>\n<li>$x+y\\le 21$ (irrelevante, pois a restri\u00e7\u00e3o est\u00e1 j\u00e1 contida nas duas anteriores)<br \/>\nO\u00a0n\u00famero m\u00e1ximo de avi\u00f5es a alugar \u00e9 21;<\/li>\n<li>$200x+100y\\ge 1600\\Leftrightarrow 2x+y\\ge 16$<br \/>\nO n\u00famero de lugares a alugar ter\u00e1 de ser superior ou igual ao n\u00famero de pessoas a transportar (1600).<\/li>\n<li>$6x+15y\\ge 90\\Leftrightarrow 2x+5y\\ge 30$<br \/>\nA capacidade de carga a alugar ter\u00e1 de ser superior \u00e0 bagagem a transportar (90 t).<\/li>\n<\/ol>\n<p>Cada\u00a0avi\u00e3o do tipo A \u00e9 alugado por 4 M\u20ac e cada do tipo B \u00e9 alugado por 1 M\u20ac.<br \/>\nPortanto, a <strong>fun\u00e7\u00e3o objetivo<\/strong> \u00e9: $C=4x+y\\Leftrightarrow y=-4x+C$.<\/p>\n<p>Os pontos admiss\u00edveis, isto \u00e9, os pontos cujas coordenadas satisfazem as condi\u00e7\u00f5es impostas, s\u00e3o os pontos de coordenadas inteiras pertencentes ao quadril\u00e1tero [GHIJ], no seu interior ou sobre os seus lados.<\/p>\n<p>Validemos as coordenadas dos pontos G e H, assinalados no referencial apresentado:<\/p>\n<p>$\\begin{matrix}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n2x+y=16\u00a0 \\\\<br \/>\n2x+5y=30\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n2x+y=16\u00a0 \\\\<br \/>\n4y=14\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=\\frac{25}{4}\u00a0 \\\\<br \/>\ny=\\frac{7}{2}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\to\u00a0 &amp; H(\\frac{25}{4},\\frac{7}{2})\u00a0 \\\\<br \/>\n\\end{matrix}$<\/p>\n<p>$\\begin{matrix}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n2x+y=16\u00a0 \\\\<br \/>\ny=9\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n2x=7\u00a0 \\\\<br \/>\ny=9\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=\\frac{7}{2}\u00a0 \\\\<br \/>\ny=9\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\to\u00a0 &amp; G(\\frac{7}{2},9)\u00a0 \\\\<br \/>\n\\end{matrix}$<\/p>\n<\/p>\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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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': 1,'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>\n<p>A solu\u00e7\u00e3o \u00f3tima encontra-se na periferia da regi\u00e3o de validez, pr\u00f3xima do ponto $G(\\frac{7}{2},9)$, o qual n\u00e3o \u00e9 ponto admiss\u00edvel.<\/p>\n<p>Os dois pontos admiss\u00edveis mais pr\u00f3ximos de G s\u00e3o ${{G}_{1}}(4,8)$\u00a0e ${{G}_{2}}(4,9)$.<\/p>\n<p>Graficamente, a solu\u00e7\u00e3o \u00f3tima parece ser o par (4, 8), isto \u00e9, as coordenadas do ponto ${{G}_{1}}$.<\/p>\n<p>Vamos, ent\u00e3o, verificar analiticamente se essa suposi\u00e7\u00e3o \u00e9 verdadeira:<\/p>\n<ul>\n<li><span style=\"color: #0000ff;\">${{C}_{{{G}_{1}}}}=4\\times 4+8=24$<\/span><\/li>\n<li>${{C}_{{{G}_{2}}}}=4\\times 4+9=25$<\/li>\n<\/ul>\n<p>Confirma-se, portanto, que a solu\u00e7\u00e3o \u00f3tima \u00e9 o par ordenado (4, 8).<\/p>\n<p>Assim, para tornar m\u00ednimo o custo da ponte a\u00e9rea, devem ser alugados\u00a04 avi\u00f5es do tipo A e 8 do tipo B.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/01\/Proposta73.pdf\">Proposta de resolu\u00e7\u00e3o em formato PDF<\/a><\/li>\n<\/ul>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6433' onClick='GTTabs_show(0,6433)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Pretende-se organizar uma ponte a\u00e9rea para transportar 1600 pessoas e 90 toneladas de bagagem. Os avi\u00f5es dispon\u00edveis s\u00e3o de dois tipos: 12 do tipo A e 9 do tipo B. Com&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20812,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,122],"tags":[124,123],"series":[],"class_list":["post-6433","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-programacao-linear","tag-funcao-objectivo","tag-programacao-linear-2"],"views":5203,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/01\/11V1Pag161-73_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6433","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=6433"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6433\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20812"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6433"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}