{"id":7166,"date":"2011-11-13T20:55:45","date_gmt":"2011-11-13T20:55:45","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7166"},"modified":"2022-01-15T21:12:52","modified_gmt":"2022-01-15T21:12:52","slug":"proporcionalidade-inversa","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7166","title":{"rendered":"Proporcionalidade inversa"},"content":{"rendered":"<p><ul id='GTTabs_ul_7166' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7166' class='GTTabs_curr'><a  id=\"7166_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7166' ><a  id=\"7166_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<li id='GTTabs_li_2_7166' ><a  id=\"7166_2\" onMouseOver=\"GTTabsShowLinks('S\u00edntese'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>S\u00edntese<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7166'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<table style=\"width: 20%; height: 189px;\" border=\"0\" align=\"right\">\n<tbody>\n<tr style=\"height: 45px;\">\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 45px;\">Base ($x$)<\/td>\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 45px;\">Altura ($y$)<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\">1<\/td>\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\"><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\">2<\/td>\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\"><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\">3<\/td>\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\"><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\">4<\/td>\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\"><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\">5<\/td>\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\"><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\">6<\/td>\n<td style=\"text-align: center; border: 1px solid #8b0000; height: 24px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Pretende-se construir ret\u00e2ngulos diferentes, mas todos de \u00e1rea 36 unidades.<\/p>\n<ol>\n<li>Completa a tabela, com os comprimentos adequados para a altura.<\/li>\n<li>Se aumentarmos o comprimento da base, o que acontece ao comprimento da altura?<\/li>\n<li>Se duplicarmos o comprimento da base, o que acontece ao comprimento da altura? E se o triplicarmos?<\/li>\n<li>Multiplica os valores correspondentes das vari\u00e1veis $x$ e $y$. O que obt\u00e9ns?<\/li>\n<li>Representa graficamente y em fun\u00e7\u00e3o de $x$, para os valores considerados na tabela.<\/li>\n<li>Trata-se de uma fun\u00e7\u00e3o de proporcionalidade direta? Porqu\u00ea?<\/li>\n<li>Escreve uma express\u00e3o anal\u00edtica que exprima $y$ em fun\u00e7\u00e3o de $x$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7166' onClick='GTTabs_show(1,7166)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7166'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><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\":544,\r\n\"height\":290,\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 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ \"material_id\":12345,\r\n\"ggbBase64\":\"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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><br \/>\n<table style=\"width: 20%;\" border=\"0\" align=\"right\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">Base ($x$)<\/td>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">Altura ($y$)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">36<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">2<\/td>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">18<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">3<\/td>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">12<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">4<\/td>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">9<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">5<\/td>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">7,2<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">6<\/td>\n<td style=\"text-align: center; border: #8b0000 1px solid;\">6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>A tabela, completada, est\u00e1 apresentada ao lado.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Se aumentarmos o comprimento da base, o comprimento da altura diminui.<br \/>\n\u00ad<\/li>\n<li>Se duplicarmos o comprimento da base, o comprimento da altura passa a metade.<br \/>\nE se o triplicarmos, o comprimento da altura passa a um ter\u00e7o.<br \/>\n\u00ad<\/li>\n<li>Multiplicando os valores correspondentes das vari\u00e1veis $x$ e $y$, obtenhos um produto constante:<br \/>\n<table class=\" aligncenter\" style=\"width: 25%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$1\\times 36=36$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$2\\times 18=36$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$3\\times 12=36$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$4\\times 9=36$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$5\\times 7,2=36$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$6\\times 6=36$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00ad<\/p>\n<\/li>\n<li>A fun\u00e7\u00e3o pedida est\u00e1 representada graficamente no referencial abaixo.<br \/>\n\u00ad<br \/>\n<script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet2\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":717,\r\n\"height\":510,\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 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"allowStyleBar\":false,\r\n\"preventFocus\":false,\r\n\"showZoomButtons\":true,\r\n\"capturingThreshold\":3,\r\n\/\/ add code here to run when the applet starts\r\n\"appletOnLoad\":function(api){ \/* api.evalCommand('Segment((1,2),(3,4))');*\/ },\r\n\"showFullscreenButton\":true,\r\n\"scale\":1,\r\n\"disableAutoScale\":false,\r\n\"allowUpscale\":false,\r\n\"clickToLoad\":false,\r\n\"appName\":\"classic\",\r\n\"buttonRounding\":0.7,\r\n\"buttonShadows\":false,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from geogebra.org\r\n\/\/ \"material_id\":\"RHYH3UQ8\",\r\n\/\/ use this instead of ggbBase64 to load a .ggb file\r\n\/\/ \"filename\":\"myfile.ggb\",\r\n\"ggbBase64\":\"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\",\r\n};\r\n\/\/ 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, macro=Macros\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'macro': 0};\r\nvar applet2 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')}; applet2.inject('ggbApplet2');\r\n<\/script>\r\n<br \/>\n\u00ad<\/li>\n<li>N\u00e3o se trata de uma fun\u00e7\u00e3o de proporcionalidade direta, pois o gr\u00e1fico n\u00e3o \u00e9 um conjunto de pontos pertencentes a uma reta que cont\u00e9m a origem do referencial.<br \/>\n\u00ad<\/li>\n<li>Uma express\u00e3o anal\u00edtica que exprime $y$ em fun\u00e7\u00e3o de $x$ \u00e9\u00a0$y=\\frac{36}{x}$, pois $x.y=36\\Leftrightarrow y=\\frac{36}{x}$.<br \/>\n\u00ad<\/li>\n<\/ol>\n<blockquote>\n<p>Duas vari\u00e1veis, $x$ e $y$, dizem-se <strong>inversamente proporcionais<\/strong> se \u00e9 constante (e diferente de zero) o produto dos valores correspondentes:\u00a0\\[x\\times y=k\\,\\,\\,(k\\ne 0)\\] $k$ \u00e9 a <strong>constante de proporcionalidade inversa<\/strong>.<\/p>\n<\/blockquote>\n<blockquote>\n<p>Uma fun\u00e7\u00e3o do tipo \\[x\\to \\frac{k}{x}\\,\\,\\,(k\\ne 0)\\] \u00e9 uma <strong>fun\u00e7\u00e3o de proporcionalidade inversa<\/strong>, em que o n\u00famero $k$ \u00e9 a <strong>constante de proporcionalidade inversa<\/strong>.<\/p>\n<\/blockquote>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7166' onClick='GTTabs_show(0,7166)'>&lt;&lt; Enunciado<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7166' onClick='GTTabs_show(2,7166)'>S\u00edntese &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_7166'>\n<span class='GTTabs_titles'><b>S\u00edntese<\/b><\/span><\/p>\n<p>Voltando ao problema dos ret\u00e2ngulos de \u00e1rea 36 unidades, podemos considerar infinitas solu\u00e7\u00f5es para os comprimentos da base e altura.<\/p>\n<p>Exploremos, ent\u00e3o, a anima\u00e7\u00e3o seguinte.<\/p>\n<p style=\"text-align: center;\"><div id=\"ggbApplet3\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet3\",\r\n\"width\":696,\r\n\"height\":400,\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 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ \"material_id\":12345,\r\n\"ggbBase64\":\"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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 applet3 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet'); applet2.inject('ggbApplet2'); applet3.inject('ggbApplet3')};\r\n<\/script><\/p>\n<p>\u00ad<br \/>\nAo representarmos graficamente os pontos cujas coordenadas s\u00e3o solu\u00e7\u00e3o do problema, obtemos uma infinidade de pontos que parecem constituir uma linha curva. Essa linha curva (infinita) \u00e9 parte de uma <strong>hip\u00e9rbole,<\/strong> designando-se, por isso, <strong>ramo de hip\u00e9rbole<\/strong>.<br \/>\n\u00ad<\/p>\n<blockquote>\n<p>Duas vari\u00e1veis, $x$ e $y$, dizem-se <strong>inversamente proporcionais<\/strong> se \u00e9 constante (e diferente de zero) o produto dos valores correspondentes:\u00a0\\[x\\times y=k\\,\\,\\,(k\\ne 0)\\] $k$ \u00e9 a <strong>constante de proporcionalidade inversa<\/strong>.<\/p>\n<\/blockquote>\n<blockquote>\n<p>Uma fun\u00e7\u00e3o do tipo \\[x\\to \\frac{k}{x}\\,\\,\\,(k\\ne 0)\\] \u00e9 uma <strong>fun\u00e7\u00e3o de proporcionalidade inversa<\/strong>, em que o n\u00famero $k$ \u00e9 a <strong>constante de proporcionalidade inversa<\/strong>.<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<p>Apresenta-se, seguidamente, a hip\u00e9rbole definida por $y=\\frac{36}{x}$, para todo o valor de $x\\ne 0$:<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv1.gif\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7167\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7167\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv1.gif\" data-orig-size=\"590,447\" 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\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv1.gif\" class=\"aligncenter size-full wp-image-7167\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv1.gif\" alt=\"\" width=\"590\" height=\"447\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv1.gif 590w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv1-300x227.gif 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv1-150x113.gif 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv1-400x303.gif 400w\" sizes=\"auto, (max-width: 590px) 100vw, 590px\" \/><\/a><\/p>\n<\/p>\n<p>A t\u00edtulo de exemplo, apresentam-se ainda mais algumas fun\u00e7\u00f5es cujos gr\u00e1ficos s\u00e3o hip\u00e9rboles:<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv2.gif\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7168\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7168\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv2.gif\" data-orig-size=\"590,445\" 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\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/wolframalpha-propinv2.gif\" class=\"aligncenter size-full wp-image-7168\" title=\"Gr\u00e1ficos\" 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