{"id":6764,"date":"2011-04-12T01:25:39","date_gmt":"2011-04-12T00:25:39","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6764"},"modified":"2026-06-05T00:07:37","modified_gmt":"2026-06-04T23:07:37","slug":"a-patricia-usando-o-geogebra","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6764","title":{"rendered":"A Patr\u00edcia, usando o GeoGebra"},"content":{"rendered":"<p><ul id='GTTabs_ul_6764' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6764' class='GTTabs_curr'><a  id=\"6764_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6764' ><a  id=\"6764_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_6764'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/PatriciaGeogebra.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6765\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6765\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/PatriciaGeogebra.png\" data-orig-size=\"291,396\" 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=\"Os gr\u00e1ficos da Patr\u00edcia\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/PatriciaGeogebra.png\" class=\"alignright size-full wp-image-6765\" title=\"Os gr\u00e1ficos da Patr\u00edcia\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/PatriciaGeogebra.png\" alt=\"\" width=\"204\" height=\"277\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/PatriciaGeogebra.png 291w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/PatriciaGeogebra-220x300.png 220w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/PatriciaGeogebra-110x150.png 110w\" sizes=\"auto, (max-width: 204px) 100vw, 204px\" \/><\/a>A Patr\u00edcia, usando o GeoGebra, construiu os gr\u00e1ficos das fun\u00e7\u00f5es <em>per\u00edmetro<\/em> e <em>\u00e1rea<\/em> do tri\u00e2ngulo [OBD], como mostra a figura.<\/p>\n<p>O ponto D \u00e9 um ponto m\u00f3vel sobre a semicircunfer\u00eancia, cujo di\u00e2metro mede 4 cm, e x \u00e9 o comprimento de [BD].<\/p>\n<ol>\n<li>A Patr\u00edcia esqueceu-se de identificar as fun\u00e7\u00f5es.<br \/>\nQual da fun\u00e7\u00f5es, f ou g, \u00e9 o <em>per\u00edmetro<\/em> e qual delas \u00e9 a <em>\u00e1rea<\/em>?<br \/>\nIdentifique-as, usando argumentos baseados na figura.<\/li>\n<li>Escreva as correspondentes express\u00f5es anal\u00edticas e, recorrendo a uma tabela obtida na calculadora, indique os valores de x para os quais cada uma das fun\u00e7\u00f5es \u00e9 m\u00e1xima.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6764' onClick='GTTabs_show(1,6764)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6764'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><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\":288,\r\n\"height\":388,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 | 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 73 , 14 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\/\/ 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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><\/p>\n<ol>\n<li>Quando o ponto D tende a coincidir com o ponto B e quando tende a coincidir com o ponto O, isto \u00e9, quando $x\\to 0$ e quando $x\\to 4$, a <em>\u00e1rea<\/em> tende para zero e o <em>per\u00edmetro<\/em> tende para 8.\n<p>Logo, f \u00e9 uma representa\u00e7\u00e3o gr\u00e1fica do <em>per\u00edmetro<\/em> e g uma representa\u00e7\u00e3o gr\u00e1fica da <em>\u00e1rea<\/em>.<\/p>\n<\/li>\n<li>Como $\\overline{OD}=\\sqrt{16-{{x}^{2}}}$, o per\u00edmetro e a \u00e1rea, em cent\u00edmetros quadrados, s\u00e3o dados, respetivamente, por:\n<p>$f(x)=4+x+\\sqrt{16-{{x}^{2}}}\\,,\\,x\\in \\left] 0,4 \\right[$<\/p>\n<p>$g(x)=\\frac{x\\sqrt{16-{{x}^{2}}}}{2}\\,,\\,x\\in \\left] 0,4 \\right[$<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6766\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6766\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-1.png\" data-orig-size=\"198,134\" 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=\"G1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-1.png\" class=\"alignnone size-full wp-image-6766\" title=\"G1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-1.png\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-1.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-1-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a>\u00a0 <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6767\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6767\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-2.png\" data-orig-size=\"198,134\" 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=\"G2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-2.png\" class=\"alignnone size-full wp-image-6767\" title=\"G2\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-2.png\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-2.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/G99-2-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a><br \/>\n<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/T99-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6768\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6768\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/T99-1.png\" data-orig-size=\"198,134\" 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=\"T1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/T99-1.png\" class=\"alignnone size-full wp-image-6768\" title=\"T1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/T99-1.png\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/T99-1.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/T99-1-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a>\u00a0 <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/C99-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6769\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6769\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/C99-1.png\" data-orig-size=\"198,134\" 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=\"C1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/C99-1.png\" class=\"alignnone size-full wp-image-6769\" title=\"C1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/C99-1.png\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/C99-1.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/C99-1-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a><\/p>\n<p>O per\u00edmetro e a \u00e1rea \u00e9 m\u00e1xima para $x=2\\sqrt{2}\\,cm$.<\/p>\n<\/li>\n<\/ol>\n<ul>\n<li><a href=\"https:\/\/www.wolframalpha.com\/input\/?i=Maximize%5B4+%2B+x+%2B+Sqrt%5B16+-+x%5E2%5D%2C+%7Bx%7D%5D\" target=\"_blank\" rel=\"noopener\">Ver maximizante e m\u00e1ximo do per\u00edmetro<\/a><\/li>\n<li><a href=\"https:\/\/www.wolframalpha.com\/input\/?i=Maximize%5B%28x+Sqrt%5B16+-+x%5E2%5D%29%2F2%2C+%7Bx%7D%5D\" target=\"_blank\" rel=\"noopener\">Ver maximizante e m\u00e1ximo da \u00e1rea<\/a><\/li>\n<\/ul>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6764' onClick='GTTabs_show(0,6764)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A Patr\u00edcia, usando o GeoGebra, construiu os gr\u00e1ficos das fun\u00e7\u00f5es per\u00edmetro e \u00e1rea do tri\u00e2ngulo [OBD], como mostra a figura. O ponto D \u00e9 um ponto m\u00f3vel sobre a semicircunfer\u00eancia, cujo&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20895,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,157],"tags":[158],"series":[],"class_list":["post-6764","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-com-radicais","tag-funcoes-com-radicais-2"],"views":3384,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/11V2Pag209-99_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6764","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=6764"}],"version-history":[{"count":2,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6764\/revisions"}],"predecessor-version":[{"id":27872,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6764\/revisions\/27872"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20895"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6764"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6764"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6764"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6764"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}