{"id":10108,"date":"2012-10-05T22:08:54","date_gmt":"2012-10-05T21:08:54","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10108"},"modified":"2022-01-20T22:21:59","modified_gmt":"2022-01-20T22:21:59","slug":"de-quanto-deve-ser-o-aumento-para-que-a-area-duplique","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10108","title":{"rendered":"De quanto deve ser o aumento para que a \u00e1rea duplique?"},"content":{"rendered":"<p><ul id='GTTabs_ul_10108' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10108' class='GTTabs_curr'><a  id=\"10108_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10108' ><a  id=\"10108_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_10108'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Aumentou-se igualmente os lados do ret\u00e2ngulo da figura ao lado.<\/p>\n<p>De quanto deve ser o aumento para que a \u00e1rea duplique?<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10110\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10110\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg\" data-orig-size=\"248,178\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349471275&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=\"Ret\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg\" class=\"wp-image-10110 aligncenter\" title=\"Ret\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg\" alt=\"\" width=\"149\" height=\"107\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg 248w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado-150x107.jpg 150w\" sizes=\"auto, (max-width: 149px) 100vw, 149px\" \/><\/a><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10108' onClick='GTTabs_show(1,10108)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10108'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"10110\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10110\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg\" data-orig-size=\"248,178\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1349471275&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=\"Ret\u00e2ngulo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg\" class=\"alignright  wp-image-10110\" title=\"Ret\u00e2ngulo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg\" alt=\"\" width=\"149\" height=\"107\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado.jpg 248w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/retangampliado-150x107.jpg 150w\" sizes=\"auto, (max-width: 149px) 100vw, 149px\" \/><\/a>O ret\u00e2ngulo tem $2$ unidades de \u00e1rea.<\/p>\n<p>A \u00e1rea do ret\u00e2ngulo ampliado pode ser expressa por: $\\left( {x + 1} \\right)\\left( {x + 2} \\right)$.<\/p>\n<p>O valor de $x$ procurado \u00e9 a solu\u00e7\u00e3o positiva da equa\u00e7\u00e3o seguinte:<\/p>\n<p>$$\\begin{array}{*{20}{l}} {\\left( {x + 1} \\right)\\left( {x + 2} \\right) = 2 \\times 2}&amp; \\Leftrightarrow &amp;{{x^2} + 3x + 2 &#8211; 4 = 0}\\\\ {}&amp; \\Leftrightarrow &amp;{x = \\frac{{ &#8211; 3 \\pm \\sqrt {{3^2} &#8211; 4 \\times 1 \\times \\left( { &#8211; 2} \\right)} }}{{2 \\times 1}}}\\\\ {}&amp; \\Leftrightarrow &amp;{x = \\frac{{ &#8211; 3 \\pm \\sqrt {17} }}{2}}\\\\ {}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}} {x = \\frac{{ &#8211; 3 &#8211; \\sqrt {17} }}{2}}&amp; \\vee &amp;{x = \\frac{{ &#8211; 3 + \\sqrt {17} }}{2}} \\end{array}} \\end{array}$$<\/p>\n<\/p>\n<p>Portanto, para que a \u00e1rea do ret\u00e2ngulo duplique, o aumento deve ser de ${\\frac{{ &#8211; 3 + \\sqrt {17} }}{2}}$ unidades de comprimento.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10108' onClick='GTTabs_show(0,10108)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Aumentou-se igualmente os lados do ret\u00e2ngulo da figura ao lado. De quanto deve ser o aumento para que a \u00e1rea duplique? Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":20782,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,108,67,325],"series":[],"class_list":["post-10108","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-modulo-inicial","tag-10-o-ano","tag-area","tag-geometria","tag-modulo-inicial"],"views":2490,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag037-23_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10108","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=10108"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10108\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20782"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10108"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10108"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10108"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}