{"id":14573,"date":"2018-04-15T22:40:30","date_gmt":"2018-04-15T21:40:30","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14573"},"modified":"2022-01-07T22:47:08","modified_gmt":"2022-01-07T22:47:08","slug":"o-desenho-do-joao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14573","title":{"rendered":"O desenho do Jo\u00e3o"},"content":{"rendered":"<p><ul id='GTTabs_ul_14573' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14573' class='GTTabs_curr'><a  id=\"14573_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14573' ><a  id=\"14573_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_14573'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag94-10.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14574\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14574\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag94-10.png\" data-orig-size=\"354,417\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Desenho\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag94-10.png\" class=\"alignright size-medium wp-image-14574\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag94-10-255x300.png\" alt=\"\" width=\"255\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag94-10-255x300.png 255w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag94-10.png 354w\" sizes=\"auto, (max-width: 255px) 100vw, 255px\" \/><\/a>O Jo\u00e3o fez um desenho numa folha de papel com 22 cm por 28 cm e colocou-lhe uma moldura de cartolina de largura constante, como v\u00eas na figura.<\/p>\n<p>A \u00e1rea dessa moldura era de 336 cm<sup>2<\/sup>.<br \/>\nQual foi a largura da moldura que o Jo\u00e3o colocou?<br \/>\nExplica o teu racioc\u00ednio<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14573' onClick='GTTabs_show(1,14573)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14573'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/2017-18-MF9P2-pag95-10.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14578\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14578\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/2017-18-MF9P2-pag95-10.png\" data-orig-size=\"233,309\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Desenho\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/2017-18-MF9P2-pag95-10.png\" class=\"alignright size-medium wp-image-14578\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/2017-18-MF9P2-pag95-10-226x300.png\" alt=\"\" width=\"226\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/2017-18-MF9P2-pag95-10-226x300.png 226w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/2017-18-MF9P2-pag95-10.png 233w\" sizes=\"auto, (max-width: 226px) 100vw, 226px\" \/><\/a>Seja x, em cent\u00edmetros, a largura da moldura.<\/p>\n<p>Comecemos por exprimir a \u00e1rea da moldura, em fun\u00e7\u00e3o de <em>x<\/em>, utilizando uma decomposi\u00e7\u00e3o em 4 ret\u00e2ngulos e 4 quadrados:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{A_{Moldura}}}&amp; = &amp;{2 \\times \\left( {28 \\times x} \\right) + 2 \\times \\left( {22 \\times x} \\right) + 4 \\times \\left( {x \\times x} \\right)}\\\\{}&amp; = &amp;{56x + 44x + 4{x^2}}\\\\{}&amp; = &amp;{4{x^2} + 100x}\\end{array}\\]<\/p>\n<p>Como a\u00a0\u00e1rea dessa moldura era de 336 cm<sup>2<\/sup>, vem:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{A_{Moldura}} = 336}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{4{x^2} + 100x = 336}&amp; \\wedge &amp;{x &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{4{x^2} + 100x &#8211; 336 = 0}&amp; \\wedge &amp;{x &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{{x^2} + 25x &#8211; 84 = 0}&amp; \\wedge &amp;{x &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{x = \\frac{{ &#8211; 25 \\mp \\sqrt {625 + 336} }}{2}}&amp; \\wedge &amp;{x &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{\\left( {\\begin{array}{*{20}{c}}{x = \\frac{{ &#8211; 25 &#8211; 31}}{2}}&amp; \\vee &amp;{x = \\frac{{ &#8211; 25 + 31}}{2}}\\end{array}} \\right)}&amp; \\wedge &amp;{x &gt; 2}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{x = 3}\\end{array}\\]<\/p>\n<p>Portanto, a moldura colocada pelo Jo\u00e3o tinha 3 cm de largura.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14573' onClick='GTTabs_show(0,14573)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado O Jo\u00e3o fez um desenho numa folha de papel com 22 cm por 28 cm e colocou-lhe uma moldura de cartolina de largura constante, como v\u00eas na figura. A \u00e1rea dessa&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14576,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,303],"tags":[426,108,306,304],"series":[],"class_list":["post-14573","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-equacoes-do-2-o-grau","tag-9-o-ano","tag-area","tag-equacao-do-2-o-grau","tag-formula-resolvente"],"views":2954,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag94-10a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14573","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=14573"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14573\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14576"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14573"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14573"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14573"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14573"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}