{"id":8610,"date":"2012-04-19T23:18:10","date_gmt":"2012-04-19T22:18:10","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8610"},"modified":"2022-01-16T11:38:47","modified_gmt":"2022-01-16T11:38:47","slug":"o-comprimento-do-lado-do-quadrado","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8610","title":{"rendered":"O comprimento do lado do quadrado"},"content":{"rendered":"<p><ul id='GTTabs_ul_8610' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8610' class='GTTabs_curr'><a  id=\"8610_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8610' ><a  id=\"8610_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_8610'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8611\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8611\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg\" data-orig-size=\"331,302\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"Quadrado\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg\" class=\"alignright  wp-image-8611\" title=\"Quadrado\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg\" alt=\"\" width=\"199\" height=\"181\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg 331w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9-300x273.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9-150x136.jpg 150w\" sizes=\"auto, (max-width: 199px) 100vw, 199px\" \/><\/a>De um quadrado de cart\u00e3o, de lado $x$ cent\u00edmetros, foi retirado, em cada canto, um quadradinho com 2 cent\u00edmetros de lado, como mostra a figura.<\/p>\n<ol>\n<li>Calcula o valor de $x$, sabendo que a figura restante tem \u00e1rea 65 cm<sup>2<\/sup>.<\/li>\n<li>Depois de cortado o cart\u00e3o, constru\u00edmos uma caixa aberta.<br \/>\nDetermina o valor de $x$ de modo que o volume da caixa seja $50\\,c{m^3}$.<\/li>\n<\/ol>\n<p>\u00a0<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8610' onClick='GTTabs_show(1,8610)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8610'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8611\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8611\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg\" data-orig-size=\"331,302\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"Quadrado\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg\" class=\"alignright wp-image-8611\" title=\"Quadrado\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg\" alt=\"\" width=\"199\" height=\"181\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9.jpg 331w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9-300x273.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9pag69-9-150x136.jpg 150w\" sizes=\"auto, (max-width: 199px) 100vw, 199px\" \/>Ora,<br \/>\n$$\\begin{array}{*{20}{r}}<br \/>\n{{\\text{\u00c1rea do quadrado:}}}&amp;{{x^2}} \\\\<br \/>\n{{\\text{\u00c1rea de um quadradinho:}}}&amp;{{2^2}}<br \/>\n\\end{array}$$<\/p>\n<p>Tendo em conta a decomposi\u00e7\u00e3o de \u00e1reas apresentada, temos:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{{x^2} &#8211; 4 \\times {2^2} = 65}&amp; \\Leftrightarrow &amp;{{x^2} = 81} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x =\u00a0 &#8211; 9}&amp; \\vee &amp;{x = 9}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, ${x = 9}$ (cent\u00edmetros), pois traduz um comprimento.<\/p>\n<p>\u00ad<\/p>\n<\/li>\n<li>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/caixa2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11697\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11697\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/caixa2.png\" data-orig-size=\"523,244\" 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=\"Caixa\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Caixa&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/caixa2.png\" class=\" wp-image-11697   alignright\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/caixa2.png\" alt=\"Caixa\" width=\"366\" height=\"171\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/caixa2.png 523w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/caixa2-300x139.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/caixa2-150x69.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/caixa2-400x186.png 400w\" sizes=\"auto, (max-width: 366px) 100vw, 366px\" \/><\/a><\/p>\n<p>O volume da caixa, em cent\u00edmetros c\u00fabicos, \u00e9 dado por: ${V_{Caixa}} = \\left( {x &#8211; 4} \\right) \\times \\left( {x &#8211; 4} \\right) \\times 2$.<\/p>\n<p>Como a caixa tem $50\\,c{m^3}$ de volume, tem-se:<\/p>\n<p>$\\begin{array}{*{20}{l}}<br \/>\n{{V_{Caixa}} = 50}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{\\left( {x &#8211; 4} \\right) \\times \\left( {x &#8211; 4} \\right) \\times 2 = 50}&amp; \\wedge &amp;{x &gt; 4}<br \/>\n\\end{array}}<br \/>\n\\end{array}$.<\/p>\n<p>Resolvendo a equa\u00e7\u00e3o do 2.\u00ba grau, vem:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{\\left( {x &#8211; 4} \\right) \\times \\left( {x &#8211; 4} \\right) \\times 2 = 50}&amp; \\Leftrightarrow &amp;{\\left( {x &#8211; 4} \\right) \\times \\left( {x &#8211; 4} \\right) = 25} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{{\\left( {x &#8211; 4} \\right)}^2} = 25} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x &#8211; 4 =\u00a0 &#8211; 5}&amp; \\vee &amp;{x &#8211; 4 = 5}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x =\u00a0 &#8211; 1}&amp; \\vee &amp;{x = 9}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, o volume da caixa \u00e9 $50\\,c{m^3}$ para ${x = 9}$ (cm).<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8610' onClick='GTTabs_show(0,8610)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado De um quadrado de cart\u00e3o, de lado $x$ cent\u00edmetros, foi retirado, em cada canto, um quadradinho com 2 cent\u00edmetros de lado, como mostra a figura. Calcula o valor de $x$, sabendo&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20353,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,303],"tags":[426,306],"series":[],"class_list":["post-8610","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-equacao-do-2-o-grau"],"views":2321,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/9V2Pag069-9_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8610","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=8610"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8610\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20353"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8610"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8610"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8610"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8610"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}