{"id":8022,"date":"2012-04-08T16:51:43","date_gmt":"2012-04-08T15:51:43","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8022"},"modified":"2012-04-08T22:21:04","modified_gmt":"2012-04-08T21:21:04","slug":"uma-trave-de-madeira","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8022","title":{"rendered":"Uma trave de madeira"},"content":{"rendered":"<p><ul id='GTTabs_ul_8022' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8022' class='GTTabs_curr'><a  id=\"8022_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8022' ><a  id=\"8022_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_8022'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8024\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8024\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73.jpg\" data-orig-size=\"282,205\" 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=\"Trave de madeira\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73.jpg\" class=\"alignright  wp-image-8024\" title=\"Trave de madeira\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73.jpg\" alt=\"\" width=\"197\" height=\"144\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73.jpg 282w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73-150x109.jpg 150w\" sizes=\"auto, (max-width: 197px) 100vw, 197px\" \/><\/a>Num canto de um terreno murado pretende-se delimitar com uma trave de madeira a maior \u00e1rea de terreno poss\u00edvel.<\/p>\n<p>Sabendo que a trave mede 5 metros, em que posi\u00e7\u00e3o deve ser colocada?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8022' onClick='GTTabs_show(1,8022)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8022'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73b.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8028\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8028\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73b.jpg\" data-orig-size=\"282,205\" 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=\"Trave de madeira\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73b.jpg\" class=\"aligncenter  wp-image-8028\" title=\"Trave de madeira\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73b.jpg\" alt=\"\" width=\"169\" height=\"123\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73b.jpg 282w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73b-150x109.jpg 150w\" sizes=\"auto, (max-width: 169px) 100vw, 169px\" \/><\/a><\/p>\n<p>Para $0 &lt; x &lt; 5$ e $0 &lt; y &lt; 5$, temos: $y = \\sqrt {25 &#8211; {x^2}} $.<\/p>\n<p>Logo, a \u00e1rea do terreno pode ser expressa por $$A(x) = \\frac{{x\\sqrt {25 &#8211; {x^2}} }}{2},{\\text{com }}0 &lt; x &lt; 5$$<\/p>\n<p>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n\u00a0 {A'(x)}&amp; = &amp;{{{\\left( {\\frac{{x\\sqrt {25 &#8211; {x^2}} }}{2}} \\right)}^&#8217;}} \\\\<br \/>\n\u00a0 {}&amp; = &amp;{\\frac{1}{2}\\sqrt {25 &#8211; {x^2}}\u00a0 + \\frac{1}{2} \\times x \\times \\frac{1}{2}{{\\left( {25 &#8211; {x^2}} \\right)}^{ &#8211; \\frac{1}{2}}} \\times \\left( { &#8211; 2x} \\right)} \\\\<br \/>\n\u00a0 {}&amp; = &amp;{\\frac{{\\sqrt {25 &#8211; {x^2}} }}{2} &#8211; \\frac{{{x^2}}}{{2\\sqrt {25 &#8211; {x^2}} }}} \\\\<br \/>\n\u00a0 {}&amp; = &amp;{\\frac{{25 &#8211; {x^2} &#8211; {x^2}}}{{2\\sqrt {25 &#8211; {x^2}} }}} \\\\<br \/>\n\u00a0 {}&amp; = &amp;{\\frac{{25 &#8211; 2{x^2}}}{{2\\sqrt {25 &#8211; {x^2}} }}}<br \/>\n\\end{array}$$<\/p>\n<p>&nbsp;<\/p>\n<p>Como $25 &#8211; 2{x^2} = 0 \\Leftrightarrow {x^2} = \\frac{{25}}{2} \\Leftrightarrow x =\u00a0 \\pm \\sqrt {\\frac{{25}}{2}}\u00a0 \\Leftrightarrow x =\u00a0 \\pm \\frac{{5\\sqrt 2 }}{2}$, temos:<\/p>\n<table border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$x$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$\\frac{{5\\sqrt 2 }}{2}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$5$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${25 &#8211; 2{x^2}}$<\/td>\n<td style=\"background-color: #a9a9a9; border: #00008b 1px solid;\">\u00a0<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"background-color: #a9a9a9; border: #00008b 1px solid;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${2\\sqrt {25 &#8211; {x^2}} }$<\/td>\n<td style=\"background-color: #a9a9a9; border: #00008b 1px solid;\">\u00a0<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"background-color: #a9a9a9; border: #00008b 1px solid;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Sinal de $A'(x) = \\frac{{25 &#8211; 2{x^2}}}{{2\\sqrt {25 &#8211; {x^2}} }}$<\/td>\n<td style=\"background-color: #a9a9a9; border: #00008b 1px solid;\">\u00a0<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"background-color: #a9a9a9; border: #00008b 1px solid;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Varia\u00e7\u00e3o de $A$<\/td>\n<td style=\"background-color: #a9a9a9; border: #00008b 1px solid;\">\u00a0<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">\u00a0$\\frac{{25}}{4}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\searrow $<\/td>\n<td style=\"background-color: #a9a9a9; border: #00008b 1px solid;\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>$$A(\\frac{{5\\sqrt 2 }}{2}) = \\frac{{5\\sqrt 2 }}{2} \\times \\frac{{\\sqrt {25 &#8211; \\frac{{25}}{2}} }}{2} = \\frac{{5\\sqrt 2 }}{2} \\times \\frac{5}{{2\\sqrt 2 }} = \\frac{{25}}{4}$$<\/p>\n<p>\u00a0Portanto, a trave deve ser colocada de forma a obter um tri\u00e2ngulo ret\u00e2ngulo is\u00f3sceles: $x = y = \\frac{{5\\sqrt 2 }}{2}$ metros.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><applet name=\"ggbApplet\" code=\"geogebra.GeoGebraApplet\" archive=\"geogebra.jar\"\r\n\tcodebase=\"http:\/\/www.geogebra.org\/webstart\/4.0\/unsigned\/\"\r\n\twidth=\"355\" height=\"431\">\r\n\t<param name=\"ggbBase64\" 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Sabendo que a trave mede 5 metros, em que posi\u00e7\u00e3o deve ser&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":8024,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97],"tags":[427,145,144],"series":[],"class_list":["post-8022","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","tag-12-o-ano","tag-derivadas-2","tag-extremos-relativos"],"views":2383,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag222-73.jpg","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8022","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=8022"}],"version-history":[{"count":1,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8022\/revisions"}],"predecessor-version":[{"id":8055,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8022\/revisions\/8055"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/8024"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8022"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8022"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8022"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8022"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}