{"id":8102,"date":"2012-04-09T16:58:23","date_gmt":"2012-04-09T15:58:23","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8102"},"modified":"2022-01-30T19:50:17","modified_gmt":"2022-01-30T19:50:17","slug":"uma-viga-de-aco","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8102","title":{"rendered":"Uma viga de a\u00e7o"},"content":{"rendered":"<p><ul id='GTTabs_ul_8102' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8102' class='GTTabs_curr'><a  id=\"8102_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8102' ><a  id=\"8102_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_8102'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Uma viga de a\u00e7o com 255 dec\u00edmetros de comprimento est\u00e1 assente sobre dois pilares com 150 dec\u00edmetros de altura cada.<\/p>\n<p>Quando, a $d$ dec\u00edmetros do 1.\u00ba pilar, se coloca um peso de 115 kg sobre a viga, esta sofre uma depress\u00e3o de valor $s$ (em dec\u00edmetros) que nos \u00e9 dada pela fun\u00e7\u00e3o assim definida: $$s(d) = 8,5 \\times {10^{ &#8211; 7}}{d^2}\\left( {255 &#8211; d} \\right)$$<\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12-pag225-83.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8105\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8105\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12-pag225-83.jpg\" data-orig-size=\"406,362\" 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=\"Pilares\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12-pag225-83.jpg\" class=\"alignright  wp-image-8105\" title=\"Pilares\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12-pag225-83.jpg\" alt=\"\" width=\"146\" height=\"130\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12-pag225-83.jpg 406w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12-pag225-83-300x267.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12-pag225-83-150x133.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12-pag225-83-400x356.jpg 400w\" sizes=\"auto, (max-width: 146px) 100vw, 146px\" \/><\/a>Entre que valores pode variar $d$?<\/li>\n<li>Recorrendo \u00e0 calculadora, determine a que dist\u00e2ncia do 1.\u00ba pilar se deve colocar o peso, para que a depress\u00e3o seja de 1 dm. Aproxime o resultado ao cent\u00edmetro.<\/li>\n<li>Qual \u00e9 o maior valor que a depress\u00e3o pode tomar? A que dist\u00e2ncia do primeiro pilar deve ser colocado o peso para que isso aconte\u00e7a?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8102' onClick='GTTabs_show(1,8102)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8102'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Uma viga de a\u00e7o com 255 dec\u00edmetros de comprimento est\u00e1 assente sobre dois pilares com 150 dec\u00edmetros de altura cada.<\/p>\n<p>Quando, a $d$ dec\u00edmetros do 1.\u00ba pilar, se coloca um peso de 115 kg sobre a viga, esta sofre uma depress\u00e3o de valor $s$ (em dec\u00edmetros) que nos \u00e9 dada pela fun\u00e7\u00e3o assim definida: $$s(d) = 8,5 \\times {10^{ &#8211; 7}}{d^2}\\left( {255 &#8211; d} \\right)$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Podemos considerar que $d \\in \\left] {0,255} \\right[$, em dec\u00edmetros.<br \/>\n\u00ad<\/li>\n<li>Para que a depress\u00e3o seja de 1 dec\u00edmetro, o peso deve ser colocado a $82,6\\,dm$ ou $233,4\\,dm$ do 1.\u00ba pilar:<br \/>\n<table style=\"width: 600px;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8112\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8112\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp1.jpg\" data-orig-size=\"264,136\" 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=\"Gr\u00e1fico 1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp1.jpg\" class=\"aligncenter size-full wp-image-8112\" title=\"Gr\u00e1fico 1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp1.jpg\" alt=\"\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp1.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp1-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8113\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8113\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp2.jpg\" data-orig-size=\"264,136\" 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=\"Gr\u00e1fico 2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp2.jpg\" class=\"aligncenter size-full wp-image-8113\" title=\"Gr\u00e1fico 2\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp2.jpg\" alt=\"\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp2.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp2-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Como $$\\begin{array}{*{20}{l}}<br \/>\n{s'(d)}&amp; = &amp;{\\left( {8,5 \\times {{10}^{ &#8211; 7}}{d^2}\\left( {255 &#8211; d} \\right)} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; 3 \\times 8,5 \\times {{10}^{ &#8211; 7}}{d^2} + 2 \\times 255 \\times 8,5 \\times {{10}^{ &#8211; 7}}d} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; 2,55 \\times {{10}^{ &#8211; 6}}{d^2} + 4,335 \\times {{10}^{ &#8211; 4}}d}<br \/>\n\\end{array}$$<br \/>\ntemos: $$\\begin{array}{*{20}{l}}<br \/>\n{s'(d) = 0}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{ &#8211; 2,55 \\times {{10}^{ &#8211; 6}}{d^2} + 4,335 \\times {{10}^{ &#8211; 4}}d = 0}&amp; \\wedge &amp;{d \\in \\left] {0,255} \\right[}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{d\\left( {4,335 \\times {{10}^{ &#8211; 4}} &#8211; 2,55 \\times {{10}^{ &#8211; 6}}d} \\right) = 0}&amp; \\wedge &amp;{d \\in \\left] {0,255} \\right[}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{d = 170}<br \/>\n\\end{array}$$<br \/>\nLogo:<\/p>\n<table border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$d$<\/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=\"text-align: center; border: #00008b 1px solid;\">$170$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$255$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Sinal\u00a0de $s'(d)$<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; 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;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; border: #00008b 1px solid;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Varia\u00e7\u00e3o de $s$<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\searrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$2,088025$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; border: #00008b 1px solid;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>$$s(170) = 8,5 \\times {10^{ &#8211; 7}} \\times {170^2}\\left( {255 &#8211; 170} \\right) = 2,088025$$<br \/>\n\u00c9 de aproximadamente 2 dec\u00edmetros o maior valor que a depress\u00e3o pode tomar, colocando o peso a 170 dec\u00edmetros do primeiro pilar.<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp3.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8122\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8122\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp3.jpg\" data-orig-size=\"264,136\" 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=\"Gr\u00e1fico 3\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp3.jpg\" class=\"aligncenter size-full wp-image-8122\" title=\"Gr\u00e1fico 3\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp3.jpg\" alt=\"\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp3.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/Disp3-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a><\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8102' onClick='GTTabs_show(0,8102)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma viga de a\u00e7o com 255 dec\u00edmetros de comprimento est\u00e1 assente sobre dois pilares com 150 dec\u00edmetros de altura cada. Quando, a $d$ dec\u00edmetros do 1.\u00ba pilar, se coloca um peso&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21147,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,292],"tags":[427,145,144],"series":[],"class_list":["post-8102","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-calculo-diferencial","tag-12-o-ano","tag-derivadas-2","tag-extremos-relativos"],"views":2012,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V2Pag225-83_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8102","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=8102"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8102\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21147"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8102"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8102"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8102"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}