{"id":6358,"date":"2010-12-08T22:25:52","date_gmt":"2010-12-08T22:25:52","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6358"},"modified":"2022-01-21T18:41:27","modified_gmt":"2022-01-21T18:41:27","slug":"uma-escada-com-tres-degraus","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6358","title":{"rendered":"Uma escada com tr\u00eas degraus"},"content":{"rendered":"<p><ul id='GTTabs_ul_6358' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6358' class='GTTabs_curr'><a  id=\"6358_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6358' ><a  id=\"6358_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_6358'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6359\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6359\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg\" data-orig-size=\"348,201\" 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=\"Escadas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg\" class=\"alignright wp-image-6359\" title=\"Escadas\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg\" alt=\"\" width=\"320\" height=\"185\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg 348w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas-300x173.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas-150x86.jpg 150w\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" \/><\/a>A figura representa uma escada com tr\u00eas degraus em madeira.<\/p>\n<p>Sabe-se que a largura da escada \u00e9 80 cm e que se gasta 1m<sup>2<\/sup> de madeira para a construir [parte colorida].<\/p>\n<p>Sabendo que o \u00e2ngulo \u03b1 \u00e9 tal que $tg\\,\\alpha =\\frac{3}{4}$, determine <strong>x<\/strong>, <strong>p<\/strong> e <strong>h<\/strong> com aproxima\u00e7\u00e3o ao cent\u00edmetro, por excesso.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6358' onClick='GTTabs_show(1,6358)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6358'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6359\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6359\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg\" data-orig-size=\"348,201\" 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=\"Escadas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg\" class=\"alignright wp-image-6359\" title=\"Escadas\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg\" alt=\"\" width=\"320\" height=\"185\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas.jpg 348w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas-300x173.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/escadas-150x86.jpg 150w\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" \/><\/a>Considerando as unidades em metros, o problema pode ser equacionado pelo seguinte sistema (Porqu\u00ea?):<\/p>\n<p style=\"text-align: center;\">$\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(3p+3h)\\times 0,8=1\u00a0 \\\\<br \/>\n\\frac{h}{p}=\\frac{3}{4}\u00a0 \\\\<br \/>\nx=4p\u00a0 \\\\<br \/>\n\\end{array} \\right.$<\/p>\n<p>Resolvendo o sistema, vem: \\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(3p+3h)\\times 0,8=1\u00a0 \\\\<br \/>\n\\frac{h}{p}=\\frac{3}{4}\u00a0 \\\\<br \/>\nx=4p\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}}<br \/>\n(-1\\times )\u00a0 \\\\<br \/>\n(+)\u00a0 \\\\<br \/>\n{}\u00a0 \\\\<br \/>\n\\end{array}\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n3p+3h=\\frac{5}{4}\u00a0 \\\\<br \/>\n3p-4h=0\u00a0 \\\\<br \/>\nx=4p\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n3p-4h=0\u00a0 \\\\<br \/>\n-7h=-\\frac{5}{4}\u00a0 \\\\<br \/>\nx=4p\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nh=\\frac{5}{28}\u00a0 \\\\<br \/>\np=\\frac{4\\times \\frac{5}{28}}{3}\u00a0 \\\\<br \/>\nx=4p\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nh=\\frac{5}{28}\u00a0 \\\\<br \/>\np=\\frac{5}{21}\u00a0 \\\\<br \/>\nx=\\frac{20}{21}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, $h\\simeq 18\\,cm$, $p\\simeq 24\\,cm$ e $x\\simeq 95\\,cm$.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6358' onClick='GTTabs_show(0,6358)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A figura representa uma escada com tr\u00eas degraus em madeira. Sabe-se que a largura da escada \u00e9 80 cm e que se gasta 1m2 de madeira para a construir [parte colorida].&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20830,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67],"series":[],"class_list":["post-6358","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-geometria-analitica","tag-11-o-ano","tag-geometria"],"views":2691,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag183-40_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6358","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=6358"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6358\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20830"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6358"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}