{"id":6646,"date":"2011-03-22T18:54:53","date_gmt":"2011-03-22T18:54:53","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6646"},"modified":"2022-01-25T00:14:47","modified_gmt":"2022-01-25T00:14:47","slug":"a-area-de-um-triangulo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6646","title":{"rendered":"A \u00e1rea de um tri\u00e2ngulo"},"content":{"rendered":"<p><ul id='GTTabs_ul_6646' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6646' class='GTTabs_curr'><a  id=\"6646_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6646' ><a  id=\"6646_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_6646'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6647\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6647\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg\" data-orig-size=\"236,217\" 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=\"Tri\u00e2ngulo inscrito\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg\" class=\"alignright wp-image-6647\" title=\"Tri\u00e2ngulo inscrito\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg\" alt=\"\" width=\"200\" height=\"184\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg 236w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51-150x137.jpg 150w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/><\/a>O tri\u00e2ngulo [ABC] est\u00e1 inscrito num semic\u00edrculo de di\u00e2metro 15 cm.<\/p>\n<ol>\n<li>Exprima a \u00e1rea de tri\u00e2ngulo [ABC] em fun\u00e7\u00e3o do cateto de medida x.<\/li>\n<li>Determine um valor aproximado de x para o qual a \u00e1rea \u00e9 m\u00e1xima.<br \/>\nQual \u00e9 o valor dessa \u00e1rea?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6646' onClick='GTTabs_show(1,6646)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6646'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6647\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6647\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg\" data-orig-size=\"236,217\" 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=\"Tri\u00e2ngulo inscrito\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg\" class=\"alignright wp-image-6647\" title=\"Tri\u00e2ngulo inscrito\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg\" alt=\"\" width=\"200\" height=\"184\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51.jpg 236w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag199-51-150x137.jpg 150w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/><\/a>Admitindo que o cateto de medida x \u00e9 [BC], temos $\\overline{AC}=\\sqrt{{{\\overline{AB}}^{2}}-{{\\overline{BC}}^{2}}}=\\sqrt{{{15}^{2}}-{{x}^{2}}}$, com $0&lt;x&lt;15$.\n<p>Assim, a \u00e1rea do tri\u00e2ngulo [ABC], em cent\u00edmetros quadrados, pode ser expressa por: \u00a0\\[\\begin{matrix}<br \/>\na(x)=\\frac{x\\sqrt{225-{{x}^{2}}}}{2} &amp; (0&lt;x&lt;15)\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<\/li>\n<li>Dado que \u00e9 complexa a obten\u00e7\u00e3o da express\u00e3o da derivada da fun\u00e7\u00e3o a ($a'(x)=\\frac{225-2{{x}^{2}}}{2\\sqrt{225-{{x}^{2}}}}$), vamos resolver a quest\u00e3o com aux\u00edlio da calculadora gr\u00e1fica.\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/22-03-2011Ecra006.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6648\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6648\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/22-03-2011Ecra006.png\" data-orig-size=\"690,468\" 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\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/22-03-2011Ecra006.png\" class=\"alignnone size-full wp-image-6648 aligncenter\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/22-03-2011Ecra006.png\" alt=\"\" width=\"552\" height=\"374\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/22-03-2011Ecra006.png 690w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/22-03-2011Ecra006-300x203.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/22-03-2011Ecra006-150x101.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/22-03-2011Ecra006-400x271.png 400w\" sizes=\"auto, (max-width: 552px) 100vw, 552px\" \/><\/a><\/p>\n<p>A \u00e1rea m\u00e1xima \u00e9 $56,25\\,c{{m}^{2}}$, obtida para $x\\approx 10,6\\,cm$ (o valor exacto \u00e9 $\\frac{15\\sqrt{2}}{2}$).<\/p>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><strong><span style=\"text-decoration: underline;\">Interprete a tabela apresentada a seguir<\/span><\/strong>:<\/p>\n<p>Utilizando a express\u00e3o $a'(x)=\\frac{225-2{{x}^{2}}}{2\\sqrt{225-{{x}^{2}}}}$, obt\u00e9m-se:<\/p>\n<table class=\"aligncenter\" style=\"width: 70%;\" border=\"2\" cellspacing=\"4\" cellpadding=\"4\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$x$<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$\\frac{15\\sqrt{2}}{2}$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$15$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$225-2{{x}^{2}}$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$2\\sqrt{225-{{x}^{2}}}$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$a'(x)=\\frac{225-2{{x}^{2}}}{2\\sqrt{225-{{x}^{2}}}}$<\/td>\n<td style=\"text-align: center;\">n.d.<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">n.d.<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$a(x)$<\/td>\n<td style=\"text-align: center;\">n.d.<\/td>\n<td style=\"text-align: center;\">\u00a0\u00a0\u00a0 \u00a0$\\nearrow $<\/td>\n<td style=\"text-align: center;\">$56,25$<\/td>\n<td style=\"text-align: center;\">\u00a0\u00a0\u00a0\u00a0 $\\searrow $<\/td>\n<td style=\"text-align: center;\">n.d.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6646' onClick='GTTabs_show(0,6646)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado O tri\u00e2ngulo [ABC] est\u00e1 inscrito num semic\u00edrculo de di\u00e2metro 15 cm. Exprima a \u00e1rea de tri\u00e2ngulo [ABC] em fun\u00e7\u00e3o do cateto de medida x. Determine um valor aproximado de x para&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20937,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,134],"tags":[144],"series":[],"class_list":["post-6646","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-derivadas","tag-extremos-relativos"],"views":2782,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11V2Pag199-51_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6646","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=6646"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6646\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20937"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6646"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6646"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6646"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6646"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}