{"id":8749,"date":"2012-04-23T22:42:53","date_gmt":"2012-04-23T21:42:53","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8749"},"modified":"2022-01-30T00:21:46","modified_gmt":"2022-01-30T00:21:46","slug":"a-seccao-de-um-tunel","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8749","title":{"rendered":"A sec\u00e7\u00e3o de um t\u00fanel"},"content":{"rendered":"<p><ul id='GTTabs_ul_8749' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8749' class='GTTabs_curr'><a  id=\"8749_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8749' ><a  id=\"8749_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_8749'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8751\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8751\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg\" data-orig-size=\"338,160\" 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=\"T\u00fanel\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg\" class=\"alignright wp-image-8751\" title=\"T\u00fanel\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg\" alt=\"\" width=\"270\" height=\"128\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg 338w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel-300x142.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel-150x71.jpg 150w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>A sec\u00e7\u00e3o de um t\u00fanel \u00e9 um semic\u00edrculo com 1 hm de raio.<\/p>\n<p>No interior do t\u00fanel h\u00e1 uma estrutura com a forma de um trap\u00e9zio, como mostra a figura.<\/p>\n<p>Qual \u00e9 o valor de $\\theta $ $\\left( {0 &lt; \\theta\u00a0 &lt; \\frac{\\pi }{2}} \\right)$ que torna m\u00e1xima a \u00e1rea da sec\u00e7\u00e3o da estrutura trapezoidal?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8749' onClick='GTTabs_show(1,8749)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8749'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8751\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8751\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg\" data-orig-size=\"338,160\" 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=\"T\u00fanel\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg\" class=\"alignright wp-image-8751\" title=\"T\u00fanel\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg\" alt=\"\" width=\"270\" height=\"128\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel.jpg 338w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel-300x142.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/tunel-150x71.jpg 150w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>Seja $C&#8217;$ a proje\u00e7\u00e3o ortogonal do ponto C sobre [AD].<\/p>\n<p>Ora, $$\\operatorname{sen} \\theta\u00a0 = \\frac{{\\overline {CC&#8217;} }}{{\\overline {OC} }} \\Leftrightarrow \\overline {CC&#8217;}\u00a0 = \\overline {OC}\u00a0 \\times \\operatorname{sen} \\theta $$ e $$\\cos \\theta\u00a0 = \\frac{{\\overline {OC&#8217;} }}{{\\overline {OC} }} \\Leftrightarrow \\overline {OC&#8217;}\u00a0 = \\overline {OC}\u00a0 \\times \\cos \\theta $$<\/p>\n<p>Como $\\overline {OC}\u00a0 = 1$, vem: $\\overline {CC&#8217;}\u00a0 = \\operatorname{sen} \\theta $ e $\\overline {OC&#8217;}\u00a0 = \\cos \\theta $.<br \/>\n\u00ad<\/p>\n<p>Assim, a \u00e1rea da estrutura trapezoidal \u00e9 dada por: $$\\begin{array}{*{20}{l}}<br \/>\n{A\\left( \\theta\u00a0 \\right)}&amp; = &amp;{\\frac{{\\overline {AD}\u00a0 + \\overline {BC} }}{2} \\times \\overline {CC&#8217;} } \\\\<br \/>\n{}&amp; = &amp;{\\frac{{2 + 2\\cos \\theta }}{2} \\times \\operatorname{sen} \\theta } \\\\<br \/>\n{}&amp; = &amp;{\\operatorname{sen} \\theta\u00a0 + \\operatorname{sen} \\theta\u00a0 \\times \\cos \\theta }<br \/>\n\\end{array}$$ com ${0 &lt; \\theta\u00a0 &lt; \\frac{\\pi }{2}}$.<br \/>\n\u00ad<\/p>\n<p>Como $$\\begin{array}{*{20}{l}}<br \/>\n{A&#8217;\\left( \\theta\u00a0 \\right)}&amp; = &amp;{\\left( {\\operatorname{sen} \\theta\u00a0 + \\operatorname{sen} \\theta\u00a0 \\times \\cos \\theta } \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{\\cos \\theta\u00a0 + {{\\cos }^2}\\theta\u00a0 &#8211; {{\\operatorname{sen} }^2}\\theta } \\\\<br \/>\n{}&amp; = &amp;{\\cos \\theta\u00a0 + {{\\cos }^2}\\theta\u00a0 &#8211; \\left( {1 &#8211; {{\\cos }^2}\\theta } \\right)} \\\\<br \/>\n{}&amp; = &amp;{2{{\\cos }^2}\\theta\u00a0 + \\cos \\theta\u00a0 &#8211; 1}<br \/>\n\\end{array}$$ com ${0 &lt; \\theta\u00a0 &lt; \\frac{\\pi }{2}}$, ent\u00e3o $$\\begin{array}{*{20}{l}}<br \/>\n{A&#8217;\\left( \\theta\u00a0 \\right) = 0}&amp; \\Leftrightarrow &amp;{2{{\\cos }^2}\\theta\u00a0 + \\cos \\theta\u00a0 &#8211; 1 = 0 \\wedge 0 &lt; \\theta\u00a0 &lt; \\frac{\\pi }{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\cos \\theta\u00a0 = \\frac{{ &#8211; 1 \\pm \\sqrt {1 + 8} }}{4} \\wedge 0 &lt; \\theta\u00a0 &lt; \\frac{\\pi }{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\cos \\theta\u00a0 = \\frac{{ &#8211; 1 \\pm 3}}{4} \\wedge 0 &lt; \\theta\u00a0 &lt; \\frac{\\pi }{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\left( {\\cos \\theta\u00a0 = 1 \\vee \\cos \\theta\u00a0 = \\frac{1}{2}} \\right) \\wedge 0 &lt; \\theta\u00a0 &lt; \\frac{\\pi }{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\theta\u00a0 = \\frac{\\pi }{3}}<br \/>\n\\end{array}$$<\/p>\n<p>Assim, temos:<\/p>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$\\theta $<\/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;\">${\\frac{\\pi }{3}}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${\\frac{\\pi }{2}}$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$A&#8217;\\left( \\theta\u00a0 \\right) = 2{\\cos ^2}\\theta\u00a0 + \\cos \\theta\u00a0 &#8211; 1$<\/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 $A$<\/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;\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${\\frac{{3\\sqrt 3 }}{4}}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\searrow $<\/td>\n<td style=\"text-align: center; background-color: #a9a9a9; border: #00008b 1px solid;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{A\\left( {\\frac{\\pi }{3}} \\right)}&amp; = &amp;{\\operatorname{sen} \\frac{\\pi }{3} + \\operatorname{sen} \\frac{\\pi }{3} \\times \\cos \\frac{\\pi }{3}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\sqrt 3 }}{2} + \\frac{{\\sqrt 3 }}{2} \\times \\frac{1}{2}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{3\\sqrt 3 }}{4}}<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, a \u00e1rea \u00e9 m\u00e1xima para ${\\theta\u00a0 = \\frac{\\pi }{3}}$.<br \/>\n\u00ad<\/p>\n<blockquote><p><strong>Sobre o preenchimento da tabela relativamente ao sinal de $A&#8217;$<\/strong>:<\/p>\n<p>Como $2{\\cos ^2}0 + \\cos 0 &#8211; 1 = 2 + 1 &#8211; 1 = 2 &gt; 0$ e $2{\\cos ^2}\\frac{\\pi }{2} + \\cos \\frac{\\pi }{2} &#8211; 1 = 0 + 0 &#8211; 1 =\u00a0 &#8211; 1 &lt; 0$ e dado que a fun\u00e7\u00e3o $x \\to 2{\\cos ^2}x + \\cos x &#8211; 1$ \u00e9 cont\u00ednua em $\\mathbb{R}$, esta n\u00e3o pode mudar de sinal sem anular.<\/p><\/blockquote>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-8.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8763\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8763\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-8.png\" data-orig-size=\"568,531\" 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\/2012\/04\/12pag128-8.png\" class=\"aligncenter wp-image-8763 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-8.png\" alt=\"\" width=\"568\" height=\"531\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-8.png 568w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-8-300x280.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-8-150x140.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-8-400x373.png 400w\" sizes=\"auto, (max-width: 568px) 100vw, 568px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8749' onClick='GTTabs_show(0,8749)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A sec\u00e7\u00e3o de um t\u00fanel \u00e9 um semic\u00edrculo com 1 hm de raio. No interior do t\u00fanel h\u00e1 uma estrutura com a forma de um trap\u00e9zio, como mostra a figura. Qual&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21100,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,144,293,307],"series":[],"class_list":["post-8749","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-seno-co-seno-e-tangente","tag-12-o-ano","tag-extremos-relativos","tag-funcao-derivada","tag-funcoes-trigonometricas"],"views":2662,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V3Pag128-8_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8749","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=8749"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8749\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21100"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8749"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8749"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8749"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8749"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}