{"id":14020,"date":"2018-03-10T18:06:04","date_gmt":"2018-03-10T18:06:04","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14020"},"modified":"2022-01-11T02:17:50","modified_gmt":"2022-01-11T02:17:50","slug":"a-altura-de-uma-torre","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14020","title":{"rendered":"A altura de uma torre"},"content":{"rendered":"<p><ul id='GTTabs_ul_14020' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14020' class='GTTabs_curr'><a  id=\"14020_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14020' ><a  id=\"14020_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_14020'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14021\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14021\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b.png\" data-orig-size=\"405,290\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Torre\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b.png\" class=\"alignright wp-image-14021 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b-300x215.png\" alt=\"\" width=\"300\" height=\"215\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b-300x215.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b.png 405w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>De um ponto <em>O<\/em> v\u00ea-se o topo de uma torre sob um \u00e2ngulo de 35 graus.<br \/>\nAvan\u00e7ando 10 metros em dire\u00e7\u00e3o \u00e0 torre, o \u00e2ngulo passa a ser de 58 graus.<\/p>\n<p>Determina a altura da torre arredondada \u00e0s d\u00e9cimas.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14020' onClick='GTTabs_show(1,14020)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14020'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14021\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14021\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b.png\" data-orig-size=\"405,290\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Torre\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b.png\" class=\"alignright wp-image-14021 size-medium\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b-300x215.png\" alt=\"\" width=\"300\" height=\"215\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b-300x215.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11b.png 405w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Do tri\u00e2ngulo ret\u00e2ngulo [<em>ABC<\/em>], temos:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm tg}\\nolimits} 58^\\circ = \\frac{{\\overline {AB} }}{{\\overline {BC} }}}&amp; \\Leftrightarrow &amp;{{\\mathop{\\rm tg}\\nolimits} 58^\\circ = \\frac{x}{y}}\\\\{}&amp; \\Leftrightarrow &amp;{y = \\frac{x}{{{\\mathop{\\rm tg}\\nolimits} 58^\\circ }}}\\end{array}\\]<\/p>\n<p>Do tri\u00e2ngulo [<em>OAB<\/em>], temos:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{\\mathop{\\rm tg}\\nolimits} 35^\\circ = \\frac{{\\overline {AB} }}{{\\overline {OB} }}}&amp; \\Leftrightarrow &amp;{{\\mathop{\\rm tg}\\nolimits} 35^\\circ = \\frac{x}{{10 + y}}}\\\\{}&amp; \\Leftrightarrow &amp;{x = 10 \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ + y \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ }\\end{array}\\]<\/p>\n<p>\u00ad<br \/>\nVamos agora eliminar a vari\u00e1vel <strong><em>y<\/em><\/strong> nesta equa\u00e7\u00e3o, por substitui\u00e7\u00e3o da express\u00e3o final obtida na resolu\u00e7\u00e3o feita acima na primeira equa\u00e7\u00e3o considerada:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{x = 10 \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ + \\frac{x}{{{\\mathop{\\rm tg}\\nolimits} 58^\\circ }} \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ }&amp; \\Leftrightarrow &amp;{x \\times {\\mathop{\\rm tg}\\nolimits} 58^\\circ = 10 \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ \\times {\\mathop{\\rm tg}\\nolimits} 58^\\circ + x \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ }\\\\{}&amp; \\Leftrightarrow &amp;{x \\times \\left( {{\\mathop{\\rm tg}\\nolimits} 58^\\circ &#8211; {\\mathop{\\rm tg}\\nolimits} 35^\\circ } \\right) = 10 \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ \\times {\\mathop{\\rm tg}\\nolimits} 58^\\circ }\\\\{}&amp; \\Leftrightarrow &amp;{x = \\frac{{10 \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ \\times {\\mathop{\\rm tg}\\nolimits} 58^\\circ }}{{{\\mathop{\\rm tg}\\nolimits} 58^\\circ &#8211; {\\mathop{\\rm tg}\\nolimits} 35^\\circ }}}\\end{array}\\]<\/p>\n<p>\u00ad<br \/>\nOra,\u00a0\\(\\overline {AB} = \\frac{{10 \\times {\\mathop{\\rm tg}\\nolimits} 35^\\circ \\times {\\mathop{\\rm tg}\\nolimits} 58^\\circ }}{{{\\mathop{\\rm tg}\\nolimits} 58^\\circ &#8211; {\\mathop{\\rm tg}\\nolimits} 35^\\circ }} \\approx 12,4\\).<br \/>\nPortanto, a torre tem, aproximadamente, 12,4 metros de altura.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14020' onClick='GTTabs_show(0,14020)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado De um ponto O v\u00ea-se o topo de uma torre sob um \u00e2ngulo de 35 graus. Avan\u00e7ando 10 metros em dire\u00e7\u00e3o \u00e0 torre, o \u00e2ngulo passa a ser de 58 graus.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14022,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,489],"tags":[426,493,492],"series":[],"class_list":["post-14020","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-trigonometria-9--ano","tag-9-o-ano","tag-razao-trigonometrica","tag-tangente"],"views":3834,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/9V2Pag052-11.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14020","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=14020"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14020\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14022"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14020"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14020"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14020"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14020"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}