{"id":2957,"date":"2010-09-27T15:41:48","date_gmt":"2010-09-27T14:41:48","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=2957"},"modified":"2022-01-21T01:09:30","modified_gmt":"2022-01-21T01:09:30","slug":"2957","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=2957","title":{"rendered":"Uma chamin\u00e9 de cozinha"},"content":{"rendered":"<p><ul id='GTTabs_ul_2957' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_2957' class='GTTabs_curr'><a  id=\"2957_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_2957' ><a  id=\"2957_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_2957'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"2962\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=2962\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg\" data-orig-size=\"930,402\" 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=\"Chamin\u00e9\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg\" class=\"alignright wp-image-2962 size-full\" title=\"Chamin\u00e9\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg\" alt=\"\" width=\"651\" height=\"281\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg 930w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine-300x129.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine-150x64.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine-400x172.jpg 400w\" sizes=\"auto, (max-width: 651px) 100vw, 651px\" \/><\/a>Uma chamin\u00e9 de cozinha tem a forma de um tronco de pir\u00e2mide com bases retangulares.<\/p>\n<p>As faces [ADHE] e [ABFE] s\u00e3o perpendiculares \u00e0s duas bases.<\/p>\n<p>Na figura, as dimens\u00f5es est\u00e3o expressas em mil\u00edmetros.<\/p>\n<p>Calcule:<\/p>\n<ol>\n<li>as alturas de [BF] e de [DH] dos trap\u00e9zios [BCGF] e [CDHG], com aproxima\u00e7\u00e3o ao mil\u00edmetro;<\/li>\n<li>a \u00e1rea de cada uma das faces [BCGF] e [CDHG], com aproxima\u00e7\u00e3o ao cm<sup>2<\/sup>;<\/li>\n<li>as medidas dos \u00e2ngulos \u03b1 e \u03b2, com aproxima\u00e7\u00e3o \u00e0 d\u00e9cima de grau;<\/li>\n<li>o comprimento da aresta [GC], com aproxima\u00e7\u00e3o ao mil\u00edmetro.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_2957' onClick='GTTabs_show(1,2957)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_2957'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"2962\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=2962\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg\" data-orig-size=\"930,402\" 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=\"Chamin\u00e9\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg\" class=\"size-full wp-image-2962 aligncenter\" title=\"Chamin\u00e9\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg\" alt=\"\" width=\"651\" height=\"281\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine.jpg 930w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine-300x129.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine-150x64.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/chamine-400x172.jpg 400w\" sizes=\"auto, (max-width: 651px) 100vw, 651px\" \/><\/a>\u00ad<\/p>\n<ol>\n<li>Seja B&#8217; a proje\u00e7\u00e3o ortogonal do ponto B sobre [FE].<br \/>\nAplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [BFB&#8217;], temos:<br \/>\n\\[\\overline{BF}=\\sqrt{{{\\overline{BB&#8217;}}^{2}}+{{\\overline{FB&#8217;}}^{2}}}=\\sqrt{{{800}^{2}}+{{200}^{2}}}=\\sqrt{680000}=100\\sqrt{68}=200\\sqrt{17}\\]<br \/>\nLogo,\u00a0$\\overline{BF}\\approx 815\\ mm$.<\/p>\n<p>Seja D&#8217; a proje\u00e7\u00e3o ortogonal do ponto D sobre [HE].<br \/>\nAplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [DHD&#8217;], temos:<br \/>\n\\[\\overline{DH}=\\sqrt{{{\\overline{DD&#8217;}}^{2}}+{{\\overline{HD&#8217;}}^{2}}}=\\sqrt{{{800}^{2}}+{{300}^{2}}}=\\sqrt{730000}=100\\sqrt{73}\\]<br \/>\nLogo,\u00a0$\\overline{DH}\\approx 854\\ mm$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Ora, \\[{{A}_{[BCGF]}}=\\frac{\\overline{GF}+\\overline{CB}}{2}\\times \\overline{BF}=\\frac{700+400}{2}\\times 200\\sqrt{17}=110000\\sqrt{17}\\approx 453542\\]<br \/>\nLogo, ${{A}_{[BCGF]}}\\approx 4535\\ c{{m}^{2}}$.<\/p>\n<p>De forma an\u00e1loga, \\[{{A}_{[CDHG]}}=\\frac{\\overline{GH}+\\overline{CD}}{2}\\times \\overline{DH}=\\frac{700+500}{2}\\times 100\\sqrt{73}=60000\\sqrt{73}\\approx 512640\\]<br \/>\nLogo, ${{A}_{[CDHG]}}\\approx 5126\\ c{{m}^{2}}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Como $tg\\,\\alpha =\\frac{\\overline{BB&#8217;}}{\\overline{FB&#8217;}}=\\frac{800}{200}=4$, ent\u00e3o $\\alpha =t{{g}^{-1}}(4)\\approx 76,0{}^\\text{o}$.<br \/>\nComo $tg\\,\\beta =\\frac{\\overline{DD&#8217;}}{\\overline{HD&#8217;}}=\\frac{800}{300}=\\frac{8}{3}$, ent\u00e3o $\\beta =t{{g}^{-1}}(\\frac{8}{3})\\approx 69,4{}^\\text{o}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Seja C&#8217; a proje\u00e7\u00e3o ortogonal do ponto C sobre [GF].<br \/>\nAplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo ret\u00e2ngulo [CGC&#8217;], temos: \\[\\overline{CG}=\\sqrt{{{\\overline{BF}}^{2}}+{{\\overline{G{C}&#8217;}}^{2}}}=\\sqrt{({{800}^{2}}+{{200}^{2}})+{{300}^{2}}}=\\sqrt{770000}=100\\sqrt{77}\\]<br \/>\nLogo, $\\overline{CG}\\approx 877\\ mm$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_2957' onClick='GTTabs_show(0,2957)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma chamin\u00e9 de cozinha tem a forma de um tronco de pir\u00e2mide com bases retangulares. As faces [ADHE] e [ABFE] s\u00e3o perpendiculares \u00e0s duas bases. Na figura, as dimens\u00f5es est\u00e3o expressas&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20792,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,99],"tags":[422,423],"series":[],"class_list":["post-2957","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-trigonometria","tag-11-o-ano","tag-trigonometria"],"views":2547,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/09\/11V1Pag089-25_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/2957","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=2957"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/2957\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20792"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2957"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2957"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2957"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=2957"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}