{"id":6371,"date":"2010-12-13T00:02:58","date_gmt":"2010-12-13T00:02:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6371"},"modified":"2022-01-21T21:56:55","modified_gmt":"2022-01-21T21:56:55","slug":"uma-piramide-quadrangular-regular-3","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6371","title":{"rendered":"Uma pir\u00e2mide quadrangular regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_6371' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6371' class='GTTabs_curr'><a  id=\"6371_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6371' ><a  id=\"6371_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_6371'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6372\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6372\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg\" data-orig-size=\"362,278\" 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=\"Pir\u00e2mide\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg\" class=\"alignright wp-image-6372\" title=\"Pir\u00e2mide\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg\" alt=\"\" width=\"270\" height=\"207\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg 362w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47-300x230.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47-150x115.jpg 150w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>Considere no referencial o.n. do espa\u00e7o (Oxyz), a pir\u00e2mide quadrangular regular de v\u00e9rtice V e base [ABCO], assente no plano xOy.<\/p>\n<p>Sabendo que a pir\u00e2mide tem 5 unidades de altura e que C (0,4,0):<\/p>\n<ol>\n<li>determine $\\alpha $ de modo que $(3\\alpha ,\\alpha +2,1-\\alpha )$ perten\u00e7a ao plano VBC;<\/li>\n<li>determine o \u00e2ngulo das retas VB e VC;<\/li>\n<li>supondo que a unidade considerada \u00e9 o cent\u00edmetro, determine a \u00e1rea total da pir\u00e2mide.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6371' onClick='GTTabs_show(1,6371)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6371'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $A\\,(4,0,0)$, $B\\,(4,4,0)$, $C\\,(0,4,0)$ e $V\\,(2,2,5)$.\n<p>Logo, $\\overrightarrow{VB}=(2,2,-5)$ e $\\overrightarrow{VC}=(-2,2,-5)$.<\/p>\n<p>Seja $\\vec{n}\\,(a,b,c)$\u00a0 um vetor gen\u00e9rico perpendicular aos vetores anteriores. Assim, vem:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(a,b,c).(2,2,-5)=0\u00a0 \\\\<br \/>\n(a,b,c).(-2,2,-5)=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n2a+2b-5c=0\u00a0 \\\\<br \/>\n-2a+2b-5c=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n2a+2b-5c=0\u00a0 \\\\<br \/>\n4b-10c=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=\\frac{5c-2b}{2}\u00a0 \\\\<br \/>\nb=\\frac{5}{2}c\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=0\u00a0 \\\\<br \/>\nb=\\frac{5}{2}c\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n<img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6372\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6372\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg\" data-orig-size=\"362,278\" 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=\"Pir\u00e2mide\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg\" class=\"alignright wp-image-6372\" title=\"Pir\u00e2mide\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg\" alt=\"\" width=\"270\" height=\"207\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47.jpg 362w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47-300x230.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag185-47-150x115.jpg 150w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/>Portanto,\u00a0$\\overrightarrow{{{n}_{1}}}(0,5,2)$ \u00e9 um vector normal ao plano VBC.<\/p>\n<p>Assim, uma equa\u00e7\u00e3o do plano VBC \u00e9 da forma $5y+2z+d=0$. Como o ponto C, por exemplo, pertence a este plano, ent\u00e3o as suas coordenadas t\u00eam de verificar a equa\u00e7\u00e3o anterior. Logo, $5\\times 4+2\\times 0+d=0\\Leftrightarrow d=-20$.<\/p>\n<p>Portanto, uma equa\u00e7\u00e3o do plano VBC \u00e9 $5y+2z-20=0$.<\/p>\n<p>Logo, para que $(3\\alpha ,\\alpha +2,1-\\alpha )$ perten\u00e7a ao plano VBC ter\u00e1 de ser: \\[5\\times (\\alpha +2)+2\\times (1-\\alpha )-20=0\\Leftrightarrow 5\\alpha +10+2-2\\alpha -20=0\\Leftrightarrow \\alpha =\\frac{8}{3}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Ora,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\cos (B\\hat{V}C) &amp; = &amp; \\frac{\\overrightarrow{VB}.\\overrightarrow{VC}}{\\left\\| \\overrightarrow{VB} \\right\\|\\times \\left\\| \\overrightarrow{VC} \\right\\|}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{(2,2,-5).(-2,2,-5)}{\\sqrt{4+4+25}\\times \\sqrt{4+4+25}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{-4+4+25}{33}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{25}{33}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nLogo, a amplitude do \u00e2ngulo das retas VB e VC \u00e9 $B\\hat{V}C={{\\cos }^{-1}}(\\frac{25}{33})\\simeq 40,7{}^\\text{o}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Seja $M(4,2,0)$ o ponto m\u00e9dio do segmento [AB].<\/p>\n<p>Logo, $\\overline{VM}=\\sqrt{{{(4-2)}^{2}}+{{0}^{2}}+{{(0-5)}^{2}}}=\\sqrt{29}$.<\/p>\n<p>Assim, temos:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nA &amp; = &amp; {{A}_{b}}+4\\times {{A}_{[ABV]}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 4\\times 4+4\\times \\frac{4\\times \\sqrt{29}}{2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; (16+8\\sqrt{29})\\,c{{m}^{2}}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6371' onClick='GTTabs_show(0,6371)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere no referencial o.n. do espa\u00e7o (Oxyz), a pir\u00e2mide quadrangular regular de v\u00e9rtice V e base [ABCO], assente no plano xOy. Sabendo que a pir\u00e2mide tem 5 unidades de altura e&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20834,"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-6371","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":3046,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag185-47_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6371","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=6371"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6371\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20834"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6371"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6371"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6371"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6371"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}