{"id":6416,"date":"2010-12-21T18:54:01","date_gmt":"2010-12-21T18:54:01","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6416"},"modified":"2022-01-22T00:02:41","modified_gmt":"2022-01-22T00:02:41","slug":"um-cubo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6416","title":{"rendered":"Um cubo"},"content":{"rendered":"<p><ul id='GTTabs_ul_6416' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6416' class='GTTabs_curr'><a  id=\"6416_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6416' ><a  id=\"6416_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_6416'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6417\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6417\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69.jpg\" data-orig-size=\"375,354\" 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=\"Cubo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69.jpg\" class=\"alignright wp-image-6417 size-medium\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69-300x283.jpg\" alt=\"\" width=\"300\" height=\"283\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69-300x283.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69-150x141.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69.jpg 375w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Na figura est\u00e1 representado um cubo, em referencial o.n. Oxyz.<\/p>\n<p>Sabe-se que:<\/p>\n<ul>\n<li>a face [OPQR] est\u00e1 contida no plano xOy;<\/li>\n<li>a face [OSVR] est\u00e1 contida no plano xOz;<\/li>\n<li>a face [OSTP] est\u00e1 contida no plano yOz;<\/li>\n<li>uma equa\u00e7\u00e3o do plano VTQ \u00e9 $x+y+z=6$.<\/li>\n<\/ul>\n<ol>\n<li>Mostre que o volume do cubo \u00e9 27.<\/li>\n<li>Determine uma equa\u00e7\u00e3o da superf\u00edcie esf\u00e9rica, tal que:<br \/>\n&#8211; o centro \u00e9 o sim\u00e9trico de U, em rela\u00e7\u00e3o ao plano xOy;<br \/>\n&#8211; o ponto Q pertence a essa superf\u00edcie esf\u00e9rica.<\/li>\n<li>Seja $\\alpha $ o plano que cont\u00e9m o ponto S e \u00e9 paralelo ao plano VTQ.<br \/>\nProve que a recta RP est\u00e1 contida em $\\alpha $.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6416' onClick='GTTabs_show(1,6416)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6416'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6417\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6417\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69.jpg\" data-orig-size=\"375,354\" 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=\"Cubo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69.jpg\" class=\"alignright wp-image-6417 size-medium\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69-300x283.jpg\" alt=\"\" width=\"300\" height=\"283\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69-300x283.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69-150x141.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-69.jpg 375w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>As coordenadas do ponto V s\u00e3o da forma $(a,0,a)$, com $a&gt;0$.\n<p>Como V pertence ao plano VTQ, vem $a+0+a=6\\Leftrightarrow a=3$.<\/p>\n<p>Logo, a aresta do cubo tem 3 unidades de comprimento, pelo que o seu volume \u00e9 27 unidades c\u00fabicas.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O ponto sim\u00e9trico de U, em rela\u00e7\u00e3o ao plano xOy, \u00e9 o ponto $U&#8217;\\,(3,3,-3)$.\n<p>O raio da superf\u00edcie esf\u00e9rica \u00e9 $r=\\overline{U&#8217;Q}=3$.<\/p>\n<p>Logo, ${{(x-3)}^{2}}+{{(y-3)}^{2}}+{{(z+3)}^{2}}=9$ \u00e9 uma equa\u00e7\u00e3o da superf\u00edcie esf\u00e9rica considerada.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>A equa\u00e7\u00e3o do plano $\\alpha $ \u00e9 da forma $x+y+z+d=0$.\n<p>Como S \u00e9 um ponto desse plano, ent\u00e3o $0+0+3+d=0\\Leftrightarrow d=-3$. Logo, $x+y+z-3=0$ \u00e9 uma equa\u00e7\u00e3o do plano $\\alpha $.<\/p>\n<p>Os pontos R e P (distintos) pertencem a este plano: $3+0+0-3=0$ e $0+3+0-3=0$ s\u00e3o ambas proposi\u00e7\u00f5es verdadeiras.<\/p>\n<p>Consequentemente, a reta RP est\u00e1 contida no plano $\\alpha $, pois dois pontos distintos dessa reta pertencem a esse plano.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6416' onClick='GTTabs_show(0,6416)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado um cubo, em referencial o.n. Oxyz. Sabe-se que: a face [OPQR] est\u00e1 contida no plano xOy; a face [OSVR] est\u00e1 contida no plano xOz; a face [OSTP]&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20853,"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-6416","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":3807,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag191-69_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6416","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=6416"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6416\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20853"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6416"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6416"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6416"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6416"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}