{"id":6414,"date":"2010-12-21T18:08:14","date_gmt":"2010-12-21T18:08:14","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6414"},"modified":"2022-01-21T23:59:14","modified_gmt":"2022-01-21T23:59:14","slug":"um-cone-de-revolucao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6414","title":{"rendered":"Um cone de revolu\u00e7\u00e3o"},"content":{"rendered":"<p><ul id='GTTabs_ul_6414' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6414' class='GTTabs_curr'><a  id=\"6414_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6414' ><a  id=\"6414_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_6414'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6415\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6415\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68.jpg\" data-orig-size=\"354,336\" 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=\"Cone\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68.jpg\" class=\"alignright wp-image-6415\" title=\"Cone\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68-300x284.jpg\" alt=\"\" width=\"240\" height=\"228\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68-300x284.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68-150x142.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68.jpg 354w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Na figura est\u00e1 representado, num referencial o.n. Oxyz, um cone de revolu\u00e7\u00e3o.<\/p>\n<p>Sabe-se que:<\/p>\n<ul>\n<li>A base do cone est\u00e1 contida no plano xOy e tem o seu centro na origem do referencial.<\/li>\n<li>[AC] e [BD] s\u00e3o di\u00e2metros da base.<\/li>\n<li>O ponto A pertence ao semieixo positivo Ox.<\/li>\n<li>O ponto B pertence ao semieixo positivo Oy.<\/li>\n<li>O v\u00e9rtice V pertence ao semieixo positivo Oz.<\/li>\n<\/ul>\n<ol>\n<li>Sabendo que uma equa\u00e7\u00e3o do plano ABV \u00e9 $4x+4y+3z=12$, mostre que o comprimento do raio da base \u00e9 3 e a altura do cone \u00e9 4.<\/li>\n<li>Determine uma condi\u00e7\u00e3o que defina a esfera cujo centro \u00e9 o ponto V e cuja intersec\u00e7\u00e3o com o plano xOy \u00e9 a base do cone.<\/li>\n<li>Designando por $\\alpha $ a amplitude do \u00e2ngulo BVD, determine o valor de $sen\\,\\alpha $.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6414' onClick='GTTabs_show(1,6414)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6414'>\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-191-68.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6415\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6415\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68.jpg\" data-orig-size=\"354,336\" 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=\"Cone\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68.jpg\" class=\"alignright wp-image-6415\" title=\"Cone\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68-300x284.jpg\" alt=\"\" width=\"240\" height=\"228\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68-300x284.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68-150x142.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-191-68.jpg 354w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Para $x=0\\wedge z=0$, vem $4\\times 0+4y+3\\times 0=12\\Leftrightarrow y=3$. Logo, $B\\,(0,3,0)$.\n<p>Para $y=0\\wedge z=0$, vem $4x+4\\times 0+3\\times 0=12\\Leftrightarrow x=3$. Logo, $A\\,(3,0,0)$.<\/p>\n<p>Para $x=0\\wedge y=0$, vem $4\\times 0+4\\times 0+3z=12\\Leftrightarrow z=4$. Logo, $V\\,(0,0,4)$.<\/p>\n<p>Logo, o raio da base \u00e9 $\\overline{OA}=3$ e a altura $\\overline{OV}=4$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O ponto B \u00e9 um ponto da superf\u00edcie da esfera.<br \/>\nLogo, o raio dessa esfera \u00e9 $r=\\overline{VB}=\\sqrt{{{3}^{2}}+{{4}^{2}}}=5$.<\/p>\n<p>Assim, uma condi\u00e7\u00e3o que define essa esfera \u00e9 ${{x}^{2}}+{{y}^{2}}+{{(z-4)}^{2}}\\le 25$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora,<br \/>\n\\[\\cos \\alpha =\\frac{\\overrightarrow{VB}.\\overrightarrow{VD}}{\\left\\| \\overrightarrow{VB} \\right\\|\\times \\left\\| \\overrightarrow{VD} \\right\\|}=\\frac{(0,3,-4).(0,-3,-4)}{\\sqrt{25}\\times \\sqrt{25}}=\\frac{-9+16}{25}=\\frac{7}{25}\\]<\/p>\n<p>Logo, \\[sen\\,\\alpha =+\\sqrt{1-{{\\left( \\frac{7}{25} \\right)}^{2}}}=\\sqrt{\\frac{625-49}{625}}=\\frac{\\sqrt{576}}{25}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6414' onClick='GTTabs_show(0,6414)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado, num referencial o.n. Oxyz, um cone de revolu\u00e7\u00e3o. Sabe-se que: A base do cone est\u00e1 contida no plano xOy e tem o seu centro na origem do&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20852,"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-6414","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":5088,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag191-68_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6414","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=6414"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6414\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20852"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6414"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6414"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6414"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6414"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}