{"id":6400,"date":"2010-12-19T23:20:40","date_gmt":"2010-12-19T23:20:40","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6400"},"modified":"2022-01-21T23:32:38","modified_gmt":"2022-01-21T23:32:38","slug":"um-paralelepipedo-de-altura-variavel","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6400","title":{"rendered":"Um paralelep\u00edpedo de altura vari\u00e1vel"},"content":{"rendered":"<p><ul id='GTTabs_ul_6400' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6400' class='GTTabs_curr'><a  id=\"6400_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6400' ><a  id=\"6400_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_6400'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6401\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6401\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62.jpg\" data-orig-size=\"371,399\" 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=\"Paralelep\u00edpedo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62.jpg\" class=\"alignright wp-image-6401\" title=\"Paralelep\u00edpedo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62-278x300.jpg\" alt=\"\" width=\"240\" height=\"258\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62-278x300.jpg 278w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62-139x150.jpg 139w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62.jpg 371w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>A figura representa um paralelep\u00edpedo de altura vari\u00e1vel, sendo:<\/p>\n<ul>\n<li>$\\overline{AB}=4\\,cm$<\/li>\n<li>$\\overline{BC}=3\\,cm$<\/li>\n<li>$\\overrightarrow{AF}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{AC}=\\alpha $<\/li>\n<\/ul>\n<ol>\n<li>Mostre que o volume do paralelep\u00edpedo \u00e9 dado por $V(\\alpha )=60\\,tg\\,\\alpha $.<\/li>\n<li>Determine o valor exato do volume do s\u00f3lido quando $\\cos (\\frac{\\pi }{2}+\\alpha )$ \u00e9 igual a $-\\frac{2}{3}$.<\/li>\n<li>Calcule o valor do produto escalar $\\overrightarrow{AF}\\,.\\,\\overrightarrow{BC}$.<\/li>\n<li>Se, no referencial o.n. representado, G for o ponto $(0,0,8)$, quais s\u00e3o as coordenadas dos pontos m\u00e9dios das arestas laterais do paralelep\u00edpedo?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6400' onClick='GTTabs_show(1,6400)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6400'>\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-189-62.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6401\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6401\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62.jpg\" data-orig-size=\"371,399\" 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=\"Paralelep\u00edpedo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62.jpg\" class=\"alignright wp-image-6401\" title=\"Paralelep\u00edpedo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62-278x300.jpg\" alt=\"\" width=\"240\" height=\"258\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62-278x300.jpg 278w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62-139x150.jpg 139w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-189-62.jpg 371w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Ora, $tg\\,\\alpha =\\frac{\\overline{FC}}{\\overline{AC}}$.\n<p>Como $\\overline{AC}=\\sqrt{{{4}^{2}}+{{3}^{2}}}=5$, ent\u00e3o $\\overline{FC}=5\\times tg\\,\\alpha $.<\/p>\n<p>Assim, $V(\\alpha )=4\\times 3\\times 5\\times tg\\,\\alpha =60\\,tg\\,\\alpha $.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Ora,<br \/>\n\\[\\begin{array}{*{20}{l}}{\\cos (\\frac{\\pi }{2} + \\alpha ) = &#8211; \\frac{2}{3}}&amp; \\Leftrightarrow &amp;{\\cos (\\frac{\\pi }{2} &#8211; ( &#8211; \\alpha )) = &#8211; \\frac{2}{3}}\\\\{}&amp; \\Leftrightarrow &amp;{sen{\\mkern 1mu} ( &#8211; \\alpha ) = &#8211; \\frac{2}{3}}\\\\{}&amp; \\Leftrightarrow &amp;{sen{\\mkern 1mu} \\alpha = \\frac{2}{3}}\\end{array}\\]<\/p>\n<p>Como $\\alpha \\in \\left] 0,\\frac{\\pi }{2} \\right[$, ent\u00e3o \\[\\cos \\alpha =+\\sqrt{1-{{(\\frac{2}{3})}^{2}}}=\\frac{\\sqrt{5}}{3}\\] e \\[tg\\,\\alpha =\\frac{\\frac{2}{3}}{\\frac{\\sqrt{5}}{3}}=\\frac{2}{\\sqrt{5}}=\\frac{2\\sqrt{5}}{5}\\]<\/p>\n<p>Logo, $V=60\\times \\frac{2\\sqrt{5}}{5}=24\\sqrt{5}$ unidades de volume.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Como $A\\,(3,0,0)$, $B\\,(3,4,0)$, $C\\,(0,4,0)$ e $F\\,(0,4,5\\,tg\\,\\alpha )$, ent\u00e3o $\\overrightarrow{AF}=(-3,4,5\\,tg\\,\\alpha )$ e $\\overrightarrow{BC}=(-3,0,0)$.<\/p>\n<p>Logo, $\\overrightarrow{AF}\\,.\\,\\overrightarrow{BC}=-3\\times (-3)+0+0=9$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Se, no referencial o.n. representado, G for o ponto $(0,0,8)$, as coordenadas dos pontos m\u00e9dios das arestas laterais do paralelep\u00edpedo s\u00e3o: $(3,0,4)$, $(3,4,4)$, $(0,4,4)$ e $(0,0,4)$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6400' onClick='GTTabs_show(0,6400)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A figura representa um paralelep\u00edpedo de altura vari\u00e1vel, sendo: $\\overline{AB}=4\\,cm$ $\\overline{BC}=3\\,cm$ $\\overrightarrow{AF}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{AC}=\\alpha $ Mostre que o volume do paralelep\u00edpedo \u00e9 dado por $V(\\alpha )=60\\,tg\\,\\alpha $. Determine o valor exato do&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20847,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,121],"series":[],"class_list":["post-6400","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","tag-produto"],"views":2331,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag189-62_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6400","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=6400"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6400\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20847"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6400"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6400"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6400"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6400"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}