{"id":6670,"date":"2011-03-30T02:59:47","date_gmt":"2011-03-30T01:59:47","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6670"},"modified":"2022-01-25T00:25:52","modified_gmt":"2022-01-25T00:25:52","slug":"surf-fresco-e-natural","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6670","title":{"rendered":"SURF, Fresco e Natural"},"content":{"rendered":"<p><ul id='GTTabs_ul_6670' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6670' class='GTTabs_curr'><a  id=\"6670_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6670' ><a  id=\"6670_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_6670'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Uma nova empresa de refrigerantes pretende lan\u00e7ar no mercado embalagens de sumo de fruta, com capacidade de dois litros.<\/p>\n<p>Por quest\u00f5es de marketing, as embalagens dever\u00e3o ter a forma de um prisma quadrangular regular.<\/p>\n<ol>\n<li>Mostre que a \u00e1rea total da embalagem, em dm<sup>2<\/sup>, \u00e9 dada por \\[A(x)=2{{x}^{2}}+\\frac{8}{x}\\]<br \/>\n<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/SURF.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6671\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6671\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/SURF.png\" data-orig-size=\"116,196\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&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=\"SURF\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/SURF.png\" class=\"alignright size-full wp-image-6671\" title=\"SURF\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/SURF.png\" alt=\"\" width=\"116\" height=\"196\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/SURF.png 116w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/SURF-88x150.png 88w\" sizes=\"auto, (max-width: 116px) 100vw, 116px\" \/><\/a>(x \u00e9 o comprimento da aresta da base, em dm)<br \/>\n<strong>Nota<\/strong>: Recorde que $1\\ litro=1\\ d{{m}^{3}}$.<\/li>\n<li>Utilizando m\u00e9todos exclusivamente anal\u00edticos, mostre que existe um valor de x para o qual a \u00e1rea total da embalagem \u00e9 m\u00ednima e determine-o.<\/li>\n<li>H\u00e1 exatamente dois valores de x para os quais a \u00e1rea total da embalagem \u00e9 igual a 12 dm<sup>2<\/sup>.<br \/>\nRecorrendo \u00e0 sua calculadora, resolva graficamente este problema.<br \/>\nApresente as solu\u00e7\u00f5es com aproxima\u00e7\u00e3o \u00e0s d\u00e9cimas. Explique como procedeu, apresentando o gr\u00e1fico, ou gr\u00e1ficos, em que se baseou para dar a sua resposta.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6670' onClick='GTTabs_show(1,6670)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6670'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>A \u00e1rea de uma das bases do prisma \u00e9 ${{A}_{b}}={{x}^{2}}$ e a altura do prisma \u00e9 $h=\\frac{2}{{{x}^{2}}}$, pois o seu volume \u00e9 2.<br \/>\nLogo, a \u00e1rea de uma das faces laterais \u00e9 ${{A}_{fL}}=x.\\frac{2}{{{x}^{2}}}=\\frac{2}{x}$.<br \/>\nAssim, \\[A(x)=2\\times {{x}^{2}}+4\\times \\frac{2}{x}=2{{x}^{2}}+\\frac{8}{x}\\] para $x&gt;0$.<br \/>\n\u00ad<\/li>\n<li>Ora, \\[A'(x)=4x-\\frac{8}{{{x}^{2}}}=\\frac{4{{x}^{3}}-8}{{{x}^{2}}}=\\frac{4.({{x}^{3}}-2)}{{{x}^{2}}}\\] para $x&gt;0$.\n<p>Como ${{x}^{3}}-2=0\\Leftrightarrow x=\\sqrt[3]{2}$, vem:<\/p>\n<table class=\"aligncenter\" style=\"width: 70%;\" border=\"2\" cellspacing=\"4\" cellpadding=\"4\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$x$<\/td>\n<td style=\"text-align: center;\">\u00a0\u00a0\u00a0 $0$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$\\sqrt[3]{2}$<\/td>\n<td style=\"text-align: right;\">$+\\infty $<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$4.({{x}^{3}}-2)$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">${{x}^{2}}$<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$A'(x)$<\/td>\n<td style=\"text-align: center;\">n.d.<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$A(x)$<\/td>\n<td style=\"text-align: center;\">n.d.<\/td>\n<td style=\"text-align: center;\">\u00a0\u00a0\u00a0\u00a0\u00a0 $\\searrow $<\/td>\n<td style=\"text-align: center;\">$A(\\sqrt[3]{2})$<\/td>\n<td style=\"text-align: center;\">\u00a0\u00a0\u00a0\u00a0\u00a0 $\\nearrow $<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Portanto, o valor de x para o qual a \u00e1rea total da embalagem \u00e9 m\u00ednima \u00e9 $\\sqrt[3]{2}$.<\/p>\n<\/li>\n<li>As solu\u00e7\u00f5es do problema s\u00e3o as abcissas dos pontos de intersec\u00e7\u00e3o do gr\u00e1fico da fun\u00e7\u00e3o A com a reta de equa\u00e7\u00e3o $y=12$. Com recurso \u00e0 calculadora, podemos obter parte do gr\u00e1fico da fun\u00e7\u00e3o A, parte da reta de equa\u00e7\u00e3o $y=12$, bem como as abcissas dos pontos de intersec\u00e7\u00e3o do gr\u00e1fico da fun\u00e7\u00e3o com a referida reta:\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/J1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6672\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6672\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/J1.png\" data-orig-size=\"129,65\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&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=\"J1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/J1.png\" class=\"alignnone size-full wp-image-6672\" title=\"J1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/J1.png\" alt=\"\" width=\"129\" height=\"65\" \/><\/a>\u00a0 <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/G1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6673\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6673\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/G1.png\" data-orig-size=\"129,65\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&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=\"G1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/G1.png\" class=\"alignnone size-full wp-image-6673\" title=\"G1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/G1.png\" alt=\"\" width=\"129\" height=\"65\" \/><\/a>\u00a0 <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/G2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6674\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6674\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/G2.png\" data-orig-size=\"129,65\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&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=\"G2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/G2.png\" class=\"alignnone size-full wp-image-6674\" title=\"G2\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/G2.png\" alt=\"\" width=\"129\" height=\"65\" \/><\/a><\/p>\n<p>As solu\u00e7\u00e3o do problema, com aproxima\u00e7\u00e3o \u00e0s d\u00e9cimas, s\u00e3o $0,7$ e $2,0$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6670' onClick='GTTabs_show(0,6670)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma nova empresa de refrigerantes pretende lan\u00e7ar no mercado embalagens de sumo de fruta, com capacidade de dois litros. Por quest\u00f5es de marketing, as embalagens dever\u00e3o ter a forma de um&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20939,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,134],"tags":[136,144],"series":[],"class_list":["post-6670","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-derivadas","tag-derivada","tag-extremos-relativos"],"views":2683,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-SURF.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6670","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=6670"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6670\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20939"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6670"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6670"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6670"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6670"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}