{"id":11759,"date":"2014-02-05T21:14:19","date_gmt":"2014-02-05T21:14:19","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11759"},"modified":"2022-01-22T01:38:18","modified_gmt":"2022-01-22T01:38:18","slug":"uma-peca-de-forma-cilindrica","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11759","title":{"rendered":"Uma pe\u00e7a de forma cil\u00edndrica"},"content":{"rendered":"<p><ul id='GTTabs_ul_11759' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11759' class='GTTabs_curr'><a  id=\"11759_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11759' ><a  id=\"11759_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_11759'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Uma empresa de alum\u00ednio pretende fabricar uma pe\u00e7a de forma cil\u00edndrica, com capacidade de $500$ cm<sup>3<\/sup>.<\/p>\n<p>As tampas superior e inferior s\u00e3o feitas de alum\u00ednio especial que custa $5$ c\u00eantimos por cent\u00edmetro quadrado.<\/p>\n<p>A superf\u00edcie lateral \u00e9 feita de material mais barato, que custa $2$ c\u00eantimos por cent\u00edmetro quadrado.<\/p>\n<ol>\n<li>Exprima o custo $C$ da pe\u00e7a em fun\u00e7\u00e3o do raio $r$ da base.<\/li>\n<li>Fa\u00e7a uma representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o $C$.<\/li>\n<li>Qual \u00e9 o de $r$ para o qual a pe\u00e7a \u00e9 mais barata?<br \/>\nIndique o valor aproximado ao mil\u00edmetro.<\/li>\n<li>Qual \u00e9 o custo m\u00ednimo da pe\u00e7a?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11759' onClick='GTTabs_show(1,11759)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11759'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Considerando que a pe\u00e7a tem capacidade de $500$ cm<sup>3<\/sup> e designando por $r$ e por $h$, respetivamente, o raio da base e a altura da pe\u00e7a, em cent\u00edmetros, temos: \\[\\pi {r^2}h = 500 \\Leftrightarrow h = \\frac{{500}}{{\\pi {r^2}}}\\]<br \/>\nEm cent\u00edmetros quadrados, a \u00e1rea da superf\u00edcie total da pe\u00e7a pode ser expressa por: \\[{A_T} = 2\\pi {r^2} + 2\\pi r \\times \\frac{{500}}{{\\pi {r^2}}} = \\underbrace {2\\pi {r^2}}_{{A_{2B}}} + \\underbrace {\\frac{{1000}}{r}}_{{A_L}}\\]<br \/>\nAssim, o custo da pe\u00e7a, em c\u00eantimos, \u00e9 dado por: \\[C\\left( r \\right) = 10\\pi {r^2} + \\frac{{2000}}{r}\\]<br \/>\n\u00ad<\/li>\n<li>\n<table border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/janela.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11763\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11763\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/janela.png\" data-orig-size=\"198,134\" 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=\"Janela de visualiza\u00e7\u00e3o\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/janela.png\" class=\"aligncenter size-full wp-image-11763\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/janela.png\" alt=\"Janela de visualiza\u00e7\u00e3o\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/janela.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/janela-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/graf.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11764\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11764\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/graf.png\" data-orig-size=\"198,134\" 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=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/graf.png\" class=\"aligncenter size-full wp-image-11764\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/graf.png\" alt=\"Gr\u00e1fico\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/graf.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/graf-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a><\/td>\n<td><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/min.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11765\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11765\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/min.png\" data-orig-size=\"198,134\" 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=\"M\u00ednimo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/min.png\" class=\"aligncenter size-full wp-image-11765\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/min.png\" alt=\"M\u00ednimo\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/min.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/min-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00ad<\/p>\n<\/li>\n<li>A pe\u00e7a mais barata \u00e9 obtida para $r \\approx 3,2$ cm.<br \/>\n\u00ad<\/li>\n<li>O custo m\u00ednimo da pe\u00e7a \u00e9, aproximadamente, $9,47$ euros.<\/li>\n<\/ol>\n<p style=\"text-align: center;\">\u00ad<br \/>\n<script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": 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