{"id":11841,"date":"2014-03-17T22:54:17","date_gmt":"2014-03-17T22:54:17","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11841"},"modified":"2022-01-25T01:09:02","modified_gmt":"2022-01-25T01:09:02","slug":"uma-escultura-em-cimento","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11841","title":{"rendered":"Uma escultura em cimento"},"content":{"rendered":"<p><ul id='GTTabs_ul_11841' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11841' class='GTTabs_curr'><a  id=\"11841_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11841' ><a  id=\"11841_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_11841'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11842\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11842\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg\" data-orig-size=\"286,382\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1395097677&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=\"Escultura\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg\" class=\"alignright  wp-image-11842\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg\" alt=\"Escultura\" width=\"103\" height=\"137\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg 286w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura-224x300.jpg 224w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura-112x150.jpg 112w\" sizes=\"auto, (max-width: 103px) 100vw, 103px\" \/><\/a>Na figura, est\u00e1 representado um projeto de uma escultura em cimento para o jardim de uma escola, constitu\u00edda por uma esfera colocada sobre um cubo.<\/p>\n<p>Pretende-se que a escultura tenha uma altura total de $2$ metros.<\/p>\n<p>Apresentam-se, a seguir, as vistas de frente de tr\u00eas poss\u00edveis concretiza\u00e7\u00f5es do projeto.<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11843\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11843\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas.jpg\" data-orig-size=\"1037,252\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1395097739&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=\"Vistas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas-1024x248.jpg\" class=\"aligncenter  wp-image-11843\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas.jpg\" alt=\"Vistas\" width=\"622\" height=\"151\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas.jpg 1037w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas-300x72.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas-1024x248.jpg 1024w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas-150x36.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Vistas-400x97.jpg 400w\" sizes=\"auto, (max-width: 622px) 100vw, 622px\" \/><\/a><\/p>\n<p>Designemos por $x$ o raio da esfera (em metros).<\/p>\n<ol>\n<li>Indique, na forma de intervalo de n\u00fameros reais, o conjunto dos valores que a vari\u00e1vel $x$ pode assumir.<\/li>\n<li>Mostre que o volume total, em metros c\u00fabicos, da escultura \u00e9 dado, em fun\u00e7\u00e3o de $x$, por:<br \/>\n\\[V\\left( x \\right) = \\frac{{4\\pi\u00a0 &#8211; 24}}{3}{x^3} + 24{x^2} &#8211; 24x + 8\\]<\/li>\n<li>Determine o raio da esfera e a aresta do cubo de modo que o volume total da escultura seja m\u00ednimo.<br \/>\nApresente os resultados em metros, arredondados \u00e0s cent\u00e9simas.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11841' onClick='GTTabs_show(1,11841)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11841'>\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\/2014\/03\/Escultura.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11842\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11842\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg\" data-orig-size=\"286,382\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1395097677&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=\"Escultura\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg\" class=\"alignright  wp-image-11842\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg\" alt=\"Escultura\" width=\"103\" height=\"137\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura.jpg 286w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura-224x300.jpg 224w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Escultura-112x150.jpg 112w\" sizes=\"auto, (max-width: 103px) 100vw, 103px\" \/><\/a>Como a escultura \u00e9 constitu\u00edda por uma esfera colocada sobre um cubo, com uma altura total de $2$ metros, resulta: $0 &lt; 2x &lt; 2 \\Leftrightarrow 0 &lt; x &lt; 1$.<br \/>\nLogo, ${D_V} = \\left] {0,1} \\right[$.<br \/>\n\u00ad<\/li>\n<li>Para $x \\in \\left] {0,1} \\right[$, vem:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{V\\left( x \\right)}&amp; = &amp;{{V_{Cubo}} + {V_{Esfera}}} \\\\<br \/>\n{}&amp; = &amp;{{{\\left( {2 &#8211; 2x} \\right)}^3} + \\frac{4}{3}\\pi {x^3}} \\\\<br \/>\n{}&amp; = &amp;{\\left( {4 &#8211; 8x + 4{x^2}} \\right)\\left( {2 &#8211; 2x} \\right) + \\frac{4}{3}\\pi {x^3}} \\\\<br \/>\n{}&amp; = &amp;{8 &#8211; 8x &#8211; 16x + 16{x^2} + 8{x^2} &#8211; 8{x^3} + \\frac{4}{3}\\pi {x^3}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{4\\pi\u00a0 &#8211; 24}}{3}{x^3} + 24{x^2} &#8211; 24x + 8}<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/li>\n<li>Para $x \\in \\left] {0,1} \\right[$, vem:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{V&#8217;\\left( x \\right)}&amp; = &amp;{\\frac{d}{{dx}}\\left( {\\frac{{4\\pi\u00a0 &#8211; 24}}{3}{x^3} + 24{x^2} &#8211; 24x + 8} \\right)} \\\\<br \/>\n{}&amp; = &amp;{3 \\times \\frac{{4\\pi\u00a0 &#8211; 24}}{3}{x^2} + 48x &#8211; 24} \\\\<br \/>\n{}&amp; = &amp;{\\left( {4\\pi\u00a0 &#8211; 24} \\right){x^2} + 48x &#8211; 24}<br \/>\n\\end{array}\\]<br \/>\nDeterminemos os zeros de $V&#8217;$:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{V&#8217;\\left( x \\right) = 0}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{x = \\frac{{ &#8211; 48 \\pm \\sqrt {{{48}^2} + 96\\left( {4\\pi\u00a0 &#8211; 24} \\right)} }}{{8\\pi\u00a0 &#8211; 48}}}&amp; \\wedge &amp;{x \\in \\left] {0,1} \\right[}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{x = \\frac{{ &#8211; 48 \\pm \\sqrt {384\\pi } }}{{8\\pi\u00a0 &#8211; 48}}}&amp; \\wedge &amp;{x \\in \\left] {0,1} \\right[}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{x = \\frac{{ &#8211; 6 \\pm \\sqrt {6\\pi } }}{{\\pi\u00a0 &#8211; 6}}}&amp; \\wedge &amp;{x \\in \\left] {0,1} \\right[}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{x = \\frac{{6 \\mp \\sqrt {6\\pi } }}{{6 &#8211; \\pi }}}&amp; \\wedge &amp;{x \\in \\left] {0,1} \\right[}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{{6 &#8211; \\sqrt {6\\pi } }}{{6 &#8211; \\pi }}}<br \/>\n\\end{array}\\]<br \/>\nNote que o \u00fanico zero de $V&#8217;$ \u00e9:\\[{x_1} = \\frac{{6 &#8211; \\sqrt {6\\pi } }}{{6 &#8211; \\pi }} \\times \\frac{{6 + \\sqrt {6\\pi } }}{{6 + \\sqrt {6\\pi } }} = \\frac{{36 &#8211; 6\\pi }}{{\\left( {6 &#8211; \\pi } \\right)\\left( {6 + \\sqrt {6\\pi } } \\right)}} = \\frac{{36 &#8211; 6\\pi }}{{\\left( {6 &#8211; \\pi } \\right)\\left( {6 + \\sqrt {6\\pi } } \\right)}} = \\frac{6}{{6 + \\sqrt {6\\pi } }} = \\frac{{\\sqrt 6\u00a0 \\times \\sqrt 6 }}{{\\sqrt 6\u00a0 \\times \\sqrt 6\u00a0 + \\sqrt 6\u00a0 \\times \\sqrt \\pi\u00a0 }} = \\frac{{\\sqrt 6 }}{{\\sqrt 6\u00a0 + \\sqrt \\pi\u00a0 }}\\]<br \/>\nsendo \\[V\\left( {{x_1}} \\right) = \\frac{{8\\pi }}{{{{\\left( {\\sqrt \\pi\u00a0\u00a0 + \\sqrt 6 } \\right)}^2}}}\\]<br \/>\nO outro zero da equa\u00e7\u00e3o do 2.\u00ba grau \u00e9:<br \/>\n\\[{x_2} = \\frac{{6 + \\sqrt {6\\pi } }}{{6 &#8211; \\pi }} = \\frac{{\\sqrt 6 }}{{\\sqrt 6\u00a0 &#8211; \\sqrt \\pi\u00a0 }}\\]<br \/>\nConstruindo um quadro de sinal de $V&#8217;$, vem:<\/p>\n<table class=\" aligncenter\" style=\"width: 60%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"border: 1px solid #000080; width: 80px; text-align: center;\">$x$<\/td>\n<td style=\"border: 1px solid #000080; width: 40px; text-align: center;\">$0$<\/td>\n<td style=\"border: 1px solid #000080; width: 120px;\"><\/td>\n<td style=\"border: 1px solid #000080; width: 40px; text-align: center;\">$\\frac{{\\sqrt 6 }}{{\\sqrt 6\u00a0 + \\sqrt \\pi\u00a0 }}$<\/td>\n<td style=\"border: 1px solid #000080; width: 120px;\"><\/td>\n<td style=\"border: 1px solid #000080; width: 40px; text-align: center;\">$1$<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #000080; width: 80px; text-align: center;\">${V&#8217;\\left( x \\right)}$<\/td>\n<td style=\"border: 1px solid #000080; width: 40px; text-align: center;\">n.d.<\/td>\n<td style=\"border: 1px solid #000080; width: 120px; text-align: center;\">$ &#8211; $<\/td>\n<td style=\"border: 1px solid #000080; width: 40px; text-align: center;\">$0$<\/td>\n<td style=\"border: 1px solid #000080; width: 120px; text-align: center;\">$ + $<\/td>\n<td style=\"border: 1px solid #000080; width: 40px; text-align: center;\">n.d.<\/td>\n<\/tr>\n<tr>\n<td style=\"border: 1px solid #000080; width: 80px; text-align: center;\">${V\\left( x \\right)}$<\/td>\n<td style=\"border: 1px solid #000080; width: 40px; text-align: center;\">n.d.<\/td>\n<td style=\"border: 1px solid #000080; width: 120px; text-align: center;\">$ \\searrow $<\/td>\n<td style=\"border: 1px solid #000080; width: 40px;\">$\\frac{{8\\pi }}{{{{\\left( {\\sqrt \\pi\u00a0\u00a0 + \\sqrt 6 } \\right)}^2}}}$<\/td>\n<td style=\"border: 1px solid #000080; width: 120px; text-align: center;\">$ \\nearrow $<\/td>\n<td style=\"border: 1px solid #000080; width: 40px; text-align: center;\">n.d.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Portanto, o volume total da escultura \u00e9 m\u00ednimo para ${x_1} = \\frac{{\\sqrt 6 }}{{\\sqrt 6\u00a0 + \\sqrt \\pi\u00a0 }} \\approx 0,58$ m (raio da esfera) e $a = 2 &#8211; 2 \\times \\frac{{\\sqrt 6 }}{{\\sqrt 6\u00a0 + \\sqrt \\pi\u00a0 }} \\approx 0,84$ m (aresta do cubo).<br \/>\n\u00ad<\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><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\": \"ggbApplet\",\r\n\"width\":900,\r\n\"height\":580,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"language\":\"pt\",\r\n\/\/ use 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