{"id":10055,"date":"2012-10-05T01:16:03","date_gmt":"2012-10-05T00:16:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=10055"},"modified":"2022-01-20T22:16:04","modified_gmt":"2022-01-20T22:16:04","slug":"duas-bolas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=10055","title":{"rendered":"Duas bolas"},"content":{"rendered":"<p><ul id='GTTabs_ul_10055' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_10055' class='GTTabs_curr'><a  id=\"10055_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_10055' ><a  id=\"10055_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_10055'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Uma bola com $30$ cm de di\u00e2metro est\u00e1 apoiada no solo e encostada a uma parede.<\/p>\n<p>Poder\u00e1 uma bola de $5$ cm de di\u00e2metro passar entre a parede e o solo sem tocar na bola maior?<\/p>\n<div id=\"attachment_10057\" style=\"width: 188px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2bolas.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-10057\" data-attachment-id=\"10057\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=10057\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2bolas.jpg\" data-orig-size=\"297,279\" 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;1349396219&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=\"Bolas\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Duas bolas&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2bolas.jpg\" class=\"wp-image-10057 \" title=\"Bolas\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2bolas.jpg\" alt=\"\" width=\"178\" height=\"167\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2bolas.jpg 297w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/2bolas-150x140.jpg 150w\" sizes=\"auto, (max-width: 178px) 100vw, 178px\" \/><\/a><p id=\"caption-attachment-10057\" class=\"wp-caption-text\">Duas bolas<\/p><\/div>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_10055' onClick='GTTabs_show(1,10055)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_10055'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\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\":715,\r\n\"height\":380,\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 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/p>\n<\/p>\n<p>A bola mais pequena poder\u00e1 passar entre a parede e o solo sem tocar na bola maior se e s\u00f3 se $\\overline {OP}\u00a0 &lt; \\overline {OQ} $.<\/p>\n<p>\u00c9\u00a0conveniente reparar que $\\left[ {OA} \\right]$ e $\\left[ {OB} \\right]$ s\u00e3o diagonais de quadrados, cujos comprimentos dos lados s\u00e3o, respetivamente, $15$ cm e $r$ cm.<\/p>\n<\/p>\n<p>Assim, para $r = \\frac{5}{2}$, tem-se:<\/p>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{\\overline {OA}\u00a0 = 15\\sqrt 2 }&amp;{\\rm{e}}&amp;{\\overline {OB}\u00a0 = \\frac{5}{2}\\sqrt 2 }<br \/>\n\\end{array}$$<\/p>\n<p>Logo, vem:<\/p>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{\\overline {OQ}\u00a0 = \\overline {OA}\u00a0 &#8211; \\overline {QA}\u00a0 = 15\\sqrt 2\u00a0 &#8211; 15 = 15\\left( {\\sqrt 2\u00a0 &#8211; 1} \\right)}&amp;{\\rm{e}}&amp;{\\overline {OP}\u00a0 = \\overline {OB}\u00a0 + \\overline {BP}\u00a0 = \\frac{5}{2}\\sqrt 2\u00a0 + \\frac{5}{2} = \\frac{5}{2}\\left( {\\sqrt 2\u00a0 + 1} \\right)}\\\\<br \/>\n{\\overline {OQ}\u00a0 \\approx 6,213}&amp;e&amp;{\\overline {OP}\u00a0 \\approx 6,036}<br \/>\n\\end{array}$$<\/p>\n<\/p>\n<p>Como $\\overline {OP} &lt; \\overline {OQ} $,\u00a0ent\u00e3o a bola mais pequena poder\u00e1 passar entre a parede e o solo sem tocar na bola maior.<\/p>\n<\/p>\n<p><strong>Nota 1<\/strong>:<br \/>\nVerifique esta possibilidade na aplica\u00e7\u00e3o apresentada acima, fazendo $r = 2,5$.<\/p>\n<\/p>\n<p><strong>Nota 2<\/strong>:<br \/>\nSejam $c$ e $d$, respetivamente, os comprimentos do lado de um quadrado e da sua diagonal. De acordo com o Teorema de Pit\u00e1goras, tem-se:<\/p>\n<p>$$d = \\sqrt {{c^2} + {c^2}}\u00a0 = \\sqrt {2{c^2}}\u00a0 = c\\sqrt 2 $$<\/p>\n<blockquote>\n<p style=\"text-align: center;\">Isto \u00e9, <strong>o comprimento da diagonal de um quadrado \u00e9 $\\sqrt 2 $ vezes o comprimento do seu lado<\/strong>.<\/p>\n<\/blockquote>\n<p>J\u00e1 agora, determine a rela\u00e7\u00e3o entre o comprimento da diagonal espacial de um cubo e o comprimento da sua aresta.<\/p>\n<p>(Resposta: ${d_e} = a\\sqrt 3 $) &#8211; Porqu\u00ea?<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_10055' onClick='GTTabs_show(0,10055)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma bola com $30$ cm de di\u00e2metro est\u00e1 apoiada no solo e encostada a uma parede. Poder\u00e1 uma bola de $5$ cm de di\u00e2metro passar entre a parede e o solo&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20780,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,322],"tags":[429,67,430],"series":[],"class_list":["post-10055","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-modulo-inicial","tag-10-o-ano","tag-geometria","tag-modulo-inicial"],"views":3792,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/10V1Pag036-17_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10055","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=10055"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/10055\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20780"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10055"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10055"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10055"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=10055"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}