{"id":14499,"date":"2018-04-14T21:17:00","date_gmt":"2018-04-14T20:17:00","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=14499"},"modified":"2022-01-07T22:08:19","modified_gmt":"2022-01-07T22:08:19","slug":"dois-circulos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=14499","title":{"rendered":"Dois c\u00edrculos"},"content":{"rendered":"<p><ul id='GTTabs_ul_14499' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_14499' class='GTTabs_curr'><a  id=\"14499_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_14499' ><a  id=\"14499_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_14499'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14503\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14503\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17.png\" data-orig-size=\"385,245\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"C\u00edrculos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17.png\" class=\"alignright size-medium wp-image-14503\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17-300x191.png\" alt=\"\" width=\"300\" height=\"191\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17-300x191.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17.png 385w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Se adicionarmos 3 cm ao comprimento do raio de um c\u00edrculo, obtemos outro cuja \u00e1rea \u00e9 o qu\u00e1druplo da \u00e1rea do primeiro.<\/p>\n<p>Calcula o comprimento do raio do primeiro c\u00edrculo.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_14499' onClick='GTTabs_show(1,14499)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_14499'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"14503\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=14503\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17.png\" data-orig-size=\"385,245\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"C\u00edrculos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17.png\" class=\"alignright size-medium wp-image-14503\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17-300x191.png\" alt=\"\" width=\"300\" height=\"191\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17-300x191.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17.png 385w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Seja\u00a0\\(r\\) o comprimento, em cm, do raio do primeiro c\u00edrculo.<\/p>\n<p>As \u00e1reas dos dois c\u00edrculos podem ser expressas, em fun\u00e7\u00e3o de <em>r<\/em>, por:\u00a0\\({A_{C1}} = \\pi {r^2}\\) e\u00a0\\({A_{C2}} = \\pi {\\left( {r + 3} \\right)^2}\\).<\/p>\n<p>Tendo em conta que a \u00e1rea do segundo c\u00edrculo \u00e9 o qu\u00e1druplo da \u00e1rea do primeiro, vem:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{{A_{C2}} = 4 \\times {A_{C1}}}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{\\pi {{\\left( {r + 3} \\right)}^2} = 4 \\times \\pi {r^2}}&amp; \\wedge &amp;{r &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{{{\\left( {r + 3} \\right)}^2} = 4 \\times {r^2}}&amp; \\wedge &amp;{r &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{{r^2} + 6r + 9 = 4{r^2}}&amp; \\wedge &amp;{r &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{3{r^2} &#8211; 6r &#8211; 9 = 0}&amp; \\wedge &amp;{r &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{r = \\frac{{6 \\mp \\sqrt {36 + 108} }}{6}}&amp; \\wedge &amp;{r &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{\\left( {\\begin{array}{*{20}{c}}{r = \\frac{{6 &#8211; 12}}{6}}&amp; \\vee &amp;{r = \\frac{{6 + 12}}{6}}\\end{array}} \\right)}&amp; \\wedge &amp;{r &gt; 0}\\end{array}}\\\\{}&amp; \\Leftrightarrow &amp;{r = 3}\\end{array}\\]<\/p>\n<p>Portanto, o primeiro c\u00edrculo tem 3 cm de raio.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_14499' onClick='GTTabs_show(0,14499)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Se adicionarmos 3 cm ao comprimento do raio de um c\u00edrculo, obtemos outro cuja \u00e1rea \u00e9 o qu\u00e1druplo da \u00e1rea do primeiro. Calcula o comprimento do raio do primeiro c\u00edrculo. Resolu\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14504,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,303],"tags":[426,108,306],"series":[],"class_list":["post-14499","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-equacoes-do-2-o-grau","tag-9-o-ano","tag-area","tag-equacao-do-2-o-grau"],"views":2158,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/04\/9V2Pag89-17a.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14499","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=14499"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/14499\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14504"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14499"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14499"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14499"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=14499"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}