{"id":11673,"date":"2013-02-21T23:21:30","date_gmt":"2013-02-21T23:21:30","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11673"},"modified":"2021-12-26T18:32:59","modified_gmt":"2021-12-26T18:32:59","slug":"duas-variaveis-p-e-q","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11673","title":{"rendered":"Duas vari\u00e1veis $p$ e $q$"},"content":{"rendered":"<p><ul id='GTTabs_ul_11673' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11673' class='GTTabs_curr'><a  id=\"11673_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11673' ><a  id=\"11673_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_11673'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Na tabela seguinte, encontra valores correspondentes das vari\u00e1veis $p$ e $q$.<\/p>\n<table class=\"aligncenter\" style=\"width: 30%;\" border=\"2\" cellspacing=\"4\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$p$<\/td>\n<td style=\"text-align: center;\">$1$<\/td>\n<td style=\"text-align: center;\">$2$<\/td>\n<td style=\"text-align: center;\">$3$<\/td>\n<td style=\"text-align: center;\">$4$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$q$<\/td>\n<td style=\"text-align: center;\">$950$<\/td>\n<td style=\"text-align: center;\">$900$<\/td>\n<td style=\"text-align: center;\">$850$<\/td>\n<td style=\"text-align: center;\">$800$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Determine uma express\u00e3o de $q$ como fun\u00e7\u00e3o afim de $p$.<\/li>\n<li>Determine uma express\u00e3o de $p$ como fun\u00e7\u00e3o afim de $q$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11673' onClick='GTTabs_show(1,11673)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11673'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<table class=\"aligncenter\" style=\"width: 30%;\" border=\"2\" cellspacing=\"4\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$p$<\/td>\n<td style=\"text-align: center;\">$1$<\/td>\n<td style=\"text-align: center;\">$2$<\/td>\n<td style=\"text-align: center;\">$3$<\/td>\n<td style=\"text-align: center;\">$4$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$q$<\/td>\n<td style=\"text-align: center;\">$950$<\/td>\n<td style=\"text-align: center;\">$900$<\/td>\n<td style=\"text-align: center;\">$850$<\/td>\n<td style=\"text-align: center;\">$800$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Nesta situa\u00e7\u00e3o, $p$ \u00e9 a vari\u00e1vel independente e $q$ \u00e9 a vari\u00e1vel dependente, sendo, por isso, a express\u00e3o pretendida da forma $q = mp + b$.\n<p>Consideremos dois pontos do gr\u00e1fico dessa fun\u00e7\u00e3o: $A\\left( {1,950} \\right)$ e $B\\left( {2,900} \\right)$.<\/p>\n<p>Ora, $m = \\frac{{950 &#8211; 900}}{{1 &#8211; 2}} =\u00a0 &#8211; 50$, pelo que vem $q =\u00a0 &#8211; 50p + b$.<\/p>\n<p>Como as coordenadas de $A$ t\u00eam de verificar esta equa\u00e7\u00e3o, temos: $950 =\u00a0 &#8211; 50 \\times 1 + b \\Leftrightarrow b = 1000$.<\/p>\n<p>Portanto, $q =\u00a0 &#8211; 50p + 1000$.<\/p>\n<\/li>\n<li>Podemos obter a express\u00e3o usando um procedimento an\u00e1logo ao da al\u00ednea anterior.<br \/>\nEm alternativa, basta resolver a equa\u00e7\u00e3o anterior em ordem \u00e0 vari\u00e1vel $p$:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{q =\u00a0 &#8211; 50p + 1000}&amp; \\Leftrightarrow &amp;{50p =\u00a0 &#8211; q + 1000} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{p =\u00a0 &#8211; \\frac{1}{{50}}q + 20}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11673' onClick='GTTabs_show(0,11673)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na tabela seguinte, encontra valores correspondentes das vari\u00e1veis $p$ e $q$. $p$ $1$ $2$ $3$ $4$ $q$ $950$ $900$ $850$ $800$ Determine uma express\u00e3o de $q$ como fun\u00e7\u00e3o afim de $p$.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19353,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,343],"tags":[429,346,345,347],"series":[],"class_list":["post-11673","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-funcoes-e-graficos","tag-10-o-ano","tag-declive","tag-funcao-afim","tag-ordenada-na-origem"],"views":2408,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/Duas_variaveis_p_q_5.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11673","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=11673"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11673\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19353"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11673"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11673"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11673"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}