{"id":11779,"date":"2014-02-08T00:47:35","date_gmt":"2014-02-08T00:47:35","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11779"},"modified":"2022-01-22T02:06:00","modified_gmt":"2022-01-22T02:06:00","slug":"concentracao-do-composto","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11779","title":{"rendered":"Concentra\u00e7\u00e3o do composto"},"content":{"rendered":"<p><ul id='GTTabs_ul_11779' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11779' class='GTTabs_curr'><a  id=\"11779_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11779' ><a  id=\"11779_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_11779'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Juntou-se \u00e1cido puro a $30$ gramas de uma subst\u00e2ncia $30$% \u00e1cida.<\/p>\n<p>Seja $x$ o n\u00famero de gramas de \u00e1cido puro adicionado.<\/p>\n<ol>\n<li>Determine uma express\u00e3o que represente a concentra\u00e7\u00e3o do composto formado.<\/li>\n<li>Represente graficamente a fun\u00e7\u00e3o da al\u00ednea anterior.<\/li>\n<li>Entre que valores varia a fun\u00e7\u00e3o?<\/li>\n<li>Qual a quantidade de \u00e1cido puro que devemos adicionar para produzir uma solu\u00e7\u00e3o $75$% \u00e1cida?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11779' onClick='GTTabs_show(1,11779)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11779'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Massa de \u00e1cido (em gramas): $x + 30\\%\u00a0 \\times 30 = x + 9$.\n<p>Massa do composto (em gramas): $x + 30$.<\/p>\n<p>Portanto, a concentra\u00e7\u00e3o do composto formado pode ser expressa por: \\[C\\left( x \\right) = \\frac{{x + 9}}{{x + 30}}\\]<\/p>\n<\/li>\n<li>Se considerarmos que n\u00e3o h\u00e1 qualquer limita\u00e7\u00e3o quanto \u00e0 quantidade de \u00e1cido poss\u00edvel de adicionar, o dom\u00ednio da fun\u00e7\u00e3o \u00e9 $\\mathbb{R}_0^ + $.\n<p>Por outro lado, como \\[C\\left( x \\right) = \\frac{{x + 9}}{{x + 30}} = \\frac{{x + 30 &#8211; 21}}{{x + 30}} = 1 + \\frac{{ &#8211; 21}}{{x + 30}}\\] o gr\u00e1fico da fun\u00e7\u00e3o admite uma ass\u00edntota horizontal de equa\u00e7\u00e3o $y = 1$, quando $x \\to\u00a0 + \\infty $.<\/p>\n<p>A fun\u00e7\u00e3o est\u00e1 representada graficamente na figura abaixo.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag-52-1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11780\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11780\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag-52-1.png\" data-orig-size=\"772,417\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag-52-1.png\" class=\"aligncenter wp-image-11780 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag-52-1.png\" alt=\"Gr\u00e1fico\" width=\"772\" height=\"417\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag-52-1.png 772w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag-52-1-300x162.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag-52-1-150x81.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2pag-52-1-400x216.png 400w\" sizes=\"auto, (max-width: 772px) 100vw, 772px\" \/><\/a><\/p>\n<\/li>\n<li>$C\\left( 0 \\right) = \\frac{{0 + 9}}{{0 + 30}} = \\frac{3}{{10}}$.\n<p>Por outro lado, como a fun\u00e7\u00e3o \u00e9 cont\u00ednua, estritamente crescente e \\[\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } C\\left( x \\right) = {1^ &#8211; }\\] ent\u00e3o o contradom\u00ednio da fun\u00e7\u00e3o \u00e9 $\\left[ {\\frac{3}{{10}},1} \\right[$.<\/p>\n<p>Logo, a fun\u00e7\u00e3o assume todos os valores do intervalo $\\left[ {\\frac{3}{{10}},1} \\right[$.<\/p>\n<\/li>\n<li>Tem-se sucessivamente:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{C\\left( x \\right) = 0,75}&amp; \\Leftrightarrow &amp;{\\frac{{x + 9}}{{\\mathop {x + 30}\\limits_{\\left( 4 \\right)} }} &#8211; \\mathop {\\frac{3}{4}}\\limits_{\\left( {x + 30} \\right)}\u00a0 = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\frac{{4x + 36 &#8211; 3x &#8211; 90}}{{4x + 120}} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\frac{{x &#8211; 54}}{{4x + 120}} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x &#8211; 54 = 0}&amp; \\wedge &amp;{4x + 120 \\ne 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = 54}<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, para produzir uma solu\u00e7\u00e3o $75$% \u00e1cida, devemos adicionar $54$ gramas de \u00e1cido puro.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11779' onClick='GTTabs_show(0,11779)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Juntou-se \u00e1cido puro a $30$ gramas de uma subst\u00e2ncia $30$% \u00e1cida. Seja $x$ o n\u00famero de gramas de \u00e1cido puro adicionado. Determine uma express\u00e3o que represente a concentra\u00e7\u00e3o do composto formado.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20877,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,125],"tags":[422,160,126],"series":[],"class_list":["post-11779","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-racionais","tag-11-o-ano","tag-equacao","tag-funcao-racional"],"views":2688,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11V2Pag053-1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11779","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=11779"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11779\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20877"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11779"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11779"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11779"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11779"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}