{"id":6546,"date":"2011-03-01T02:59:40","date_gmt":"2011-03-01T02:59:40","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6546"},"modified":"2022-01-23T15:45:47","modified_gmt":"2022-01-23T15:45:47","slug":"um-estudo-sobre-audiencias-televisivas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6546","title":{"rendered":"Um estudo sobre audi\u00eancias televisivas"},"content":{"rendered":"<p><ul id='GTTabs_ul_6546' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6546' class='GTTabs_curr'><a  id=\"6546_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6546' ><a  id=\"6546_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_6546'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Um estudo sobre audi\u00eancias televisivas concluiu que, durante os 90 minutos da transmiss\u00e3o do jogo Fran\u00e7a-Portugal, do Campeonato da Europa de Futebol, em 2000, a varia\u00e7\u00e3o do n\u00famero de telespectadores, no nosso pa\u00eds, foi modelada, aproximadamente, pela fun\u00e7\u00e3o definida por:<br \/>\n\\[E(t)=-0,04t+10-\\frac{49}{t+10}\\]<br \/>\nEm que $E$ representa o n\u00famero de telespectadores (em milh\u00f5es) e $t$ o tempo (em minutos).<\/p>\n<ol>\n<li>Qual o n\u00famero de pessoas que assistiu ao fim da transmiss\u00e3o?<\/li>\n<li>Calcule as taxas m\u00e9dias de varia\u00e7\u00e3o do n\u00famero de telespectadores durante os primeiros cinco minutos e entre os 10 e os 15 minutos de jogo e interprete, no contexto da situa\u00e7\u00e3o descrita, os valores que encontrou.<\/li>\n<li>Indique, recorrendo \u00e0 sua calculadora, o momento da transmiss\u00e3o que registou a audi\u00eancia m\u00e1xima. Quantas pessoas assistiam, nesse momento, \u00e0 transmiss\u00e3o?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6546' onClick='GTTabs_show(1,6546)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6546'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $E(90)=-0,04\\times 90+10-\\frac{49}{90+10}=-3,6+10-0,49=5,91$.<br \/>\nPortanto, assistiram ao fim da transmiss\u00e3o 5,91 milh\u00f5es de pessoas.<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\ntm{{v}_{[0,5]}} &amp; = &amp; \\frac{E(5)-E(0)}{5-0}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\left( -0,04\\times 5+10-\\frac{49}{5+10} \\right)-\\left( 10-\\frac{49}{10} \\right)}{5}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -0,04-\\frac{49}{75}+\\frac{49}{50}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -\\frac{6}{150}-\\frac{98}{150}+\\frac{147}{150}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{43}{150}\u00a0 \\\\<br \/>\n{} &amp; \\approx\u00a0 &amp; 0,287\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Nos primeiros cinco minutos de jogo, o n\u00famero de telespectadores aumenta cerca de 287 mil por minuto.<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\ntm{{v}_{[10,15]}} &amp; = &amp; \\frac{E(15)-E(10)}{15-10}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\left( -0,04\\times 15+10-\\frac{49}{15+10} \\right)-\\left( -0,04\\times 10+10-\\frac{49}{10+10} \\right)}{5}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -0,04-\\frac{49}{125}+\\frac{49}{100}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -\\frac{20}{500}-\\frac{196}{500}+\\frac{245}{150}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{29}{500}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,058\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Nos cinco minutos entre o 10.\u00ba e 15.\u00ba\u00a0de jogo, o n\u00famero de telespectadores aumenta cerca de\u00a058 mil por minuto.<br \/>\n\u00ad<\/p>\n<\/li>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6547\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6547\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-1.jpg\" data-orig-size=\"264,136\" 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=\"J1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-1.jpg\" class=\"alignnone size-full wp-image-6547\" title=\"J1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-1.jpg\" alt=\"\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-1.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-1-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a>\u00a0 <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6548\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6548\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-2.jpg\" data-orig-size=\"264,136\" 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=\"G1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-2.jpg\" class=\"alignnone size-full wp-image-6548\" title=\"G1\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-2.jpg\" alt=\"\" width=\"264\" height=\"136\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-2.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11-pag193-31-2-150x77.jpg 150w\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" \/><\/a>\n<p>O n\u00famero m\u00e1ximo de telespectadores foi de 7,6 milh\u00f5es aos 25 minutos de jogo.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6546' onClick='GTTabs_show(0,6546)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um estudo sobre audi\u00eancias televisivas concluiu que, durante os 90 minutos da transmiss\u00e3o do jogo Fran\u00e7a-Portugal, do Campeonato da Europa de Futebol, em 2000, a varia\u00e7\u00e3o do n\u00famero de telespectadores, no&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20916,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,134],"tags":[135],"series":[],"class_list":["post-6546","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-derivadas","tag-taxa-media-de-variacao"],"views":2418,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11V2Pag193-31_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6546","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=6546"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6546\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20916"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6546"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6546"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6546"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6546"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}