{"id":36347,"date":"2019-12-16T16:41:55","date_gmt":"2019-12-16T19:41:55","guid":{"rendered":"https:\/\/blog.estrategiavestibulares.com.br\/?p=36347"},"modified":"2021-03-11T09:57:35","modified_gmt":"2021-03-11T12:57:35","slug":"prova-2a-fase-unesp-2020-matematica","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/prova-2a-fase-unesp-2020-matematica\/","title":{"rendered":"Resolu\u00e7\u00e3o comentada da prova de Matem\u00e1tica da 2\u00aa Fase UNESP"},"content":{"rendered":"<p>Ol&aacute;, pessoal&hellip; Tudo bem? Sou o prof. Mar&ccedil;al Ferreira, professor de Matem&aacute;tica do Estrat&eacute;gia Vestibulares. Escrevo este artigo para resolver as quest&otilde;es da prova da 2&ordf; Fase do vestibular da UNESP 2020. Nesta p&aacute;gina, voc&ecirc; vai conferir a resolu&ccedil;&atilde;o completa e ainda vai poder baixar gratuitamente os coment&aacute;rios em PDF.<h2 class=\"wp-block-heading\">UNESP 2&ordf; Fase &ndash; Matem&aacute;tica<\/h2><h3 class=\"wp-block-heading\">Quest&atilde;o 22<\/h3><p>Um grupo de cientistas estuda os h&aacute;bitos de uma esp&eacute;cie animal em uma &aacute;rea de preserva&ccedil;&atilde;o. Inicialmente, delimitou-se uma &aacute;rea plana (ABCD, figura 1), na qual dever&atilde;o ser estabelecidos dois pontos de observa&ccedil;&atilde;o. Afigura 2 apresenta um modelo matem&aacute;tico da &aacute;rea delimitada, com dois setores retangulares nos quais ser&atilde;o estabelecidos os pontos de observa&ccedil;&atilde;o, sendo que cada ponto de observa&ccedil;&atilde;o dever&aacute; pertencer a apenas um dos setores. Parte do grupo de cientistas ocupar-se-&aacute; exclusivamente com os h&aacute;bitos de reprodu&ccedil;&atilde;o dessa esp&eacute;cie e atuar&aacute; na regi&atilde;o em forma de paralelogramo, indicada na figura 3. <\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/12\/image107.png\" alt=\"\" class=\"wp-image-36353\"><\/figure><p>a) Para a constru&ccedil;&atilde;o dos dois pontos de observa&ccedil;&atilde;o, considere que a localiza&ccedil;&atilde;o do ponto do setor I dever&aacute; ser equidistante dos pontos A e B e que a localiza&ccedil;&atilde;o do ponto do setor II dever&aacute; ser equidistante dos pontos B e C. Utilizando as coordenadas do plano cartesiano da figura 2, determine uma poss&iacute;vel localiza&ccedil;&atilde;o do ponto de observa&ccedil;&atilde;o para cada um dos setores.<\/p><p>b) Dado que 1 unidade de dist&acirc;ncia dos planos cartesianos equivale a 200 metros de dist&acirc;ncia real, determine o per&iacute;metro da regi&atilde;o em que ser&atilde;o estudados os h&aacute;bitos de reprodu&ccedil;&atilde;o da esp&eacute;cie (figura 3).<\/p><h3 class=\"wp-block-heading\">Resolu&ccedil;&atilde;o<\/h3><p>a) Para a constru&ccedil;&atilde;o dos dois pontos de observa&ccedil;&atilde;o, considere que a localiza&ccedil;&atilde;o do ponto do setor I dever&aacute; ser equidistante dos pontos A e B e que a localiza&ccedil;&atilde;o do ponto do setor II dever&aacute; ser equidistante dos pontos B e C. Utilizando as coordenadas do plano cartesiano da figura 2, determine uma poss&iacute;vel localiza&ccedil;&atilde;o do ponto de observa&ccedil;&atilde;o para cada um dos setores.<\/p><p>Os pontos equidistantes de A e de B s&atilde;o dados pela reta y=4. Como, pelo enunciado, o setor I tem valores de x tais que 1&lt;x&lt;3, uma poss&iacute;vel localiza&ccedil;&atilde;o do ponto de observa&ccedil;&atilde;o &eacute; (2;4).<\/p><p>Os pontos equidistantes de B e de C s&atilde;o dados pela reta x=7,5. Como, pelo enunciado, o setor II tem valores de y tais que 5&lt;y&lt;7, uma poss&iacute;vel localiza&ccedil;&atilde;o do ponto de observa&ccedil;&atilde;o &eacute; (7,5;6).<\/p><p>b) Dado que 1 unidade de dist&acirc;ncia dos planos cartesianos equivale a 200 metros de dist&acirc;ncia real, determine o per&iacute;metro da regi&atilde;o em que ser&atilde;o estudados os h&aacute;bitos de reprodu&ccedil;&atilde;o da esp&eacute;cie (figura 3).<\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/12\/image108.png\" alt=\"\" class=\"wp-image-36356\"><\/figure><p>As coordenadas dos pontos D,E,F,G s&atilde;o:<\/p><p>D = (9; 0)<\/p><p>G = (13,5; 6)<\/p><p>F = (9; 6)<\/p><p>E = (4,5 ; 0)<\/p><p>Chamando x a dist&acirc;ncia DG, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5E2=6%5E2+%7B4,5%7D%5E2\" alt=\"\\large x^2=6^2+{4,5}^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5E2=36+20,25\" alt=\"\\large x^2=36+20,25\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5E2=56,25\" alt=\"\\large x^2=56,25\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=7,5\" alt=\"\\large x=7,5\" align=\"absmiddle\"><\/p><p>Dessa forma, o per&iacute;metro P, em metros, &eacute; dado por:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cleft(2x+2%5Ccdot4,5%5Cright)%5Ccdot200\" alt=\"\\large P=\\left(2x+2\\cdot4,5\\right)\\cdot200\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cleft(2%5Ccdot7,5+2%5Ccdot4,5%5Cright)%5Ccdot200\" alt=\"\\large P=\\left(2\\cdot7,5+2\\cdot4,5\\right)\\cdot200\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cleft(15+9%5Cright)%5Ccdot200\" alt=\"\\large P=\\left(15+9\\right)\\cdot200\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=24%5Ccdot200\" alt=\"\\large P=24\\cdot200\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=4800%5C&amp;space;m\" alt=\"\\large P=4800\\ m\" align=\"absmiddle\"><\/p><h3 class=\"wp-block-heading\">Quest&atilde;o 23<\/h3><p>A modelagem dos sistemas de cor &eacute; essencial na computa&ccedil;&atilde;o gr&aacute;fica, e um dos maiores desafios dessa &aacute;rea &eacute; a convers&atilde;o de coordenadas de diferentes sistemas. O sistema RGB pressup&otilde;e que o sistema de processamento de cor do olho humano seja baseado nas faixas vermelha (red). verde (green) e azul (blue) do espectro vis&iacute;vel J&aacute; o modelo CMY usa cores complementares, ciano (cyan), magenta (magenta) e amarelo (yellow), e foi importante no desenvolvimento de impressoras As cores no sistema CMY ficam delimitadas por um cubo, o cubo CMY, conforme ilustrado<\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/12\/image109.png\" alt=\"\" class=\"wp-image-36360\"><\/figure><p>a) A transforma&ccedil;&atilde;o de uma cor no sistema RGB, descrita por (r, g, b), para o sistema CMY, descrita por (c, m, y), &eacute; dada por <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7Dc%5C%5Cm%5C%5Cy%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D1%5C%5C1%5C%5C1%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D-%5Cleft%5B%5Cbegin%7Bmatrix%7Dr%5C%5Cg%5C%5Cb%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}c\\\\m\\\\y\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}1\\\\1\\\\1\\\\\\end{matrix}\\right]-\\left[\\begin{matrix}r\\\\g\\\\b\\\\\\end{matrix}\\right]\" align=\"absmiddle\">. Supondo que uma cor no sistema RGB seja descrita por <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft(%5Cfrac%7B1%7D%7B4%7D,%5Cfrac%7B1%7D%7B100%7D,0%5Cright)\" alt=\"\\large \\left(\\frac{1}{4},\\frac{1}{100},0\\right)\" align=\"absmiddle\">, apresente as coordenadas dessa cor o sistema CMY e indique qual das oito cores detalhadas no cubo CMY est&aacute; mais pr&oacute;xima dela.<\/p><p class=\"Questes-enunciado\">b) O sistema NTSC (National Television Standards Committee), utilizado em emiss&otilde;es para a televis&atilde;o, baseia-se na separa&ccedil;&atilde;o dos sinais de cor RGB em um sinal de luminosidade e dois sinais de cromaticidade Assim como no espa&ccedil;o RGB, as cores no espa&ccedil;o YIQ, utilizado no sistema NTSC, s&atilde;o descritas por coordenadas, sendo representadas por (y, i, q). A rela&ccedil;&atilde;o entre as cores desses dois sistemas &eacute; dada, de modo simplificado, pela express&atilde;o matricial:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7Dy%5C%5Ci%5C%5Cq%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D0,3&amp;2%5Cbeta&amp;%5Cgamma%5C%5C3%5Calpha&amp;-%5Cbeta&amp;-0,3%5C%5C%5Calpha&amp;-0,5&amp;-3%5Cgamma%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D%5Ccdot%5Cleft%5B%5Cbegin%7Bmatrix%7Dr%5C%5Cg%5C%5Cb%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}y\\\\i\\\\q\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}0,3&amp;2\\beta&amp;\\gamma\\\\3\\alpha&amp;-\\beta&amp;-0,3\\\\\\alpha&amp;-0,5&amp;-3\\gamma\\\\\\end{matrix}\\right]\\cdot\\left[\\begin{matrix}r\\\\g\\\\b\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p>Sabendo que uma cor no sistema RGB descrita por (0,2; 0,5, 0,4) est&aacute; associada a uma cor no sistema YIQ descrita por (0,4, -0,15; -0,33), determine <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Calpha\" alt=\"\\large \\alpha\" align=\"absmiddle\">, <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cbeta\" alt=\"\\large \\beta\" align=\"absmiddle\"> e <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cgamma\" alt=\"\\large \\gamma\" align=\"absmiddle\">.<\/p><h3>Resolu&ccedil;&atilde;o<\/h3><p>a) Utilizando a equa&ccedil;&atilde;o matricial fornecida no enunciado, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7Dc%5C%5Cm%5C%5Cy%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D1%5C%5C1%5C%5C1%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D-%5Cleft%5B%5Cbegin%7Bmatrix%7Dr%5C%5Cg%5C%5Cb%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}c\\\\m\\\\y\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}1\\\\1\\\\1\\\\\\end{matrix}\\right]-\\left[\\begin{matrix}r\\\\g\\\\b\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7Dc%5C%5Cm%5C%5Cy%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D1%5C%5C1%5C%5C1%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D-%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Cfrac%7B1%7D%7B4%7D%5C%5C%5Cfrac%7B1%7D%7B100%7D%5C%5C0%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}c\\\\m\\\\y\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}1\\\\1\\\\1\\\\\\end{matrix}\\right]-\\left[\\begin{matrix}\\frac{1}{4}\\\\\\frac{1}{100}\\\\0\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7Dc%5C%5Cm%5C%5Cy%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Cfrac%7B3%7D%7B4%7D%5C%5C%5Cfrac%7B99%7D%7B100%7D%5C%5C0%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}c\\\\m\\\\y\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}\\frac{3}{4}\\\\\\frac{99}{100}\\\\0\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7Dc%5C%5Cm%5C%5Cy%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D0,75%5C%5C0,99%5C%5C0%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D%5Crightarrow%5Cleft%5B%5Cbegin%7Bmatrix%7D1%5C%5C1%5C%5C1%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=Preto\" alt=\"\\large \\left[\\begin{matrix}c\\\\m\\\\y\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}0,75\\\\0,99\\\\0\\\\\\end{matrix}\\right]\\rightarrow\\left[\\begin{matrix}1\\\\1\\\\1\\\\\\end{matrix}\\right]=Preto\" align=\"absmiddle\"><\/p><p>Assim, as coordenadas dessa cor no sistema CMY s&atilde;o <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7D0,75%5C%5C0,99%5C%5C0%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}0,75\\\\0,99\\\\0\\\\\\end{matrix}\\right]\" align=\"absmiddle\"> e a cor no cubo que est&aacute; mais pr&oacute;xima dela &eacute; o preto.<\/p><p>b) Utilizando a equa&ccedil;&atilde;o matricial fornecida, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7Dy%5C%5Ci%5C%5Cq%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D0,3&amp;2%5Cbeta&amp;%5Cgamma%5C%5C3%5Calpha&amp;-%5Cbeta&amp;-0,3%5C%5C%5Calpha&amp;-0,5&amp;-3%5Cgamma%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D%5Ccdot%5Cleft%5B%5Cbegin%7Bmatrix%7Dr%5C%5Cg%5C%5Cb%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}y\\\\i\\\\q\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}0,3&amp;2\\beta&amp;\\gamma\\\\3\\alpha&amp;-\\beta&amp;-0,3\\\\\\alpha&amp;-0,5&amp;-3\\gamma\\\\\\end{matrix}\\right]\\cdot\\left[\\begin{matrix}r\\\\g\\\\b\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7D0,4%5C%5C-0,15%5C%5C-0,33%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D0,3&amp;2%5Cbeta&amp;%5Cgamma%5C%5C3%5Calpha&amp;-%5Cbeta&amp;-0,3%5C%5C%5Calpha&amp;-0,5&amp;-3%5Cgamma%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D%5Ccdot%5Cleft%5B%5Cbegin%7Bmatrix%7D0,2%5C%5C0,5%5C%5C0,4%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}0,4\\\\-0,15\\\\-0,33\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}0,3&amp;2\\beta&amp;\\gamma\\\\3\\alpha&amp;-\\beta&amp;-0,3\\\\\\alpha&amp;-0,5&amp;-3\\gamma\\\\\\end{matrix}\\right]\\cdot\\left[\\begin{matrix}0,2\\\\0,5\\\\0,4\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7D0,4%5C%5C-0,15%5C%5C-0,33%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D0,06+%5Cbeta+0,4%5Cgamma%5C%5C0,6%5Calpha-0,5%5Cbeta-0,12%5C%5C0,2%5Calpha-0,25-1,2%5Cgamma%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\large \\left[\\begin{matrix}0,4\\\\-0,15\\\\-0,33\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}0,06+\\beta+0,4\\gamma\\\\0,6\\alpha-0,5\\beta-0,12\\\\0,2\\alpha-0,25-1,2\\gamma\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D0,06+%5Cbeta+0,4%5Cgamma=0,4%5C%5C0,6%5Calpha-0,5%5Cbeta-0,12=-0,15%5C%5C0,2%5Calpha-0,25-1,2%5Cgamma=-0,33%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\large \\left\\{\\begin{matrix}0,06+\\beta+0,4\\gamma=0,4\\\\0,6\\alpha-0,5\\beta-0,12=-0,15\\\\0,2\\alpha-0,25-1,2\\gamma=-0,33\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta+0,4%5Cgamma=0,34%5C%5C0,6%5Calpha-0,5%5Cbeta=-0,03%5C%5C0,2%5Calpha-1,2%5Cgamma=-0,08%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\large \\left\\{\\begin{matrix}\\beta+0,4\\gamma=0,34\\\\0,6\\alpha-0,5\\beta=-0,03\\\\0,2\\alpha-1,2\\gamma=-0,08\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,34-0,4%5Cgamma%5C%5C0,6%5Calpha-0,5%5Cbeta=-0,03%5C%5C%5Calpha=%5Cfrac%7B-0,08+1,2%5Cgamma%7D%7B0,2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\large \\left\\{\\begin{matrix}\\beta=0,34-0,4\\gamma\\\\0,6\\alpha-0,5\\beta=-0,03\\\\\\alpha=\\frac{-0,08+1,2\\gamma}{0,2}\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Crightarrow%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,34-0,4%5Cgamma%5C%5C0,6%5Ccdot%5Cleft(%5Cfrac%7B-0,08+1,2%5Cgamma%7D%7B0,2%7D%5Cright)-0,5%5Ccdot%5Cleft(0,34-0,4%5Cgamma%5Cright)=-0,03%5C%5C%5Calpha=%5Cfrac%7B-0,08+1,2%5Cgamma%7D%7B0,2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\large \\rightarrow\\left\\{\\begin{matrix}\\beta=0,34-0,4\\gamma\\\\0,6\\cdot\\left(\\frac{-0,08+1,2\\gamma}{0,2}\\right)-0,5\\cdot\\left(0,34-0,4\\gamma\\right)=-0,03\\\\\\alpha=\\frac{-0,08+1,2\\gamma}{0,2}\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Crightarrow%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,34-0,4%5Cgamma%5C%5C-0,24+3,6%5Cgamma-0,17+0,2%5Cgamma=-0,03%5C%5C%5Calpha=%5Cfrac%7B-0,08+1,2%5Cgamma%7D%7B0,2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\large \\rightarrow\\left\\{\\begin{matrix}\\beta=0,34-0,4\\gamma\\\\-0,24+3,6\\gamma-0,17+0,2\\gamma=-0,03\\\\\\alpha=\\frac{-0,08+1,2\\gamma}{0,2}\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,34-0,4%5Cgamma%5C%5C3,8%5Cgamma=0,41-0,03%5C%5C%5Calpha=%5Cfrac%7B-0,08+1,2%5Cgamma%7D%7B0,2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\large \\left\\{\\begin{matrix}\\beta=0,34-0,4\\gamma\\\\3,8\\gamma=0,41-0,03\\\\\\alpha=\\frac{-0,08+1,2\\gamma}{0,2}\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,34-0,4%5Cgamma%5C%5C3,8%5Cgamma=0,38%5C%5C%5Calpha=%5Cfrac%7B-0,08+1,2%5Cgamma%7D%7B0,2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\large \\left\\{\\begin{matrix}\\beta=0,34-0,4\\gamma\\\\3,8\\gamma=0,38\\\\\\alpha=\\frac{-0,08+1,2\\gamma}{0,2}\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Crightarrow%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,34-0,4%5Cgamma%5C%5C%5Cgamma=0,1%5C%5C%5Calpha=%5Cfrac%7B-0,08+1,2%5Cgamma%7D%7B0,2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\large \\rightarrow\\left\\{\\begin{matrix}\\beta=0,34-0,4\\gamma\\\\\\gamma=0,1\\\\\\alpha=\\frac{-0,08+1,2\\gamma}{0,2}\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,34-0,4%5Ccdot0,1%5C%5C%5Cgamma=0,1%5C%5C%5Calpha=%5Cfrac%7B-0,08+1,2%5Ccdot0,1%7D%7B0,2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,3%5C%5C%5Cgamma=0,1%5C%5C%5Calpha=0,2%5C%5C%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\large \\left\\{\\begin{matrix}\\beta=0,34-0,4\\cdot0,1\\\\\\gamma=0,1\\\\\\alpha=\\frac{-0,08+1,2\\cdot0,1}{0,2}\\\\\\end{matrix}\\right.\\rightarrow\\left\\{\\begin{matrix}\\beta=0,3\\\\\\gamma=0,1\\\\\\alpha=0,2\\\\\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p>Assim, os valores de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Calpha\" alt=\"\\large \\alpha\" align=\"absmiddle\">, <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cbeta\" alt=\"\\large \\beta\" align=\"absmiddle\"> e <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cgamma\" alt=\"\\large \\gamma\" align=\"absmiddle\"> s&atilde;o tais que <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%5Cbeta=0,3%5C%5C%5Cgamma=0,1%5C%5C%5Calpha=0,2%5C%5C%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\large \\left\\{\\begin{matrix}\\beta=0,3\\\\\\gamma=0,1\\\\\\alpha=0,2\\\\\\end{matrix}\\right.\" align=\"absmiddle\">.<\/p><h3 class=\"wp-block-heading\">Quest&atilde;o 24<\/h3><p>A penicilina benzatina &eacute; um antibi&oacute;tico indicado no\ntratamento de certas infec&ccedil;&otilde;es, e sua meia-vida &eacute; de 336 horas. Ou seja, ap&oacute;s\nesse per&iacute;odo de tempo a quantidade de medicamento no sangue reduz-se pela\nmetade. O tratamento convencional &eacute; feito com uma aplica&ccedil;&atilde;o de 1200000 Ul do\nmedicamento e essa dose mant&eacute;m-se em quantidade adequada no sangue (isto &eacute;, n&atilde;o\ninferior a 300000 Ul) durante os 28 dias seguintes. A dosagem, o n&uacute;mero de\ndoses e o intervalo de tempo entre as doses depende da doen&ccedil;a a ser tratada.<\/p><p>a) Considere um paciente que recebeu 2 doses, cada\numa de 1200000 Ul, desse medicamento, sendo que a segunda dose foi aplicada 28\ndias ap&oacute;s a primeira dose. Fa&ccedil;a um esbo&ccedil;o gr&aacute;fico na malha presente no campo de\nResolu&ccedil;&atilde;o e Resposta, representando a quantidade desse medicamento no sangue ao\nlongo de 8 semanas de tratamento.<\/p><p>b) Considere outro caso, em que um paciente foi tratado com 2 doses, cada uma de 2400000 Ul, de penicilina benzatina, sendo a segunda dose aplicada 14 dias ap&oacute;s a primeira. Determine a quantidade desse medicamento no sangue do paciente, em Ul, logo ap&oacute;s ele tomar a segunda dose e indique durante quantos dias completos, ap&oacute;s essa segunda dose, a quantidade de medicamento permanecer&aacute; em quantidade adequada no sangue desse paciente. Adote em seus c&aacute;lculos log2 = 0,30; log3 = 0,48<\/p><h3 class=\"wp-block-heading\">Resolu&ccedil;&atilde;o<\/h3><p>O tempo de meia vida de &nbsp;horas &eacute; equivalente a <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B336%7D%7B24%7D\" alt=\"\\large \\frac{336}{24}\" align=\"absmiddle\"> dias, ou seja, 14 dias, ou ainda, &nbsp;semanas.<\/p><p>Dessa forma, a progress&atilde;o em UI a cada duas semanas &eacute; a seguinte:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;semana%5C&amp;space;0=1%5C&amp;space;200%5C&amp;space;000\" alt=\"\\large semana\\ 0=1\\ 200\\ 000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;semana%5C&amp;space;2=600%5C&amp;space;000\" alt=\"\\large semana\\ 2=600\\ 000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;semana%5C&amp;space;4=300%5C&amp;space;000+1%5C&amp;space;200%5C&amp;space;000=1%5C&amp;space;500%5C&amp;space;000\" alt=\"\\large semana\\ 4=300\\ 000+1\\ 200\\ 000=1\\ 500\\ 000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;semana%5C&amp;space;6=150%5C&amp;space;000+600%5C&amp;space;000=750%5C&amp;space;000\" alt=\"\\large semana\\ 6=150\\ 000+600\\ 000=750\\ 000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;semana%5C&amp;space;8=150%5C&amp;space;000+300%5C&amp;space;000=375%5C&amp;space;000\" alt=\"\\large semana\\ 8=150\\ 000+300\\ 000=375\\ 000\" align=\"absmiddle\"><\/p><p>Colocando esses valores no gr&aacute;fico, temos:<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/12\/image110.png\" alt=\"\" class=\"wp-image-36364\" width=\"448\" height=\"572\"><\/figure><\/div><p>b) Inicialmente, temos a seguinte progress&atilde;o:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;semana%5C&amp;space;0=2%5C&amp;space;400%5C&amp;space;000\" alt=\"\\large semana\\ 0=2\\ 400\\ 000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;semana%5C&amp;space;2=1%5C&amp;space;200%5C&amp;space;000+2%5C&amp;space;400%5C&amp;space;000=3%5C&amp;space;600%5C&amp;space;000\" alt=\"\\large semana\\ 2=1\\ 200\\ 000+2\\ 400\\ 000=3\\ 600\\ 000\" align=\"absmiddle\"><\/p><p>Assim, a quantidade desse medicamento no sangue do paciente, em UI, logo ap&oacute;s ele tomar a segunda dose &eacute; de 3 600 000 UI.<\/p><p>Calculando a quantidade n de per&iacute;odos de meia vida para que a concentra&ccedil;&atilde;o n&atilde;o seja superior a 300 000 UI:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3%5C&amp;space;600%5C&amp;space;000%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5En%5Cgeq300%5C&amp;space;000\" alt=\"\\large 3\\ 600\\ 000\\cdot\\left(\\frac{1}{2}\\right)^n\\geq300\\ 000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;12%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5En%5Cgeq1\" alt=\"\\large 12\\cdot\\left(\\frac{1}{2}\\right)^n\\geq1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5En%5Cgeq%5Cfrac%7B1%7D%7B12%7D\" alt=\"\\large \\left(\\frac{1}{2}\\right)^n\\geq\\frac{1}{12}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;log%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5En%5Cgeq%5Clog%7B%5Cleft(%5Cfrac%7B1%7D%7B12%7D%5Cright)%7D\" alt=\"\\large log\\left(\\frac{1}{2}\\right)^n\\geq\\log{\\left(\\frac{1}{12}\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5Ccdot&amp;space;log%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5Cgeq%5Clog%7B%5Cleft(%5Cfrac%7B1%7D%7B12%7D%5Cright)%7D\" alt=\"\\large n\\cdot log\\left(\\frac{1}{2}\\right)\\geq\\log{\\left(\\frac{1}{12}\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5Ccdot%5Cleft(log1-%5Clog%7B2%7D%5Cright)%5Cgeq%5Cleft(log1-%5Clog%7B12%7D%5Cright)\" alt=\"\\large n\\cdot\\left(log1-\\log{2}\\right)\\geq\\left(log1-\\log{12}\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5Ccdot%5Cleft(log1-%5Clog%7B2%7D%5Cright)%5Cgeq%5Cleft(log1-%5Clog%7B2%5E2%7D-%5Clog%7B3%7D%5Cright)\" alt=\"\\large n\\cdot\\left(log1-\\log{2}\\right)\\geq\\left(log1-\\log{2^2}-\\log{3}\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5Ccdot%5Cleft(0-0,3%5Cright)%5Cgeq%5Cleft(0-2%5Ccdot0,3-0,48%5Cright)\" alt=\"\\large n\\cdot\\left(0-0,3\\right)\\geq\\left(0-2\\cdot0,3-0,48\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;-0,3n%5Cgeq-0,6-0,48\" alt=\"\\large -0,3n\\geq-0,6-0,48\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5Cle%5Cfrac%7B-1,08%7D%7B-0,3%7D\" alt=\"\\large n\\le\\frac{-1,08}{-0,3}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5Cle3,6%5C&amp;space;periodos\" alt=\"\\large n\\le3,6\\ periodos\" align=\"absmiddle\"><\/p><p>Como cada per&iacute;odo de meia vida &eacute; de 14 dias, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5Cle3,6%5Ccdot14%5C&amp;space;dias\" alt=\"\\large n\\le3,6\\cdot14\\ dias\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5Cle50,4%5C&amp;space;dias\" alt=\"\\large n\\le50,4\\ dias\" align=\"absmiddle\"><\/p><p>Assim, a quantidade permanece adequada at&eacute; 50 dias ap&oacute;s o paciente tomar a segunda dose.<\/p><p>&Eacute; isso, pessoal! Espero que tenham curtido a resolu&ccedil;&atilde;o da prova de Matem&aacute;tica da prova do Vestibular UNESP 2020.&nbsp;Sigam-me nas redes sociais. T&ecirc;m muitas dicas l&aacute;. Mande uma mensagem, caso tenha tido alguma d&uacute;vida.<\/p><p>Voc&ecirc; tamb&eacute;m pode baixar essa resolu&ccedil;&atilde;o em PDF. Basta clicar no link a seguir. Abra&ccedil;os!<\/p><div class=\"wp-block-file\">\"&gt;Prova UNESP 2020 &ndash; Matem&aacute;tica &ndash; 2&ordf; Fase\" class=\"wp-block-file__button\" download&gt;Baixar<\/div><p>Abra&ccedil;os!<\/p><p><strong>Instagram:<\/strong>&nbsp;<a href=\"https:\/\/www.instagram.com\/professor.marcal\/\" target=\"_blank\">@professor.marcal<\/a><\/p><p class=\"has-text-align-center has-pale-cyan-blue-background-color has-background has-medium-font-size\"><a href=\"https:\/\/estrategiavestibulares.com.br\/vestibulares\/unesp\/\" target=\"_blank\" rel=\"noopener noreferrer\">CURSOS PARA UNESP<\/a><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"Ol&aacute;, pessoal&hellip; Tudo bem? Sou o prof. Mar&ccedil;al Ferreira, professor de Matem&aacute;tica do Estrat&eacute;gia Vestibulares. Escrevo este artigo&hellip;\n","protected":false},"author":18,"featured_media":36371,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[29],"tags":[],"wl_entity_type":[732],"class_list":{"0":"post-36347","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v25.9) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Resolu\u00e7\u00e3o comentada da prova de Matem\u00e1tica da 2\u00aa Fase UNESP<\/title>\n<meta name=\"description\" content=\"Voc\u00ea participou da 2\u00aa Fase da Unesp? 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