ࡱ> hjefgmShbjbj̸1 HHH\\\8xD\J4***[I]I]I]I]I]I]I$K`NdI9H*z)N***I I///*8[I/*[I//EJ0H;7\+FGII0JF\NV-BN0HNHH4**/*****II.***J****N*********  :EMBED PBrushCOLGIO PEDRO II UNIDADE ESCOLAR SO CRISTVO III PRIMEIRA ETAPA LETIVA / 2010 REGULAR COORDENADORA: MARIA HELENA M. M. BACCAR PROFESSOR: WALTER TADEU DATA: ____________NOTA:NOME: GABARITO N: ______ TURMA: _______ TESTE DE MATEMTICA II 1 SRIE (Vale 1,5 pontos) 1) (UERJ) Um atleta faz seu treinamento de corrida em uma pista circular que tem 400 metros de dimetro. Nessa pista, h seis cones de marcao indicados pelas letras A, B, C, D, E e F, que dividem a circunferncia em seis arcos, cada um medindo 60 graus. O atleta partiu do ponto correspondente ao cone A em direo a cada um dos outros cones, sempre correndo em linha reta e retornando ao cone A. Assim, seu percurso correspondeu a ABACADAEAFA. Considerando  EMBED Equation.3 , o total de metros percorridos pelo atleta nesse treino foi igual a: (A) 1480 (B) 2960 (C) 3080 (D) 3120 Soluo. Os tringulos assinalados so retngulos, pois um dos lados coincide com o dimetro da circunferncia. Logo, os ngulos so inscritos e medem a metade do ngulo central de 180. O ngulo  EMBED Equation.3  mede 60, pois inscrito e determina dois arcos que somam 120. O ngulo  EMBED Equation.3  mede 30, pois inscrito e determina um arco de 60. As distncias pedidas na parte superior da circunferncia so equivalentes s da parte inferior e podem ser calculadas aplicando a razo trigonomtrica envolvendo o cosseno. i)  EMBED Equation.3  ii)  EMBED Equation.3  Cada percurso deste foi feito duas vezes (ida e volta). Na parte superior temos: 2(200 + 340) = 1080m. O dimetro foi percorrido duas vezes: 2(400) = 800m. A parte inferior foi percorrida na mesma distncia que a superior: 1080m. Logo o total de metros ser: (1080m) + 800m + 1080m = 2960m. 2) Paulo observa da janela do seu quarto (A) uma banca de revistas (R), bem em frente ao seu prdio, segundo um ngulo de 60 com a vertical. Desejando avaliar a distncia do prdio banca, Paulo sobe seis andares (aproximadamente 16 metros) at o apartamento de seu amigo e passa a avistar a banca (do ponto B) segundo um ngulo de 30 com a vertical. Qual deve ser o valor da distncia d, aproximadamente? (Dados:  EMBED Equation.3   EMBED Equation.3 ) Soluo. Considere a altura do solo ao quarto A como x. A altura do quarto B em relao ao solo ser (16 + x). Aplicando a razo trigonomtrica da tangente, temos:  EMBED Equation.3  3) (ENEM) Considere um ponto P em uma circunferncia de raio r no plano cartesiano. Seja Q a projeo ortogonal de P sobre o eixo X, como mostra a figura, e suponha que o ponto P percorra, no sentido anti-horrio, uma distncia  EMBED Equation.3 sobre a circunferncia. Ento o ponto Q percorrer, no eixo X, uma distncia dada por: a)  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  d)  EMBED Equation.3  e)  EMBED Equation.3  Soluo. A figura mostra que o ponto P se desloca at P e sua projeo Q para Q. A distncia d percorre um arco de comprimento d = r.a, onde a o ngulo central em radianos. A distncia no eixo X, pedida,  EMBED Equation.3 . No tringulo hachurado x o cateto adjacente ao ngulo a de hipotenusa r. Aplicando a razo trigonomtrica do cosseno, temos:  EMBED Equation.3  4) Um botnico interessado em descobrir qual o comprimento da copa de uma rvore fez as observaes indicadas na figura a partir de ponto no solo. Calcule o comprimento (H), em metros, dessa copa. Use  EMBED Equation.3  Soluo. Considere a altura do solo at o incio da copa da rvore como x. A altura total da rvore em relao ao solo ser (x + H). Aplicando a razo trigonomtrica da tangente, temos:   EMBED Equation.3  5) Dois homens, H1 e H2, com 2m e 1,5m de altura, respectivamente, esto em p numa calada, em lados opostos de um poste de 5m de comprimento, iluminados por uma lmpada desse poste, como mostra a figura. Determine a distncia (em metros) entre os homens. (Use  EMBED Equation.3 ) Soluo. A figura original pode ser representada pela que mostra tringulos semelhantes. A distncia entre os homens o valor de (z + x). i) clculo de x:  EMBED Equation.3  ii) clculo de y:  EMBED Equation.3  iii) clculo de z:  EMBED Equation.3  A distncia entre os homens de 7m + 5,1m = 12,1m. 6) No tringulo retngulo ABC representado na figura seguinte, tem-se que AB =  EMBED Equation.3 cm, AD = y e CD = x. Nessas condies, responda:  Qual o valor de y? Soluo. Aplicando a razo trigonomtrica da tangente, temos:  EMBED Equation.3  Qual o valor de x? Soluo. Aplicando novamente a razo trigonomtrica da tangente, temos:  EMBED Equation.3  7) (PUC) Na figura  EMBED Equation.3 ,  EMBED Equation.3  e o comprimento do arco  EMBED Equation.3  3. Qual a medida do arco CD? a) 1,33 b) 4,50 c) 5,25 d) 6,50 e) 7,25 Soluo. O comprimento de um arco na circunferncia calculado pelo produto do ngulo central, em radianos, correspondente pelo raio desta circunferncia. Se o arco AB tem comprimento igual a 3 ento,  EMBED Equation.3 . O arco CD ainda determinado pelo ngulo central de 1,5rad. Logo,  EMBED Equation.3  8) O professor de artes em uma aula deu um desenho de uma moeda de R$1,00 para as crianas desenharem. Um dos alunos desenhou e pintou uma parte mostrada na figura. Considere OA = 10cm, OB = 8cm e o ngulo AOB = 30. Calcule o permetro da parte pintada. (Use ? = 3). Soluo. O permetro pedido a soma dos comprimentos calculados: i) (segmento AC):  EMBED Equation.3  ii) (arco AD):  EMBED Equation.3  iii) (segmento BD): mesmo comprimento de AC. Logo, BD = 2cm. iv) (arco BC):  EMBED Equation.3  O permetro ser: 2 + 5 + 2 + 4 = 13cm. 9) (FaapSP) Dois ciclistas percorrem, no mesmo sentido, uma pista circular de 50 metros de dimetro. A cada volta, o primeiro percorre 2,5m a mais do que o segundo. Supondo que mantenham o mesmo ritmo, o primeiro ciclista ter percorrido 1 radiano a mais do que o segundo aps quantas voltas? Soluo. Pela definio de radiano, quando o ngulo central medir 1 radiano, o comprimento do arco compreendido por este ngulo ser igual medida do raio. O raio mede 25m, pois o dimetro vale 50m. Como a cada volta o 1 percorre 2,5m a mais que o segundo, ele precisar de 25m 2,5m = 10 voltas para que a diferena entre eles seja de 25m. Exatamente o comprimento de arco determinado por um ngulo central de 1 radiano. Logo, a situao ocorrer aps 10 voltas. 10) (UERJ) Observe a bicicleta e tabela trigonomtrica. Os centros das rodas esto a uma distncia  EMBED Equation.3  igual a 120cm e os raios  EMBED Equation.3  e  EMBED Equation.3  medem respectivamente 25cm e 52cm. De acordo com a tabela, qual o valor do ngulo  EMBED Equation.3 ? Soluo. A figura pode ser representada pelo esquema mostrado onde o ngulo a pedido est oposto aos catetos PA e QB. Aplicando a relao trigonomtrica do seno, temos:  EMBED Equation.3  No tringulo OPA, temos:  EMBED Equation.3 . 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