GUIA DO ALUNOVolume 2

Matemática3ª série do Ensino Médio

Entre Jovens 3ª série do Ensino Médio: guia do aluno matemática.

– São Paulo: Instituto Unibanco/CAEd, 2013.

182 p.; Vol II

Coordenação do materialJuliana Irani do Amaral

Pesquisa e conteúdoCAEd – Centro de Políticas Públicas e Avaliação da Educação

Consultoria responsávelCAEd – Centro de Políticas Públicas e Avaliação da Educação

Agradecimentos especiaisSecretaria Municipal de Educação do Rio de Janeiro, Secretarias de Estado de Educação do Rio de Janeiro, São Paulo, Espírito Santo e Minas Gerais e escolas parceiras que contribuíram para a testagem e validação da Metodologia Entre Jovens.

RealizaçãoInstituto Unibanco

PresidênciaPedro Moreira Salles

Vice-PresidênciaPedro Sampaio Malan

ConselhoAntonio MatiasCláudio de Moura CastroCláudio Luiz da Silva HaddadMarcos de Barros LisboaRicardo Paes de BarrosTomas Tomislav Antonin ZinnerThomaz Souto Corrêa NettoWanda Engel

Diretoria ExecutivaFernando Marsella Chacon RuizGabriel Amado de MouraJânio GomesJosé Castro Araujo RudgeLeila Cristiane B. B. de MeloLuis Antônio RodriguesMarcelo Luis Orticelli

Superintendência ExecutivaRicardo Henriques

Gerência de Implementação de ProjetosTiago Borba

Gerência de Desenvolvimento e ConteúdoMarta Grosbaum

Gerência de Gestão do ConhecimentoMirela de Carvalho

Gerência de Administração e FinançasFábio Santiago

Assessoria de Assuntos EstratégicosChristina Fontainha

Assessoria de ComunicaçãoMarina Rosenfeld

Assessoria de VoluntariadoFabiana Mussato

SUMÁRIO

Apresentação

OFICINA 11FUNÇÃO QUADRÁTICA

OFICINA 12RAZÕES TRIGONOMÉTRICAS NO TRIÂNGULO RETÂNGULO

OFICINA 13FUNÇÕES TRIGONOMÉTRICAS

OFICINA 14APLICAÇÕES DA TRIGONOMETRIA

OFICINA 15ÁREAS DE FIGURAS PLANAS

OFICINA 16SÓLIDOS GEOMÉTRICOS

OFICINA 17NOÇÕES DE GEOMETRIA ANALÍTICA

OFICINA 18CONTAGEM

OFICINA 19PROBABILIDADE

OFICINA 20TRATAMENTO DA INFORMAÇÃO

RESPOSTAS

9

11

25

41

55

71

89

105

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135

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APRESENTAÇÃO

Prezado aluno,

Dando continuidade ao primeiro guia, organizamos esse segundo guia que complementará os temas de matemática a serem trabalhados dentro do Projeto Entre Jovens – 3º ano.

Este guia é composto de dez oficinas, cada uma a ser cumprida em uma semana. As atividades dessas oficinas serão discutidas e trabalhadas pelos tutores com você.

Conforme dito na apresentação do primeiro guia, você deverá tentar resolver todas as atividades, desenvolvendo-as ao máximo para, em seguida, discuti-las com os colegas e com o tutor. Ainda que não consiga chegar à solução, partilhe com o grupo o seu raciocínio, o seu encaminhamento parcial da atividade.

As oficinas que compõem esse guia abordarão os seguintes temas:

11. Função Quadrática.

12. Razões Trigonométricas no Triângulo Retângulo.

13. Funções Trigonométricas.

14. Aplicações da Trigonometria.

15. Áreas de Figuras Planas.

16. Sólidos Geométricos.

17. Noções de Geometria Analítica.

18. Contagem.

19. Probabilidade.

20. Tratamento da Informação.

Esperamos que os encontros sejam proveitosos para todos.

Bons estudos e ao trabalho!

FUNÇÃO QUADRÁTICAOFICINA 11

Metas:– Apresentar os principais elementos da função quadrática.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• Resolver problemas envolvendo funções quadráticas.

Pré-requisito:– Oficina 5 do Guia 1.

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Resolva as atividades propostas abaixo.

Atividade 1: Prove que não existem dois números reais cuja soma seja 2 e cujo produto seja 5.

Atividade 2: Escreva a função quadrática, definida por 2( ) 2 5 3f x x x= − + , em sua forma canônica. Determine o menor valor que a função f assume e quais são suas raízes.

Atividade 3: Mostre que a função quadrática 2( ) 2 12 19f x x x= − + não admite raiz real.

Atividade 4: Mostrar que se dois números positivos têm soma constante, seu produto é máximo quando eles são iguais.

Atividade 5: Tenho material suficiente para erguer 20 m de cerca. Com ele pretendo fazer um cercado retangular de 226m de área. Quanto devem medir os lados desse retângulo?

Atividade 6: Mostrar que se o produto de dois números positivos é constante, sua soma é mínima quando eles são iguais.

Atividade 7: Mostrar que a média aritmética de dois números positivos é sempre maior do que ou igual à média geométrica, sendo igual apenas quando esses números forem iguais.

Atividade 8: Na figura abaixo, determinar x de modo que a área do paralelogramo inscrito no retângulo seja mínima. Supõe-se que 3a b a≤ ≤ .

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Atividade 9: Dois comerciantes formam uma sociedade com o capital de 100 mil reais. Um deles trabalhou 3 dias por semana e o outro 2. Após algum tempo, desfizeram a sociedade e, considerando o capital que cada um entrou e o tempo de trabalho dedicado por cada sócio, cada um recebeu 99 mil reais. Qual foi a contribuição de cada um para o capital da sociedade?

Atividade 10: Resolva as inequações abaixo:

a) 26 1 0x x− − < .

b) 23 10 30 16 15x x x+ − ≥ + .

c) 2 2 7

13

x xx− −

≤−

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Atividade 11: Considere a função f: IR :f →� �IR, definida por ( ) 2 2 1f x x xλ λ= + + , onde . Para que valores reais de λ a função f assume somente valores positivos?

Atividade 12: Num sistema de coordenadas cartesianas, o ponto R se desloca sobre o eixo das ordenadas a partir do ponto ( )0,30 , em direção à origem O, com velocidade constante de 1 cm/s e o ponto S se desloca sobre o eixo das abscissas, partindo do ponto ( )2,0 , com o dobro da velocidade do ponto R. Eles partem no mesmo instante. Veja a figura abaixo.

Em quanto tempo o triângulo ROS atingirá área máxima?

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OFICINA 12

RAZÕES TRIGONOMÉTRICASOFICINA 12

Metas:– Introduzir a noção de ângulo.– Apresentar as razões trigonométricas num triângulo

retângulo.– Construir um teodolito doméstico para as atividades

que necessitam de medição de ângulos.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• Resolver problemas envolvendo razões trigonométricas num triângulo retângulo.

Pré-requisito:– Oficinas 8 e 9 do Guia 1.

NO TRIÂNGULO RETÂNGULO

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Resolva as atividades propostas abaixo.

Atividade 1: Sabendo que a tangente de um ângulo agudo α é igual a 2, calcule o valor de senα e cosα .

Atividade 2: Como os gregos estimaram o raio da Terra?

Atividade 3: Ao soltar uma pipa, um menino usou os 100 metros de linha de seu carretel. O ângulo que a linha forma com a horizontal é igual a 18°. A que altura está a pipa? Veja ilustração abaixo.

Atividade 4: Um piloto decola de certa cidade A com seu avião, devendo alcançar a cidade B após duas horas de voo na rota 28º (veja bússola). Porém, duas horas após a decolagem, o piloto notou que, por engano, tinha tomado a rota 280º. Supondo que o avião tenha combustível suficiente, qual deverá ser o novo rumo para que ele consiga atingir a cidade B?

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Atividade 5: Um engenheiro precisa projetar uma piscina em forma de um trapézio retângulo conforme ilustra figura abaixo. Qual deverá ser a medida do lado MN dessa piscina?

Atividade 6: Jorge é proprietário de um pequeno terreno que tem um barranco. Um corte de seu terreno está representado na figura abaixo. Ele deseja medir a largura de seu terreno que, na figura, é representada por x. Para tanto, finca uma estaca no alto do barranco na qual amarra um barbante (ponto A). Estica o barbante até um ponto B no solo de tal modo que ele faça 60º com a horizontal. Em seguida, mede a distância de B até C, encontrando 15,5 m, e o comprimento do barbante, encontrando 23 m. Qual a largura do terreno de Jorge?

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Atividade 7: Um avião levanta voo em A e sobe fazendo um ângulo constante de 15° com a horizontal. A que altura estará e qual a distância percorrida quando sobrevoar uma torre situada a 2 km do ponto de partida? (Dados: sen 15° = 0,26; cos 15° = 0,97 e tg 15° = 0,27)

Atividade 8: Determine o valor de x no seguinte caso:

Atividade 9: Com base na figura abaixo, calcule senβ , cosβ e tgβ .

Atividade 10: Pedro fabricou uma bicicleta, tendo rodas de tamanhos distintos, com o raio da roda maior (dianteira) medindo 3 dm, o raio da roda menor medindo 2 dm e a distância entre os centros A e B das rodas sendo 7 dm. As rodas da bicicleta, ao serem apoiadas no solo horizontal, podem ser representadas no plano (desprezando-se os pneus) como duas circunferências, de centros A e B, que tangenciam a reta r nos pontos P e Q, como indicado na figura.

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a) Determine a distância entre os pontos de tangência P e Q e o valor do seno do ângulo BPQ.

b) Quando a bicicleta avança, supondo que não haja deslizamento, se os raios da roda maior descrevem um ângulo de 60°, determine a medida, em graus, do ângulo descrito pelos raios da roda menor. Calcule, também, quantas voltas terá dado a roda menor quando a maior tiver rodado 80 voltas.

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FUNÇÕES TRIGONOMÉTRICAS

OFICINA 13

Meta:– Estudar a unidade radiano e o ciclo

trigonométrico.– Apresentar as principais funções trigonométricas.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• Lidar com a unidade radiano.• Reconhecer seno e cosseno de um arco no ciclo

trigonométrico.• Resolver problemas envolvendo funções

trigonométricas.

Pré-requisito:– Oficina 12 do Guia 2.

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Atividade 1: No dia do solstício de verão, Erastóstenes verificou que, ao meio dia, o sol brilhava diretamente dentro de um poço profundo em Assua e, em Alexandria, a 5000 estádios ao norte de Assuã, alguém mediu o ângulo que os raios solares faziam com a vertical, encontrando 1/50 do círculo. Com base nestes dados, calcule o raio da Terra.

Atividade 2: Encontre as três menores soluções positivas da equação

cos 3 04

xπ − =

.

Atividade 3: Verifique que as igualdades abaixo valem para todo valor de

22

x kππ≠ ± , onde k é um número inteiro qualquer (Tais desigualdades são

chamadas de identidades trigonométricas).

a)cos 1 sen

1 sen cosx xx x

−=

+

b)2

2 22

1 tgcos sen

1 tgx

x xx

−− =

+

Atividade 4: Quantos são os valores distintos de sen5kπ

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Atividade 5: Para que valores de x se tem :

a)1

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x >

b)1

sen2

x >

Atividade 6: Calcular m para que exista um ângulo x com 2

cos1

xm

=−

e tg 2x m= − .

Atividade 7: Nas desigualdades tem-se alguns valores do seno, do cosseno e tangente de certos números reais. Classifique como verdadeira ou falsa cada dessas desigualdades.

a) sen2 0>

b) cos4 0<

c) sen3 sen2>

d) cos3 cos2>

e) tg5 tg6>

f) cos cos14π>

g) cos 3 0<

Atividade 8: Determine o conjunto dos números reais x para os quais tg2 3x = .

Atividade 9: Determine para que valores de x a função 5 cos5

y xπ = − +

assume seu valor máximo.

Atividade 10: Para cada uma das funções definidas abaixo, esboce seus gráficos e dê seus respectivos conjuntos imagens.

a) f:IR:f →� �IR, definida por ( ) 1 2cos3

f x xπ = + +

b) g:IR:f →� �IR definida por ( ) 11 cos

2 4g x x

π = − + −

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OFICINA 14

APLICAÇÕES DA TRIGONOMETRIAOFICINA 14

Meta:– Apresentar algumas aplicações básicas da trigonometria.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• Resolver problemas envolvendo cálculos de distâncias ou alturas.

Pré-requisitos:– Oficinas 8 e 9 do Guia 1.– Oficinas 12 e 13 do Guia2.

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Resolva as atividades propostas abaixo.

Atividade 1: Medir a largura de um rio sem atravessá-lo.

Estando no ponto A às margens de um rio, estimar a largura do rio, sem atravessá-lo, utilizando somente trena e teodolito.

Atividade 2: Medir a largura de um rio sem atravessá-lo – parte 2.

Estando no ponto A às margens de um rio, estimar a largura do rio, sem atravessá-lo, utilizando somente trena e teodolito.

Atividade 3: Medir uma altura inacessível.

Estimar a altura h de um poste, sem escalá-lo, sendo conhecidos a altura da visão do observador, que é 1,6 m, empregando somente trena e teodolito. Veja a ilustração a seguir.

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Atividade 4: Medir uma altura inacessível – parte 2.

Estando situado no ponto A, às margens de um rio, estimar a altura de um prédio que se encontra na margem oposta desse rio, conforme ilustrado na figura abaixo, fazendo uso somente de trena e teodolito.

Atividade 5: Calcular a distância de um ponto a outro que esteja inacessível.

Estimar a distância do ponto A em uma praia a um ponto P em uma ilha, utilizando somente trena e teodolito. Veja a figura abaixo.

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Atividade 6: Determinar a distância entre dois pontos inacessíveis.

Dispondo de uma trena e de um teodolito, estimar a distância entre dois pontos P e Q, cada um em uma ilha diferente, a partir de um ponto A em uma praia de onde se possa avistar os pontos P e Q. Veja a ilustração abaixo.

Atividade 7: Um observador O situado no topo de uma montanha vê dois outros A e B situados no nível do mar. Os observadores A e B medem os ângulos α e β que as linhas AO e BO formam com o plano horizontal e o observador O mede o ângulo AÔB θ= . Conhecendo-se a distância d entre os observadores A e B, calcule a altura da montanha.

Atividade 8: Um balão foi visto simultaneamente de três estações A, B e C sob ângulos de elevação de 45º, 45º e 60º, respectivamente. Sabendo que A está 3 km a oeste de C e que B está 4 km ao norte de C, determine a altura do balão.

Atividade 9: Um corredor A está sobre uma reta r e corre sobre ela no sentido AX. Um corredor B não está em r e, correndo em linha reta, pretende alcançar A, vide figura abaixo. Sendo a partida simultânea, que direção deve tomar B se as velocidades de ambos são conhecidas?

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a) Considere 110ºBAX = , velocidade de A igual a 8m/s e velocidade de B igual a 9m/s. Determine o ângulo que a trajetória de B deve fazer com a reta BA para que o encontro seja possível.

b) Considere 110ºBAX = , velocidade de A igual a 8m/s, velocidade de B igual a 8,1 m/s e AB=50 m. Sendo B um corredor inteligente, determine que distância ele percorreu até alcançar A.

Atividade 10: Considerando ainda a figura da atividade 9, seja 60ºBAX = . O corredor A tem velocidade 15% maior que a de B. Porém, o corredor B é inteligente, planejou cuidadosamente sua trajetória, e alcançou o corredor A no ponto C da reta r. Calcule o ângulo ABC .

Observação: você vai encontrar dois valores para o ângulo ABC . Ambos são possíveis? Por que ocorre isto?

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OFICINA 15

ÁREAS DE FIGURAS PLANASOFICINA 15

Meta:– Apresentar a noção de área de uma região do plano.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• Resolver problemas envolvendo o cálculo de áreasde figuras planas.

Pré-requisitos:– Oficinas 8 e 9 do Guia 1.– Oficinas 12 e 14 do Guia 2.

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Resolva as atividades propostas abaixo.

Atividade 1: O triângulo ABC da figura abaixo tem área igual a 30 cm². O lado BC está dividido em quatro partes iguais pelos pontos D , E e F , e o lado AC está dividido em três partes iguais pelos pontos G e H . Qual é a área do triângulo GDE ?

Atividade 2: Em algum momento na primeira metade do século passado, uma pessoa chamada Afrânio tinha um valioso terreno desocupado perto do centro da cidade do Rio de Janeiro. Com a urbanização da cidade, ruas novas foram abertas e o terreno de Afrânio ficou reduzido a um triangulo ABC, retângulo em B, ainda de grande valor pois o lado AB media 156 metros. Pois bem, Afrânio morreu e em seu testamento os advogados encontraram as instruções para dividir o terreno “igualmente” entre seus dois filhos. Era assim:

“um muro deve ser construído perpendicularmente ao lado AB de forma que os dois terrenos resultantes da divisão tenham mesma área; o que tem a forma de um trapézio será do meu filho mais velho e o outro será do mais novo”.

Os advogados não foram capazes de decidir em que posição deveria ficar o murro. Em que posição relativamente ao lado AB do terreno o murro deve ser construído?

Atividade 3: Três lotes quadrados delimitam um lago em forma de um triângulo retângulo, conforme indicado na figura abaixo. Sabe-se que as medidas das áreas desses três lotes somam 800m². Qual a medida do maior dos lados desses três lotes?

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Atividade 4: Um pequeno restaurante possui um salão de jantar, uma cozinha e dois banheiros, todos em forma de quadrado. A cozinha e o salão de jantar possuem a mesma área. No exterior do restaurante há uma varanda cujas dimensões estão representadas na figura abaixo. Qual a medida da área interna desse restaurante?

Atividade 5: Um terreno pentagonal foi repartido em quatro lotes de forma a que 40BF FG= = m, 10AF = m e 60FC = m, conforme indicado na figura abaixo. Qual a área do terreno pentagonal?

Atividade 6: O logotipo de uma empresa é formado por um retângulo de dimensões 2,5cm de largura e 2 cm de altura 2cm, no qual se encontram pintados 4 triângulos retângulos idênticos, cujos catetos são paralelos aos lados do retângulo. Veja a ilustração abaixo. Qual é a área de cada um desses triângulos?

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Atividade 7: Em um supermercado existem duas câmeras de vídeo instaladas nos pontos A e B. Há duas gôndolas posicionadas perpendicularmente a parede, uma de 15 metros e a outra de 10 metros de comprimento, distantes 3 metros entre si. A região na cor cinza corresponde à área em que as câmeras não conseguem captar imagem. Veja a planta baixa na ilustração. Qual é a área da região que as câmeras não conseguem captar?

Atividade 8: Um lote retangular está dividido em quatro terrenos retangulares. As medidas das áreas de três deles estão indicadas na figura abaixo. Qual é a área do lote?

Atividade 9: O quadrado da figura abaixo foi dividido num quadrado menor rodeado por quatro retângulos iguais. O perímetro de cada um dos retângulos mede 22 cm. Qual a área do quadrado grande?

Atividade 10: No paralelogramo ABCD, o lado AD mede 15 cm, o lado AB mede 5 cm e a distância do vértice A ao lado CD é 12 cm. Veja a figura abaixo. Qual é a distância x do vértice A ao lado BC ?

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Atividade 11: Considere no paralelogramo ABCD as regiões R, S, T e U determinadas pelo segmento DM que une D ao ponto médio M de BC e pela diagonal AC. Se a área de ABCD mede 60 cm², determine as áreas das regiões R, S, T e U.

Atividade 12: Considere o triângulo equilátero e o quadrado como na figura abaixo, ambos com lados medindo a. Calcule a área da parte hachurada.

Atividade 13: A secção transversal de uma carroceria de caminhão que transporta 5 tubulações cilíndricas é um retângulo que acomoda exatamente as seções das 5 tubulações, como na figura abaixo. Sendo o raio das tubulações igual a r, quais as dimensões do retângulo dessa seção?

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Atividade 14: Na figura abaixo se encontram representadas duas circunferências. O raio da maior circunferência é 2 m. Calcule área da região hachurada.

Atividade 15: Na figura abaixo, o ponto O é o centro do círculo de raio r, AT é tangente ao círculo e MT é perpendicular a AT e o ângulo central AÔM mede

3π

radianos. Calcule a área da região hachurada.

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OFICINA 16

SÓLIDOS GEOMÉTRICOSOFICINA 16

Meta:– Calcular a área das superfícies dos principais

sólidos geométricos.– Calcular o volume dos principais sólidos geométricos.

Objetivos: Ao término dessa oficina o aluno deverá ser capaz de: • Resolver problemas envolvendo o volume e área

de superfícies dos principais sólidos geométricos.

Pré-requisito:– Oficina 15.

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Resolva as atividades propostas abaixo.

Atividade 1: Calcule o volume do bloco retangular de 5,6 cm de comprimento, 4,7 cm de largura e 2,0 cm de altura.

Atividade 2: Um artesão, para pintar a superfície de esferas de raio 2 cm, estima gastar 0,05 cm³ de tinta por cada cm² de superfície. A tinta a ser utilizada é vendida em latas em forma de cilindro circular reto, cujo diâmetro mede 4 cm e cuja altura mede 5 cm. Nessa questão, despreze todas as espessuras envolvidas.

a) Qual é o número mínimo de latas de tinta necessário para pintar 500 esferas?

b) Cada esfera será acondicionada em uma caixa cúbica cujas arestas medem 5 cm. O espaço interno da caixa, não ocupado pela esfera, será totalmente preenchido por um material de proteção. Qual é a medida do menor volume desse material necessário para acondicionar as 500 esferas?

Atividade 3: Antônio colou pelas faces 7 cubinhos idênticos conforme ilustrado na figura abaixo.

Qual o número mínimo de cubinhos, idênticos aos já utilizados, que Antônio deverá acrescentar à essa formação de maneira a completar um cubo?

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Atividade 4: Cortamos um canto de um cubo, como mostrado na seguinte figura. Qual das representações abaixo corresponde ao que restou do cubo?

Atividade 5: Para montar um cubo, Guilherme recortou um pedaço de cartolina branca e pintou de cinza algumas partes, como na figura ao lado. Qual das figuras abaixo representa o cubo construído por Guilherme?

Atividade 6: A figura ao lado mostra três dados iguais. Qual é o número da face que é base inferior da coluna de dados?

Atividade 7: Uma garrafa cilíndrica está fechada, contendo um líquido que ocupa quase completamente seu corpo, conforme mostra a figura. Suponha que, para fazer medições, você disponha apenas de uma régua milimetrada.

a) Para calcular o volume do líquido contido na garrafa, qual o número mínimo de medições que precisam ser realizadas?

b) Para calcular a capacidade total da garrafa, lembrando que você pode virá-la, qual o número mínimo de medições a serem realizadas?

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Atividade 8: Uma piscina tem 10 m de comprimento, 6 m de largura e 1,6 m de profundidade.

a) Calcule seu volume em litros.

b) Determine quantos ladrilhos quadrados com 20 cm de lado são necessários para ladrilhar essa piscina.

Atividade 9: Uma garrafa de bebida com 30 cm de altura tem uma miniatura perfeitamente semelhante com 10 cm de altura. Se a miniatura tem 50 ml de volume, qual é o volume da garrafa original?

Atividade 10: Um tanque subterrâneo tem a forma de um cone invertido com 12 m de profundidade. Este tanque está completamente cheio com 27.000 litros de água e 37.000 litros de petróleo. Calcule a altura da camada de petróleo.

Atividade 11: Um copo cilíndrico tem 3 cm de raio e 12 cm de altura. Estando inicialmente cheio d’água o copo é inclinado até que o plano de sua base faça 45º com o plano horizontal. Calcule o volume de água que permaneceu no copo.

Atividade 12: Um copo cônico de papel foi feito a partir de um setor de 12 cm de raio e ângulo central de 120º. Calcule o volume desse copo.

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OFICINA 17

NOÇÕES DE GEOMETRIAANALÍTICA

OFICINA 17

Meta:– Apresentar conceitos básicos de Geometria

Analítica Plana. – Equacionar reta e circunferência.

Objetivos: – Ao término dessa oficina o aluno deverá ser capaz de: •Equacionar uma reta, dado dois pontos ou um ponto e o

coeficiente angular.•Equacionar uma circunferência dados centro e raio ou

dados três pontos de passagem.

Pré-requisitos:– Oficinas 10 e 12.

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Resolva as atividades propostas abaixo.

Atividade 1: Determine a equação da circunferência que tem centro no ponto ( )2, 3A = − e que passa pelo ponto ( )1,1B = − .

Atividade 2: Determine quantas são as soluções do sistema de equações:

2 2

2 2

2 2 2

8 6 16

x y x y

x y x y

+ − + =

+ − − = −

Atividade 3: Obter a equação da reta que passa pelos pontos ( )2,3M = e ( )1,6N = − .

Atividade 4: Obter a equação da reta que passa pelos pontos ( )1, 3R = − e tem coeficiente angular igual a 2− .

Atividade 5: Verifique se as retas 1r e 2r , de equações 1 : 2 3 5r x y− = e

2 :15 10 3r y x= + , são paralelas.

Atividade 6: Considere a circunferência C de equação ( ) ( )2 21 2 25x y− + + = e

a reta r de equação 4x = . O objetivo dessa questão é obter a equação de uma circunferência λ , tangente exterior à circunferência C, e com centro sobre a reta r. Escolha um ponto que possa ser o centro da circunferênciaλ .

a) Determine a medida do raio da circunferência λ .

b) Encontre a equação da circunferência λ .

Atividade 7: Determine a equação da circunferência que passa pelos pontos ( )2, 1P = − − , ( )0,0Q = e ( )1,2R = .

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Atividade 8: Na malha quadriculada abaixo, cujos quadrados têm lados medindo 10 metros, encontra-se o mapa de um tesouro.

Sobre o tesouro, sabe-se que encontra-se na direção determinada pelos dois pinheiros e está a 110 metros a leste do muro. Qual a distância aproximada do tesouro até a margem do rio?

Atividade 9: Determinar o raio da circunferência inscrita em um triângulo retângulo de catetos 3 e 4.

Atividade 10: Considere a reta r determinada pelos pontos P e Q e a circunferência λ de centro C, que passa pelo ponto A, conforme representados no plano cartesiano abaixo.

Determine a equação da reta s, perpendicular à reta r, tangente à circunferência λ e que contém pontos do 2º quadrante.

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OFICINA 18

CONTAGEMOFICINA 18

Meta:– Introduzir técnicas de contagem.

Objetivos: – Ao término dessa oficina o aluno deverá ser capaz de: •Resolver problemas de contagem.

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Resolva as atividades propostas abaixo.

Atividade 1: Quantas listas de dois elementos são possíveis quando há n escolhas para o primeiro elemento e m escolhas para o segundo elemento?

Atividade 2: Com 3 homens e 3 mulheres, de quantos modos se pode formar um casal?

Atividade 3: Uma bandeira é formada por 8 listras que devem ser coloridas usando-se apenas as cores verde, azul e cinza. Se cada listra deve ter apenas uma cor e não podem ser usadas cores iguais em listras adjacentes, de quantos modos se pode colorir a bandeira?

Atividade 4: Quantos são os números formados por três dígitos distintos?

Atividade 5: O código Morse usa duas letras, ponto e traço, e as palavras têm de 1 a 4 letras. Quantas são as palavras do código Morse?

Atividade 6: Considere os divisores inteiros e positivos do número 360.

a) Quantos são eles?

b) Quantos desses divisores são pares?

c) Quantos são ímpares?

d) Quantos são quadrados perfeitos?

Atividade 7: Quantos são os números pares de três dígitos distintos?

Atividade 8: (Problema das permutações simples) De quantos modos podemos ordenar em fila n objetos distintos?

Atividade 9: Quantos são os anagramas da palavra “PRATO”? Quantos começam por consoante?

Atividade 10: De quantos modos podemos arrumar em fila 5 livros diferentes de Matemática, 3 livros diferentes de Estatística e 2 livros diferentes de Física, de modo que livros de uma mesma matéria permaneçam juntos?

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Atividade 11: De quantos modos podemos dividir 7 objetos em um grupo de 3 objetos e um de 4 objetos?

Atividade 12: (Problema das combinações simples) De quantos modos podemos selecionar p objetos distintos entre n objetos distintos dados?

Atividade 13: Com 5 homens e 4 mulheres, quantas comissões de 4 pessoas:

a) com exatamente 2 homens, podem ser formadas?

b) com pelo menos 2 homens, podem ser formadas?

Atividade 14: Quantos são os anagramas da palavra “BANANA”?

Atividade 15: Quantos são os anagramas da palavra “ANAGRAMA” que não possuem duas vogais adjacentes?

Atividade 16: Há 5 pontos sobre uma reta R e 8 pontos sobre uma reta 'R paralela a R. Quantos são os triângulos e os quadriláteros convexos com vértices nesses pontos?

Atividade 17: De quantos modos 5 crianças podem formar uma roda de ciranda?

Atividade 18: De quantos modos 5 homens e 5 mulheres podem se sentar em 5 bancos de 2 lugares, se em cada banco deve haver um homem e uma mulher?

Atividade 19: De quantos modos o número 720 pode ser decomposto em um produto de dois inteiros positivos? Aqui consideramos naturalmente, 8 90× como sendo o mesmo que 90 8× .

Atividade 20: As placas dos veículos são formadas por três letras (de um alfabeto de 26) seguidas por 4 algarismos. Quantas placas poderão ser formadas?

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Atividade 21: Quantos são os anagramas da palavra “CAPÍTULO”’:

a) possíveis?

b) que começam e terminam em vogal?

c) que têm as vogais e as consoantes intercaladas?

d) que têm as letras C, A, P juntas nessa ordem?

e) que têm as letras C, A, P juntas em qualquer ordem?

f) que têm a letra P em primeiro lugar e a letra A em segundo?

Atividade 22: De quantos modos é possível colocar 8 pessoas em fila de modo que:

a) duas dessas pessoas, Vera e Paulo, não fiquem juntas?

b) duas dessas pessoas, Vera e Paulo, não fiquem juntas e duas outras pessoas, Helena e Pedro, permaneçam juntas?

Atividade 23: Permutam-se de todas as formas possíveis os algarismos 1,2,4,6,7 e escrevem-se os números formados em ordem crescente. Determine:

a) que lugar ocupa o número 62417.

b) que número ocupa o 66º lugar.

Atividade 24: De quantos modos podemos formar uma roda de ciranda com 6 crianças, de modo que duas delas, Vera e Isadora, não fiquem juntas?

Atividade 25: Um determinado sistema de segurança tem uma tranca protegida por uma senha de 6 dígitos, formada por duas vogais, seguidas de 4 algarismos de 0 a 9. Qual o total de senhas possíveis para essa tranca?

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OFICINA 19

PROBABILIDADEOFICINA 19

Meta:– Apresentar os principais conceitos associados a probabilidade.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• Resolver problemas envolvendo probabilidade.

Pré-requisito:– Oficina 18.

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Resolva as atividades propostas abaixo.

O Jogo

Este jogo utiliza dois dados e é disputado por dois jogadores, João e Maria. Os resultados abaixo valem os pontos indicados e resultados diferentes destes não são pontuados.

Resultado Pontuação

(4; 1) ou (1; 4) 1(4; 2) ou (2; 4) 2(4; 3) ou (3; 4) 3

(4; 4) 4(4; 5) ou (5; 4) 5(4; 6) ou (6; 4) 6

Cada jogador poderá efetuar até dois lançamentos. Se não conseguir nenhuma face 4 no primeiro lançamento dos dois dados, efetua-se o segundo lançamento com os dois dados novamente. Se conseguiu pelo menos uma face 4 no primeiro lançamento, reserva este dado e decide se lança ou não o outro dado mais uma vez. Vence o jogo quem obtiver a maior pontuação.

Caso os dois jogadores obtenham a mesma pontuação o procedimento todo é repetido.

O jogador poderá não marcar pontos ou ter pontuação zero, isto ocorre se nos seus dois possíveis lançamentos ele não conseguiu nenhuma face 4.

Este jogo deve ser utilizado para resolver as atividades de 1 a 11.

Atividade 1: Considerando-se apenas o primeiro lançamento dos dois dados, João terá maior chance em conseguir 1 ponto ou 6 pontos? Justificar sua resposta.

Atividade 2: Considerando-se apenas o primeiro lançamento dos dois dados, João terá maior chance em conseguir 5 ou 4 pontos? Justificar sua resposta.

Atividade 3: Se João obteve 1 ponto no primeiro lançamento ele deverá utilizar o segundo lançamento para melhorar sua pontuação? Justificar sua resposta.

Atividade 4: Se João obteve 3 pontos no primeiro lançamento, quais são suas chances em melhorar, piorar ou manter inalterada sua pontuação se utilizar o segundo lançamento?

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Atividade 5: Qual a probabilidade de João não obter a face 4 no primeiro lançamento?

Atividade 6: Se João obteve 4 pontos no primeiro lançamento, qual a probabilidade de aumentar, diminuir ou permanecer com esta pontuação se utilizar o segundo lançamento?

Atividade 7: Qual a probabilidade de João marcar 1 ponto neste jogo?

Atividade 8: Qual a probabilidade de João marcar 4 pontos neste jogo?

Atividade 9: Qual a probabilidade de João não marcar pontos neste jogo?

Atividade 10: Considerando-se apenas o primeiro lançamento, qual a probabilidade de João marcar 3 pontos, sabendo-se que ele obteve em pelo menos um dos dois dados uma face 4?

Atividade 11: Considerando-se apenas o primeiro lançamento, qual a probabilidade de João marcar 3 pontos, sabendo-se que o número da face do primeiro dado é maior do que a do segundo?

Atividade 12: Qual é a probabilidade de se obter um resultado maior que 4 ao se lançar um dado honesto?

Atividade 13: Ao lançar um dado duas vezes, qual é a probabilidade de se obter soma 5?

Atividade 14: Em uma urna há 5 bolas vermelhas e 4 pretas, todas de mesmo tamanho e feitas do mesmo material. Retiramos duas bolas sucessivamente da urna, sem repô-las. Qual é a probabilidade de que sejam retiradas duas bolas vermelhas?

Atividade 15: Pedro e João combinaram de lançar uma moeda 4 vezes. Pedro apostou que, nestes 4 lançamentos, não apareceriam 2 caras seguidas; João aceitou a aposta. Quem tem maior chance de ganhar a aposta?

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Atividade 16: Qual é a forma justa de dividir os R$ 20,00 apostados no problema dos pontos?

Atividade 17: Uma caixa contém cinco bolas numeradas de 1 a 5. Dela são retiradas ao acaso duas bolas. Qual a probabilidade de que o maior número assim escolhido seja o 4?

Atividade 18: Uma função :f A B→ é dita crescente quando:

( ) ( )1 2 2 1 2 1, ,x x A x x f x f x∀ ∈ > ⇒ > .

Sejam { }1,2,3,4,5A = e { }1,2,3,4,5,6,7B = .

a) Quantas funções :f A B→ crescentes podem ser definidas?

b) Escolhendo aleatoriamente uma das funções :f A B→ crescentes que podem ser definidas, qual é a probabilidade de se ter ( )5 6f = ?

c) Escolhendo aleatoriamente uma das funções :f A B→ crescentes que podem ser definidas, qual é a probabilidade de se ter ( )5 4f = ?

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OFICINA 20

TRATAMENTO DA INFORMAÇÃOOFICINA 20

Meta:– Apresentar informações em gráficos e tabelas. – Utilizar medidas de tendência central.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• Ler e interpretar informações apresentadas em gráficos e/ou tabelas.

• Resolver problemas envolvendo cálculo de médias.

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Resolva as atividades propostas abaixo.

Atividade 1: João obteve nota final igual a 58 pontos em matemática. Essa nota é obtida calculando-se a média aritmética entre as quatro notas bimestrais, cada uma valendo 100 pontos. João está em recuperação em matemática, pois a nota final mínima para aprovação é 60 pontos. A nota obtida na recuperação, que vale 100 pontos, substitui a menor nota bimestral do aluno, que no caso de João é 54 pontos. Qual a menor nota que João poderá obter na recuperação para que venha a ser aprovado em matemática?

Atividade 2: Em uma classe de 20 rapazes e 30 moças, foi realizada uma avaliação. A média das notas das moças foi 8 e a média das notas dos rapazes foi 7. Qual foi a média das notas da classe?

Atividade 3: A média das alturas dos cinco jogadores titulares de um time de basquete é igual a 1,98 m. O treinador deseja substituir um jogador de modo que a média de altura do time aumente para, no mínimo, 2 m. Nessa substituição, qual deve ser a diferença mínima, em centímetros, entre a altura do jogador que entrará e a altura do jogador que sairá?

Atividade 4: Um professor fez o levantamento das notas de uma turma com 20 alunos, obtidas em uma prova cujo valor era 10 pontos. Veja o gráfico abaixo.

0

1

2

3

4

5

3 4 5 6 7 8 9

Notas

Freq

üênc

ia

Depois de confeccionado esse gráfico, o professor percebeu ter errado a nota de um dos alunos e verificou que, feita a correção dessa nota, a média das notas dessa turma aumentaria em 0,2 ponto e a moda passaria a ser 7 pontos. Qual passou a ser média das notas após a correção? Qual era o valor da nota que estava errada?

Atividade 5: Um aluno compara as notas das 6 provas de Português que fez em 2004 e de outras 6, da mesma matéria, que fez em 2005. Ele repara que em 5 provas ele obteve as mesmas notas nos dois anos. Na outra prova a nota foi 86 em 2004 e 68 em 2005. Em 2004 a média aritmética das seis notas foi 84. Qual foi a média em 2005?

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Atividade 6: O professor de Matemática aplicou em sua turma um teste com cinco questões de múltipla escolha onde cada questão valia um ponto. A nota de cada aluno no teste foi a soma das notas das questões por ele acertadas. Após corrigir o teste, o professor produziu a seguinte tabela, contendo a porcentagem de acertos em cada questão:

Questão 1 2 3 4 5

% de acertos 50% 40% 60% 20% 10%

Qual foi a média das notas nesse teste?

Atividade 7: Na sala de Pedrinho foi aplicada uma prova de matemática com 5 questões de múltipla escolha valendo um ponto cada. A distribuição dessas notas obtidas encontra-se representada no gráfico abaixo.

Distribuição de notas na prova de matemática

0

2

46

8

10

0 1 2 3 4 5Notas

Qua

ntid

ade

de a

luno

s

Pedrinho que não pôde comparecer na escola no dia da aplicação dessa prova, a realizou no mês seguinte. Qual a nota mínima que Pedrinho deverá tirar para que a média das notas dessa turma aumente?

Atividade 8: A média de idade dos 24 professores da “Escola Ensinar” é 40 anos. Rita, que tem 35 anos de idade, acaba de ser contratada como mais uma professora dessa escola. Qual a nova média de idade dentre os professores da “Escola Ensinar”?

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Atividade 9: Uma empresa fez um levantamento em todos os seus setores determinando o percentual de mulheres em cada setor da empresa. O resultado desse levantamento é mostrado no gráfico abaixo. Em quantos desses setores o percentual de homens é maior que o percentual de mulheres?

0 20 40 60 80

Setor 1

Setor 2

Setor 3

Setor 4

Setor 5

Setor 6

Setror 7

Atividade 10: Os resultados de uma pesquisa das cores de cabelo de 1200 pessoas são mostrados no gráfico abaixo. Quantas pessoas possuem o cabelo preto?

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Atividade 11: A fim de melhorar a aparência, 175 mil brasileiros enfrentaram anestesia, bisturi e o desconforto de um pós-operatório no ano de 2000. Esse desejo coletivo de conseguir um corpo mais bonito levou nosso pais ao segundo lugar no ranking dos campeões em cirurgias estéticas, só perdendo para os Estados Unidos”. Fonte: Revista Saude!,out. 2002.

O gráfico seguinte apresenta o total das cirurgias plásticas mais realizadas, no Brasil, em 2000.

a) Quantas cirurgias para redução ou aumento de mamas foram realizadas nesse ano?

b) No gráfico, qual e o ângulo central do setor circular que representa as cirurgias de face?

Atividade 12: Uma prova continha cinco questões, cada uma valendo 2 pontos. Em sua correção, foram atribuídas a cada questão apenas as notas 0 ou 2, caso a resposta estivesse, respectivamente, errada ou certa. A soma dos pontos obtidos em cada questão forneceu a nota da prova de cada aluno. Ao final da correção, produziu-se a seguinte tabela, contendo a porcentagem de acertos em cada questão:

Qual a média das notas da prova?

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Atividade 13: Em certa eleição foram obtidos os seguintes resultados:

Qual o número de votos obtidos pelo candidato vencedor?

Atividade 14: Observe os gráficos a seguir, que representam, em reais, as vendas e os lucros anuais de uma empresa no período de 1990 a 1995.

De acordo com os gráficos, calcule:

a) a média, em milhões de reais, das vendas dessa empresa no período considerado;

b) a razão entre o lucro e a venda em 1992.

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Atividade 15: Num curso de inglês, a distribuição das idades dos alunos e dada pelo gráfico seguinte.

Com base nos dados do gráfico, determine:

a) o número total de alunos do curso.

b) o número de alunos com no mínimo 19 anos.

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Anotações

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RESPOSTAS

RESPOSTAS

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Oficina 11: Função Quadrática.Atividade Resposta

022

5 1( ) 2

4 8f x x = − −

, 1

8− ,

1 e

34

.

05 Não é possível

084

a b+

09 55 mil e 45 mil reais

10 a)

1 1,

3 2 −

, b) ( ] [ ), 3 5,−∞ − +∞ , c)

( ] ( ], 1 3,4−∞ −

11 ( )0,1λ ∈

12 14,5 s

Oficina 12: Razões Trigonométricas no Triângulo Retângulo.Atividade Resposta

01 2 5sen

5α =

e

5cos

5α = .

02sen

1 senh

Rθθ

=−

.

03 32,5 m.

04 64º.

05 20 m.

06 27 m.

07 2062d ≅ m, 540h = m.

08 82 .

093

sen5

α = ,

4cos

5α = ,

3tg

4α = .

10 a) PQ = 4 3 dm, sen BPQ = 13

13.

b) 90º e 120 voltas.

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Oficina 13: Funções TrigonométricasAtividade Resposta

01 Aproximadamente 6.409 km.

027 11

, e 4 12 12π π π .

04 5.

05 a)

b)

.

06 5.

07 a) V b) V c) F d) F e) F f) V g) V

08 .

09 .

10 a) [ ]1,3− b) 3 1

,2 2

− −

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Oficina 14: Aplicações da TrigonometriaAtividade Resposta

01 tgd k β= ⋅ ou aproximadamente 61 m.

02 ou aproximadamente 42 m.

03 1,6h d tgα= + ⋅ ou aproximadamente 4,85 m.

042 2tg tg 4tg tg

2k k

hα α β α⋅ ± ⋅ + ⋅

= ou aproximadamente 47 m.

05sen

senAB

APβ

γ⋅

= ou aproximadamente 1 km.

06 ( )2 2 2 cosPQ AP AQ AP AQ α γ= + − ⋅ ⋅ ⋅ − ou aproximadamente 906 m.

07 2 2

tg tg

tg tg 2 tg tg cos

d α βα β α β θ

⋅ ⋅

+ − ⋅ ⋅ ⋅.

08 Aproximadamente: 6676 m ou 2696 m.

09 a) 8

arc sen sen110º9

b) Aproximadamente 761 metros.

10 ( )arc sen 0,575 3 .

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Oficina 15: Área de Figuras Planas.Atividade Resposta

01 5 cm².

02 110 m a partir de A.

03 20 m.

04 40 m².

05 3.400 m².

06 0,25 cm².

07 9 m².

08 143 m².

09 121 cm².

10 4 cm.

11 R = 25 cm², S = 20 cm², T = 10 cm² e U = 5 cm².

122

8 3 4a+

.

13 4r e ( )2 3 1r + .

14 2 m².

152 3 3 2

4 2 3r π

−

.

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Oficina 16: Sólidos Geométricos.Atividade Resposta

01 5,6 4,7 2,0× × cm³

02 a) 20 b) 32

500 1253π ⋅ −

cm 3

03 20.

04 E

05 C

06 4

07 a) 2 b) 3

08 a) 96.000 litros b) 2780

09 1350 ml.

10 3 m.

11 81π cm³.

12 128 2189

3π

≅ cm³.

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Oficina 17: Noções de Geometria Analítica.Atividade Resposta

01 ( ) ( )2 22 3 25x y− + + = .

02 Uma.

03 5x y+ = .

04 2 1x y+ = − .

05 São paralelas.

07 2 25 5 252 2 2

x y + + − =

.

08 46,6 m.

09 1.

10 2y x= − + .

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Oficina 18: Contagem.Atividade Resposta

01 m n× .

02 9.

03 384.

04 648.

05 30.

06 a) 24 b) 18 c) 6 d) 4

07 328.

08 n!

09 72.

10 8640.

11 35.

12 ( )!

! !n

p n p−.

13 a) 60 b) 105

14 60.

15 120.

16 220 e 280.

17 24.

18 ( )25! .

19 15.

20 3 426 10⋅ .

21 a) 8! b) 12 6!⋅ c) 2 4! 4!⋅ ⋅ d) 6! e) 3! 6!⋅ f) 6!

22 a) 8! 2 7!− ⋅ b) 2 7! 4 6!⋅ − ⋅

23 a) 81º b) 46.721

24 72.

25 2 45 10⋅ .

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Oficina 19: Probabilidade.Atividade Resposta

01 Mesma chance de conseguir 1 ou 6 pontos.

02 Maior chance de conseguir 5 pontos.

03 Deve aproveitar o segundo lançamento.

0412 ,

13

e

16

.

052536

.

0613

,

12

e

16

.

07 6,63%

08 7,48%

09 48,22%

10211

.

111

15.

1213

.

1319

.

145

18.

15 Pedro e João têm a mesma chance de vitória.

16 Um deve ficar com R$ 17,50 e o outro com R$ 2,50.

171

10.

18 a) 21 b) 521

c) 0.

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Oficina 20: Tratamento da Informação.Atividade Resposta

01 62.

02 7,6.

03 10 cm.

04 6,45 e 3.

05 81.

06 1,8.

07 3.

08 39,8.

09 5.

10 288.

11 a) 52.500 b) 72º

12 4,4.

13 182.

14 a) 3 milhões/ano b) 110

15 a) 20 b) 8