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0123 4 56784 9 2 9256 48265751285288125152287217 28 432112582 8157265 52 86585 12585 5 85657!5 2 4 882253648751 57 65855 55 788522 6 $ )58 #2 4*88(687528521 # 4"288+%11652765 57 825711857582,11582 5 4 252554 2 &78685 8 781252825556868)228'' 5 )782715 8 5 & , 52 78716' 52! 4271 88 72*7+6!21578528 227 411 5 72251225612!7585 5 21752!658617 5 5816525! #" 41,28361226875128285 2 4,7822517'8' 6 3 4 3 2 4 81176%22 57 5256!5125785 218261551 # 4767'2 25 4 885522417 2 4271578812528822157(5881251526878571658"27885725 . 28/6 #2$ 47)827582/85 50 ,)1-8251 5*8+7 2)52 2 5 5 4 2" &28771825227-556 7258727 81 8,8716!5718285821%,72655'' 5 , 7 5
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C a p ´ıtu ı tu lo 1
Proporcionalidade e F u n c¸ oes o˜ es Afins ´ ˜ PetersEm seu livro “Elementos de Algebra”, publicado em S ao Peters´ burgo em 1770, o grande matem atico Leonardo Euler prop oe o˜ e o seguinte problema: Uma lebre est a´ 50 pulos a` frent e de um cachorr cachorr o, o qua l d a´ 3 pulos pulos n o tempo tempo que que ela leva leva para dar 4. S Sabe abendo ndo que 2 pu los los do cachorro cachorro valem 3 da lebre, qua qua nt os pu los los ´ ele ele deve deve dar par a pega-la? ˜ que se refere a proporcionalidade, E s t e e´ u m exemplo exemplo de quest ao assunto que exporemos a seguir.
1
P r o po p o rc r c i o n al a l id id a d e
Diz-se que duas grandezas s a˜ o proporcionais quando existe uma correspond encia eˆ ncia x → y, que associa a cada valor x de uma delas del as um valor y bem definido da outr a, de t al m odo que sejam sejam cumpr ida s a s segu int es condic¸oes: o˜ es:
´ 1) Quan to maior maior for x, ma ior ior ser a´ y. Em termos matem aticos: s e x → y e x → y en t a˜ o x < x implica y < y .
˜ o valor 2) Se dobrarmos, triplicarmos, etc. o valor de x en t ao correspondente de y s er a´ dobra dobra do, do, tr iplic iplicado, ado, etc. etc. N a lingua lingua ´ a: se x → y en t a˜ o nx → ny para todo n N. gem matem atica: atic
∈
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4
Temas e Problemas
E x e m p l o 1 . Sejam x o volume e y o peso de um a porc¸a˜ o d e u m ´ıquido homog eneo. lıquido eˆ neo. A co corr espond espond encia eˆ ncia x → y cumpre claramente a s d u a s con dic¸oes o˜ es a cima cima , logo logo o volum volum e e´ proporcional ao peso.
lelas. Dado qualquer ret angulo ˆ E x e m p l o 2 . Sejam r e s reta s para lelas. que tenha dois lados contidos nessas retas, chamemos de x o com´ ˆ primento de um desses lados lados e z a area do ret ret angulo. s z r Figura 1
A corr corr espond espond encia eˆ ncia x → z e´ um a pr oporci porcio ona lidade. lidade. Ou seja: seja: ˆ ´ z e´ proporcioquando a altur alturaa de um ret angulo e´ fixada fixada , sua area n a l a` base x. ´ z do Com efeito, em primeiro lugar, se x < x en t a˜ o a area ˆ ´ ˆ gulo de base x mais z do ret an r et angulo de base x e´ igua igua l a` area ´ ˆ gulo de base x − x, logo z < z . a area de um ret an ˆ gulo de base n x pode ser expresEm segun segun do lugar, lugar, um r et an ˜ de n r et an ˆ gulos so como reuni ao gulos justa postos postos de ba se x (e m esma z) logo sua area area ´ ´ e´ n z.
·
·
˜ contida no Exemplo 2 e´ uma consecons eOb s e rv a c¸ a˜ o . A a fir m a c¸ ao q u¨ encia eˆ ncia imediata da f o´ rmu la que exprime a area ´ de um ret angulo ˆ como o produto da base pela altu ra . Esta e, e´ , entr etan to, to, uma justi´ ficat ficat iva iva a poster poster iori. iori. N a˜ o e´ convenient convenient e us a-la no presen te cont cont ex˜ daquela f ormula to pois, na verda de, o primeir o pass o da d edu c¸ao o´ rmula ˜ da proporcio e´ a ver ifica c¸ ao proporciona na lidade lidade acima.
ˆ AOB OB e u m a r e Consideremoss no plano um angulo E x e m p l o 3 . Consideremo t a r q u e n a˜ o e´ par alela alela a o llado ado OA n e m a OB (Figur Figur a 2). 2). Dado qualquer segment segment o de reta de co compriment o x, contido em OA a s
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˜ Afins Proporci Prop orciona onalid lidade ade e Func Fu nc¸ oes
5
B
r y
O
A
x
Figura 2
Afirmamos que a correspond encia eˆ ncia x → y e´ u ma proporcio proporciona na lilidade. Antes de just ificar ificar est a a firm ac¸ ao ˜ devemos devemos m ostra r que o co compriment primento o y depende apena s do co compriment o x m a s n a˜ o da p osic osi c¸ a˜ o do segmento tomad o sobre sobre o lado OA. (Isto signific significa que a corr espond encia eˆ ncia x → y es t a´ bem definida.) Ora, se tomarmos sobre o lado OA dois dois segmentos de mesmo ˜ na Figur ˜ par alelos comprimento x en t ao Figur a 3, onde onde MN e M N s ao alelos ˆ MNP e M N P t eˆ m , c a d a u m , u m l a d o d e a OA, o s t r iangulos ˆ M=M mesmo comprimento x, compr compr eendido entr e dois angulos ˜ tri angulos ˆ e N = N . Logo s ao congruentes e da´ da ´ı MP = M P = y. ˜ inicial, A par tir dest a observa c¸ao inicial, sempr e que t iverm iverm os x → y e x → y , se quisermos co compa ra r y com y podemos podemos supor qu e x e x ˜ medidas de segmentos com origem no v ertice ˜ fica s ao e´ rtice O. E n t ao claro que se x < x ⇒ y < y e que x = n x ⇒ y = n y, como mostra a Figur Figur a 4 (onde (onde n = 3).
·
·
E x e m p l o 4 . Investindo Investindo uma quantia x numa caderneta de poupa nc¸a , a p os o´ s o decurso de um m es eˆ s obt em-se e´ m-se um montante y. A correspond encia eˆ ncia x → y e´ uma proporcionalidade: o que se recebe n o fi m d o m es eˆ s e´ pr oporci porcio onal ao que se a plic plicou. Com Com efei efeito to,, e´ claro qu e aplic aplican do se mais receberecebe-se se ma is e investindo investindo se uma
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6
Temas e Problemas
B
P’ y M’ r
N’
x
P y M
N
x O
x
A
x Figura 3
B
B y y'
y
y O
y A
x
O
x'
Figura 4
x
x
x
A
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˜ Afins Proporci Prop orciona onalid lidade ade e Func Fu nc¸ oes
7
gera, ap o´ s um m es eˆ s de investiOb s e rv a c¸ a˜ o . Se uma qua ntia fixa gera, mento, um r etorn etorn o y, n a˜ o e´ verda verda de que ap os o´ s n meses essa mesma quan tia gere o retorno n y, mesm o que a ta xa de jur os perm an ec¸a ¸a consta nte. Pois Pois ao final de cada m es eˆ s e´ como se tivesse sido aplicada novamente uma quantia maior, igual a` existente no m es eˆ s anterior ma is os os jur os corr corr esponden esponden tes. Assim o retorno (nu (nu m per ´ıodo ı odo fixo) e´ proporcional ao capital inicial mas n a˜ o e´ proporcional ao tempo de investimento.
·
˜ mostra que a propriedade “qua E st a obser va c¸ao “qua nt o maior maior for for x, ˜ assegura a proporcionalidade entre x e y. Outro ma ior ior ser a´ y ” n ao exemplo disto e´ a correspond encia eˆ ncia x → y, onde x e´ o lado de um quadrado e y e´ sua a´ r e a .
Dian te dos exemp los los a nt eriores, podemos formu formu lar a d efinic¸a˜ o ´ a de pr oporci ˜ substim a t e m atica atic porcio ona lidade, lidade, onde a s grandezas sao ´ıdas por n um ´ eros ˜ suas medidas. t u ıdas eros r eais, eais, que s ao Estamos considerando apenas grandezas que t em eˆ m medida po´ ico sitiva, sitiva, logo logo o modelo modelo ma tem at ico da proporcio proporciona na lidade lidade leva leva em ˜ apenas ´ eros rea is positivo cons ide r a c¸ ao apenas n um positivos. s. Uma proporcionalidade (num erica) e´ rica) e´ u m a fun c¸ a˜ o f : R → R com as seguintes propriedades: ·
˜ crescente, isto e´ x < x 1) f e´ u m a fun c¸ ao quaisquer x, x R .
2) Para P ar a todo x
∈
∈R ·
⇒
f(x) < f(x ) p a r a
·
e todo n
·
∈ N tem-se f(nx) = n · f(x)
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8
Temas e Problemas
´ ´ 129, e “A Mat em atica ´ Pr ofessor essor de Ma tem atica”, p ag. do Ensino ´ 94. M edio, e´ dio, vol. 1”, p ag. ´ ´ l mostrar que f(nx) = n f(x) p a r a Na pr atica, e´ bem mais f acil aci n N do que que verificar verificar que f(cx) = c f(x) para todo c R . (Pense em c = 2 ou c = π .) . ) Por outro lado, o fato de que uma propor´ cionalidade f satisfaz satisfaz esta igualdade gualdade par a qualquer qualquer numero real positivo c tem importa importa ntes conseq u¨ encias, eˆ ncias, como como verem os agora.
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Coro l ario. ´ S e f : R → R e´ u m a proporci proporcionalida onalida de entao ˜ tem -se, para todo x > 0 , f(x) = ax , onde a = f(1). ·
·
Com Com efeito efeito,, pelo pelo Teo Teorem a F un dam ent al, par a qu aisquer x, c R , vale f(xc) = x f(c) = f(c) x. E m p a r t ic i cu l a r, r, t om o m a n d o c = 1, obtemos f(x) = a x, onde a = f(1).
·
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´ Um a fun c¸ a˜ o f : R → R definida por f(x) = ax, onde a R e uma constante, chama-se uma fun c¸ ao d o a > 0, a ˜ linear . Q u a n do fu n c¸ ao ˜ linear f(x) = ax transf transforma or ma um n umero ´ real positivo x n o ´ ˜ uma proporcionan umero positivo ax, logo defin e, p or r est r ic¸ao, lidade f : R → R . Acabam os de ver que, r ecipro eciproccam ente, toda ˜ linear proporcionalidade e´ a r es t r ic¸ a˜ o de u m a fun c¸ao linear a R . O coeficiente a cha ma-se o fator de proporcionalidade. proporcionalidade. ´ E s t a ultima observac¸a˜ o n os os p e r mi m it e con cl clu ir ir q u e s e f : R → R e´ uma proporcionalidade ent ao, ao ˜ , para quaisquer x , x com f(x ) = y , f(x ) = y , tem-se y /x = y /x . Com Com efeito, efeito, am ˜ igua bos esses quocientes s ao igua is ao fat or de proporci proporcio ona lidade lidade a.
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9
˜ Afins Proporci Prop orciona onalid lidade ade e Func Fu nc¸ oes
unidade” provem e´ m do fato de que a = f(1) e´ o valor valor de f(x) quando x = 1. Deve-se ressaltar enfaticamente que a r egr a de tr es, eˆ s, proven ient e da pr oporc¸a˜ o y /x = y/x , s o´ pode ser legitimamente empregada quan do se tem um a proporci proporcio onalidade f, sendo y = f(x ) e y = f(x ). ˜ a ser feita Ou t r a obser vac¸ao feita e´ qu e, em d iversa s situ ac¸oes o˜ es ond ond e se usa a proporci proporcio ona lidade lidade (ou (ou a regra de tr es), eˆ s), o fator de p roporroporcionalidade a e´ irrelevante e/ou complicado de se obter. No Exemplo 1, o fator de p roporcio roporciona na lidade lidade a = peso / volume, chama do a densidade d o l´ıquido ı quido (ou, mais precisamente, o peso ´ il. il. Assim, peso = densidade volume. espec´ ıfico), ıfico), e´ um conceito ut ˜ tem a menor No Exemplo 3, o fator de proporcionalidade n ao menor ˆ cia. (Por ˆ import an Por acaso ele ele e´ o quociente dos senos dos angulos que a reta r forma com os lados OA e OB, m as est a inform inform ac¸a˜ o e´ uma mera cur iosidade iosidade.) .) No Exemplo 4, e´ costume escrever o fator de proporcionalidade sob a forma a = 1 + i, portanto tem-se y = (1 + i)x. O n u´ m e r o i cha ma-se o juro. juro. Se o investimento inicial x for mantido durante n m eses eˆ ses e os jur jur os se m an tiverem fixos, fixos, tem-se ao final do n-esimo e´ simo m es eˆ s y = (1 + i) x. ´ ˆ z de um ret angulo Qua nt o ao Exemplo 2, ele ele nos diz que a area de altura fixa fixa y (= dist an ˆ cia entre as par alelas alelas r e s) e´ pr oporcio oporciona na l a` b a s e x, logo z = A x, onde o fator de proporcionalidade A e´ a ´ ˆ area do ret angulo de mesma a ltur a y e base 1. Mas e´ clar clar o que o ½
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10
Temas e Problemas
´ 1) Quan to maior maior for x, menor ser a´ y. Em termos termos matem aticos: s e x → y e x → y en t a˜ o x < x ⇒ y < y.
˜ o valor 2) Se dobrarmos, triplicarmos, etc. o valor de x en t ao correspondente de y s er a´ dividido por dois, por tr es, eˆ s, etc. etc. Em ´ ica: linguagem linguagem m atem at ica: se x → y en t a˜ o nx → y/n, para t odo n N.
∈
Portanto, dizer que y e´ inversam ent e proporcio proporciona na l a x equivale ˜ do Teorema a dizer que y e´ proporcional a 1/x. SegueSegue-se se ent ent ao Teorema Fundamental da Proporcionalidade que se y e´ inversamente pro˜ t em-se porcional a x en t ao em-se y = a/x, onde o fator de proporcionalidade a e´ o valor de y que corresponde a x = 1. ˆ gulos ´ E x e m p l o 5 . Entr e os os ret an gulos de ba se x, altura alt ur a y e area igual a 1, tem-se y inversamente proporcional a x, com y = 1/x.
2
G r a n d e z a p r o po p o r c io i o n a l a v a´ r i a s o u t r a s
Em mu itas situa c¸ oes o˜ es tem-se uma grandeza z, de tal modo rrelaelacionada com outras, digamos x, y, u, v, w, que a cada escolha de ´ valores para estas ultimas corr corr esponde esponde u m valor valor bem determina ´ x, y, u, v, w do par a z. E n t a˜ o z cha ma-se uma fun c¸ a˜ o das variaveis e escreve-se z = f(x,y,u,v,w ). Ne st a s con con dic¸oes, o˜ es, diz-se diz-se qu e z e´ (diretamen (diretamen te) proporcional a x quando:
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˜ Afins Proporcionalid Proporcio nalidade ade e Func¸ oes
11
2’) 2’) Par a n N e x, y, u, v, w quaisquer t em-se em-se f(nx,y,u,v,w) = 1 f(x,y,u,v,w ). n
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SegueSegue-se se do Teorema Teorema Fun damen tal da Pr oporci porcio ona lidade lidade que as propriedades 2) e 2’) 2’) acima acima valem pa ra c > 0 real qualquer em lugar de n N. Isto tem a seguinte conseq u¨ encia: eˆ ncia:
∈
S e z = f(x,y,u,v,w ) e´ (diretamente) proporcional a x e y e inversamente proporcional a u, v e w en t a˜ o, tomando-se a = f(1,1,1,1), tem-se
f(x,y,u,v,w ) = a Com efeito,
· u x· v· y· w ·
f(x,y,u,v,w ) = f(x 1,y,u,v,w ) = x f(1,y,u,v,w ) xy f(1,1,1,v,w ) = xy f(1,1,u,v,w ) = u xy xy f(1,1,1,1,w) = f(1,1,1,1,1) = uv uvw xy =a uvw
· · ·
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˜ un iversal, iversal, de Newton, Newton, afirm a que E x e m p l o 6 . A lei da gr a vit a c¸ ao dois corpos, de massas m e m respectivamente, situados a uma ˆ d um do outr dist ancia outr o, se at ra em segundo uma forc¸a cuja intensidade F e´ proporcional a essas massas e inversamente propor
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12
Temas e Problemas
˜ de dois 2 ) S e o s olido o´ lido Y e´ a r e u n i ao dois s olidos o´ lidos adjacentes X e X ˜ vol en t ao vol (Y ) = vol (X) + vol (X ).
˜ d e pr oporcio Dessa s du as pr oprieda des d o vol volum um e, e da defin ic¸ao oporcio-na lidade lidade acima acima da da, resulta que se X e´ um bloco bloco ret an gular cujas ˜ o volum arestas medem medem x, y e z respectiv respectivamen amen te ent ao volum e de X e´ proporcional a x, y e z. Portanto vol (X) = a xyz, onde a e´ o volum volum e do blo bloco reta ngular cujas tr es eˆ s arestas medem medem 1. Mas ta l blo bloco e´ o cubo cubo d e a res ta 1 e, por defin ic¸ao, ˜ seu volume e´ igual a 1. Logo vol (X) = xyz.
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3
F u n c¸ o˜ e s a fi n s
e´ tricas assinalam valores posiE x e m p l o 8 . As escalas termom etricas tivo tivos e n egativ egativo os. Elas se baseiam baseiam na altura de uma coluna de ´ io, mercur io, a qua l aum enta ou diminui confo conform rm e a t emperat ur a sobe o desc Na ala Celsius, Celsius, o valo valo 0 corresponde a` tempe-
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˜ Afins Proporcionalid Proporcio nalidade ade e Func¸ oes
13
´ custa a reais por km rodado E x e m p l o 9 . Uma corrida de t axi mais uma taxa fixa de b reais, chamada a “bandeirada”. Ent a˜ o o pr ec¸o ¸ o de u ma corr corr ida de x k m e´ f(x) = ax + b reais. ˜ afim f(x) = ax + b, o n u´ m e r o b = f(0) chamaNu m a fun c¸ ao s e o valor inicial e o coefici coeficient ent e a = f(1) − f(0) e´ chamado a taxa de va varia¸ ria¸ca˜ o d e f. O m otivo tivo para esta denominac denominac¸ a˜ o e´ q u e , p a r a quaisquer x, h R, com h = 0, tem-se a = [f(x + h ) − f(x)]/h , ˜ de f(x) por unidade donde a = f(x + 1) − f(x), logo a e´ a va r ia c¸ ao ˜ de x. (Compare com o exemplo acima.) de va r ia c¸ ao ˜ linear f(x) = ax e´ u m caso par ticular U m a fu n c¸ ao ticular de fun c¸a˜ o ˜ afim e´ o da s fun c¸oes afim . Outr o caso caso par ticular ticular de func¸ao o˜ es const a n t e s f(x) = b. ˜ afim f(x) = ax + b e´ crescente, isto e, Quando a > 0, a fu n c¸ ao e´ , x < x ⇒ f(x ) < f(x ). Com efeito se x < x en t a˜ o x − x > 0 logo
∈
½
¾
½
¾
½
¾
¾
½
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14
Temas e Problemas
x e x + h , logo seu C-comprimento e´ igua igua l a h . Ora, a medida deste segmento depende apenas de h m a s n a˜ o d e x e o m e s m o s e d a´ com com a difer en c¸a ¸ a f(x + h ) − f(x). Pelo Teorema Teorema de Ca ra cterizac¸a˜ o, ˜ afim: f(x) = ax + b. Sabemos que conclu ´ımos ı mos que f e´ u m a fun c¸ ao f(0) = 32 e f(100) = 212. E n t a˜ o b = 32 e 100a + 32 = 212, donde a = 1,8. Portanto P or tanto f(x) = 1,8x + 32 e´ a f ormula o´ rmula que permite passar d a t e m pe p e r a t u r a x na escala Celsius para a temper at ur a f(x) em graus Fahrenheit. A primeira per gunta do Exemplo 8 era: para qual valor valor de x tem-se f(x) = x ? Deve-se ter 1,8x + 32 = x, donde x = − 40. A resposta e: e´: − 40 graus Celsius e´ o mesmo que − 40 graus Fahrenheit. A segunda pergunta era: para qual valor de x tem-se f(x) = 2x ? E n t a˜ o 1,8x + 32 = 2x e d a ´ı x = 160. Assim 160 g r a u s Celsius Celsius equivalem a 320 graus Fahr enheit. enheit. ˜ afim e´ u m a Provaremos a seguir que o gr a´ fico de u ma fun c¸ao ˆ cia ent re dois reta. Para isso, usaremos a f ormula o´ rmula da dist an dois pont os P (x y ) e Q (x y ), segundo a qual se tem d(P Q)
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˜ Afins Proporcionalid Proporcio nalidade ade e Func¸ oes
15
a cr escimos e´ scimos iguais f(x + h ) − f(x) = f(x + h ) − f(x ). A in clin a c¸ a˜ o ˜ ao eixo horizontal e´ [f(x + h ) − f(x)]/h = d a r e t a G em r ela c¸ao Port ant o, para valores valores m aiores aiores ou ou m enoeno[a(x + h ) − ax ]/h = a. Port ˜ afim f(x) = ax + b e´ mais ou menos res de a, o gr a´ fico d a fun c¸ao inclina do em r elac¸ a˜ o a OX.
G
f ( x’ ) ( x’ +h) – – f h x+ x)) ( x +h) – f ( x b
h
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16
Temas e Problemas
ˆ AP END ICE IC E1 Te o r e m a F u n d a m e n t a l d a P r o p o r c i o n a l i d a d e . S eja f : R → R um a fu n c¸ ao ˜ com as seguintes propriedades: ·
1) x < x
·
⇒
f(x) < f(x );
2) f(nx) = n f(x) para todo n
· ∈ N e todo x ∈ R . Ent a˜ o f(cx) = c · f(x) para todo c ∈ R e todo x ∈ R . Conseq u¨ entem en en te, f(x) = ax para todo x ∈ R , com a = f(1). ·
·
·
·
´ De m on st rac¸ a˜ o : Em primeiro primeiro lugar lugar,, par a todo todo n umero racional r = m/n, com m, n N, e todo x R vale
∈
∈
·
n f(rx) = f(n rx) = f(mx) = m f(x),
·
por 2), logo f(rx) =
·
·
m f(x) = r f(x). Assim, a igualdade f(cx) = n
·
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˜ Afins Proporcionalid Proporcio nalidade ade e Func¸ oes
17
ˆ AP ENDICE 2 Teo rem a d e Cara cte riza c¸ ao ˜ da s Fu n c¸ o˜ e s A fi n s . S eja f : R → R crescente ou decrescente. S e a diferen ca ¸ f(x+h )− )− f(x) depende apenas de h m as n ao ˜ de x , en t a˜ o f e´ u m a fun c¸ao ˜ afim . emos a penas do caso caso em que f e´ crescenDe m on st rac¸ a˜ o : Tratar emos ´ go.. Pela te pois o outro e´ a n alogo alo Pela h ipotese o´ tese feita sobre f, a fun c¸ a˜ o
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18
Temas e Problemas
Problemas Propostos∗ coplana ana res. Para cada segmento segmento de reta AB 1 . Sejam r, s retas copl ˜ ort ogona cont cont ido em r, seja A B su a pr ojec¸ao gona l sobre s. Prove que o co compr iment o de A B e´ pr oporcio porciona na l a o de AB.
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˜ Afins Proporcionalid Proporcio nalidade ade e Func¸ oes
19
possu i ra c¸a˜ o s u fi ci ci e n t e p a r a a l im im e n t a r s u a s 7 . Um fazendeiro possu 16 vac vacas as duran te 62 dias. dias. Apos o´ s 14 dias, ele vende 4 vacas. Passados mais 15 dias, ele ele co compra 9 vacas. vacas. Quan tos dias, dias, no total, dur ou sua reser va de r ac¸a˜ o?
8 . Uma caravana com 7 pessoas deve atravessar o Sahara em 42 ´ dias. Seu suprimento de agua permite que cada pessoa disponha de 3,5 3,5 litros litros por por dia. dia. Ap os o´ s 12 dias, a car avana encontra 3 be-
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20
Temas e Problemas
reta r. Chamando C hamando de x e z os compr iment os de OX e XZ respectivamente, mostre que a correspond encia eˆ ncia x → z e´ uma proporcionalida de. Em qu e condic¸ oes o˜ es o fator de proporcionalidade e´ o mesmo que o da corr corr espondencia eˆ ncia x → y do Exemplo 3?
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C a p ´ıtu ı tu lo 2
´ F u n c¸ oes o˜ es Quadr aticas
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22 Temas e Problemas
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˜ Quadr aticas ´ Func¸ oes
23
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62 Temas e Problemas ia v i d a d e u m m a t e r ia ia l r a d io ioa t i vo vo q u e s of ofr e 5 . Qual e´ a m e ia ˜ de 20% de sua m assa em um per´ ´ıodo de 1 ano? de sin t egr a c¸ ao per ıodo ` 23 hocorpo de um a v´ıtima ı tima de assassinato foi descoberto as 6 . O corpo ` 23:30 r a s . O m edico e´ dico da pol´ pol´ıcia ı cia chegou as 23:30 e imediatam ente t omou ´ a temperatu ra do cad aver, aver , que era de 34,8◦. Uma hora mais tarde ele ele t omou mou a temperatura outra vez vez e enco encontrou 34,1◦ . A t e m p ee◦ r a t u r a d o qu qu a r t o e r a m a n t i da d a con s t a n t e a 20 . U s e a l ei ei d o r e s friament o de Newton Newton para estimar a hora em que se deu a m orte. Admita que a tempera tur a n orma l de uma pessoa pessoa viva viva e´ 36,5◦ . ´ de um reser vat orio o´ rio se evapora a` taxa de 10% 10% ao m es eˆ s . 7 . A agua Em quan to tempo ela ela se r eduzir eduzir a´ a um ter c¸o ¸ o do que era no in´ in ´ıcio? ıcio? famosa pelas pelas pintur as feita eita s por 8 . Em um a caverna da Fra nc¸a, famosa ˜ vehomens pr e-hist e´ -hist oricos, o´ ricos, fora fora m encontr ad os peda c¸os ¸ os de car vao vegetal, nos quais a radioatividade de C er a 0,145 vezes vezes a r adioati˜ feito hoje. Calcule a idade do carva˜ o vidade nu m peda c¸o ¸ o de car vao e d eˆ uma estimativa para a epoc e´ poca em que a s pintu ra s fo fora m feita feita s. ½
Fora m injetadas 20 mg de uma certa droga droga em um paciente. paciente. A 9 . Fora ˆ ea de elimina c¸ ao ˜ da droga, taxa instant an droga, imedi imediatam atam ente a p os o´ s a i n je c¸ a˜ o, e´ de 5 mg por por hora. Qual e´ a meia-vida da droga? (Cuidado! A resposta n a˜ o e´ 2 horas.) ˜ da Figura 29 foi desenhado utilizando-se 1 0. O gr a´ fico d a fun c¸ao uma escala logar´ logar ´ıtmica ı tmica para o eixo Y (ou seja, as ordenadas no gr afico ´ repr esen ta m o logar logar itm o decima decima l dos dos valores d a func¸ao). ˜ a) Mostre Mostre que o gr gr a´ fico de u ma fun c¸a˜ o f neste tipo de represent a c¸ a˜ o e´ uma reta se e somente se ela e´ do tipo exponencial (f(x) = ba ). Ü
˜ representa da pelo ´ o da figura ? b ) Q u a l e´ a fu n c¸ao pelo gr gr afico afic 1 1. No problema da piscina (Problema 1), verifique que a taxa ˆ ea de var iac¸ ao ˜ da quantidade de cloro no instant e t e´ instant an
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˜ Exponenci Func¸ oes Exp onenciais ais e Logar´ Lo gar´ıtmicas ıtmica s
Y 10000 1000 100 10 X
1 0
1
2
3
Figura 29
4
5
63
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C a p ´ıtu ı tu lo 4
Ap l i ca c¸ oes o˜ es da Trigonometria ´ ´ Os livros did aticos para o ensino m edio e´ dio dedic dedicam muita s p aginas ˜ fica ao ensino da trigonometria. trigonometria. En treta nto, n ao fica claro nem pa ra o aluno, nem pa ra o prof professor, essor, par a que serve este abunda nte mat erial. Vam os m ostr ar aqu i alguma s a plic plicac¸oes o˜ es em sit u a c¸ oes o˜ es reais e, par a resolver resolver os pr oblemas, necessitar emos apen as das relac¸oes o˜ es ˆ gulo ret an ˆ gulo, trigonom etricas e´ tricas n o tri an gulo, da lei dos cossenos cossenos e da lei dos senos. Ne st a s a plica c¸ oes o˜ es estaremos calculando senos, cossenos e tanˆ gent gent es de angulos e cabe aqui um esclarecimento ao leitor. Quanˆ do escrevem escrevem os por por exemplo sen 30 , querem os dizer seno do angulo ˆ cuja medida e´ 30 , ou seja, estamos identificando o angulo com ˆ gulos s u a m e di d i d a . I s t o e´ p r a´ t i co co e n a t u r a l . P a r a an gulos agudos, agudos, es˜ definidas t a s fu n c¸ oes o˜ es tr igonom igonom etricas e´ tricas s ao definidas atr aves e´ s das tradicionais ˆ ˆ r a zo˜ e s e n t r e l a d o s d e u m t r i angulo ret angulo e, par a qualquer ˆ gulo obtuso ˆ an obtuso x (quer dizer: angulo cuja medida x es t a´ e n t r e 90 e 180 ), defin imos imos sen x = s en (180 − x) e cos x = − cos (180 − x). E isto e´ tu do o que pr ecisamos. ecisamos. Desde a a ntiguidade e at e´ hoje, hoje, o homem sem pre teve a necesˆ cias inacess ˜ muito sidade de avaliar dist an inacess´´ıveis. ı veis. Na verdade, s ao ˆ poucas as dist ancias que podem ser medidas diretamente, com uma trena, por exemplo. Praticamente tudo que o desejamos saˆ ber sobre dist ancias anci as no mu ndo em que vivemo vivemoss e´ calculado com o a u x´ılio ı lio da t rigonometr rigonometr ia. ´ O problema b asico, e que estar a´ sempre presente presente em todas as ˆ s it u a c¸ oes, o˜ es, e´ o da r es oluc¸a˜ o de d e u m t r iangulo. Mas, o que significa ˆ ˜ seus lados isto isto?? Os elemento elementoss pr incip ncipais ais de um tri angulo s ao lados e ˆ gulos. ˆ gulo signific seus an gulos. Resolv Resolver er u m tr ian significaa det erm inar 3 desses ◦
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˜ da Trigonometria Apli Ap lica cac c¸ oes
65
˜ pode ser imposs´ t e r u m a u´ n ica solu c¸ao, imposs ´ıvel ı vel ou pode ter mais de ˜ e voc u m a solu c¸ ao voceˆ poder a´ ver ificar ificar isto nos nos pr oblema blema s que va mos discutir. ˆ Para medir uma dist ancia inacess´ inacess ´ıvel ı vel necessitaremos necessitaremos de uma que nada mais e´ q u e u m a fi t a m etrica e´ trica comprida que possa trena, trena , que medir dist an ˆ cias r elat elat ivamen ivamen te pequena s n o plan plan o horizo horizont nt al e de u m teodolito. ˆ teodolito. Um teodolito e´ um instrumento que mede angulos, tant o no plano horiz horizo ontal quanto n o plano plano ver vertic tical. al . Tr Trata-se ata-se de uma luneta, apoiada apoiada em um trip e´ que pode fornecer os seguintes dados:
a) Se o observador T veˆ um objeto P, ele pode determinar deter minar o ˆ gulo an gulo que que a reta TP faz com o plano h orizont rizont al.
P
T
θ
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66
Temas e Problemas
b) se o observador T veˆ um objeto A e gir girando ando a luneta v eˆ u m objeto B, ambos no plano horizontal, ele pode determinar o angulo ˆ ATB.
T
B θ
A Figura 31
˜ instrumentos equivalentes a` r egua A trena e o teodolito s ao e´ gua graduada e ao tr ansferi ansferido dorr quando trabalhamos no papel. papel. A tretr ena de hoje hoje e a da an tiguidade tiguidade diferem diferem a penas do mat erial em que ˜ o mesmo instrumenforam constru ´ıdas ı das mas essencialmente, s ao to. to. En tret ant o, o teodo teodoli lito to de h oje e´ muito mais sofisticado que o equiv equivale alente nte antigo. antigo. E neste ponto ponto est a´ a diferen diferen c¸a. ¸ a. Hoje, Hoje, poˆ ˜ muit ´ıssimo demos medir angulos com uma precis ao ı ssimo maior maior do que antigamente. Var ´ ios ios pr oblemas blemas que vamos a borda borda r fazem r efer eferˆencia eˆ ncia a` cida˜ de de do Rio de J an eiro eiro.. O m orr o do Corco Corcovado vado,, o m orr o do P ao ´ ar , o aterr o do Flamengo e sua vista Ac¸ ucar, uc vista a` cidade de Niter oi o´ i do outro lado da Ba´ Ba ´ıa ı a de Guan abar a forn eceram eceram situac¸oes o˜ es interessan tes de medidas inacess´ inacess´ıveis. ı veis. Nestes problemas as medidas s a˜ o todas r eais. eais. ´ P r o b l e m a 1 . Medir a altura do P a˜ o d e Ac¸ ucar. Medir Medir a altur a de um morro distan distan te em relac¸a˜ o a u m p la la n o horizont horizont al pr oximo o´ ximo e´ um problema problema perm an ente em toda a hist oria. o´ ria.
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˜ da Trigonometria Apli Ap lica cac c¸ oes
67
´ ssempre empre e´ poss´ poss ´ıvel. ı vel. Ma s, no caso do P a˜ o d e Ac¸ ucar o ater ro do Flamen go fo forn ece ece u m plano h orizont rizont al especial par a este objetivo objetivo.. Enunciado: Um observador est a´ em um ponto A do aterr o do Fla´ ˆ gulo de 10 com men go e v eˆ o P a˜ o d e Ac¸ ucar segundo um an com o pla no ˜ ao seu horizonta l (medido com com o teodolito) teodolito).. Ele a nd a em direc¸ ao objetivo at e´ u m p o n t o B distante distante 650 650 m de A e a g o r a veˆ o P a˜ o ´ ar segundo ˆ gulo ˜ de d e Ac¸ ucar uc segundo um an gulo de 14 . Q u a l e´ a a l t u r a d o P ao Ac¸ u´ car em r elac¸a˜ o ao p lan o de obse r va c¸a˜ o? ◦
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ˆ cia de um ponto ponto do Rio Rio de Ja neiro a P r o b l e m a 2 . Medir a dist an ´ıvel de Niter oi. um ponto vis vis´ıvel o´ i. O segundo problema importante e´ o de medir a dist ancia ˆ de um ponto ponto a outr o ina ina cess´ ess ´ıvel ı vel no plano horizontal. Para calcular a ˆ dist ancia anci a de um ponto ponto A (onde (onde est a´ o observador) a um ponto P, ´ıvel, e´ preciso que este observador possa se locomover para inacess´ inacess ıvel, um ponto ponto B no plano horizo horizonta l de onde possa t amb em e´ m ver P. Enunciado: De um ponto ponto A na praia do Flamengo Flamengo no Rio Rio de J aneiro, avista-se um ponto P n a p r a ia i a d e I ca ca r a´ a ´ı em Niter oi o´ i (estes ˜ em lados oposto dois pontos est ao opostoss do can can al de ent ra da da Ba´ Ba ´ıa ı a de Guanabara). De um ponto B na Pr aia do Flamengo, Flamengo, distan distan te 1 km d e A t a m b em e´ m se avista o ponto P (Figur Figur a 32). 32). Um observador observador n o ˆ Rio de Janeiro mediu os angulos BAP = 119 e ABP = 52 . Qual e´ ˆ a dist ancia entre A e P? ◦
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68
Temas e Problemas
Baía de
RIO DE JANEIRO
P
Guanabara
Icaraí
A
NITERÓI
Flamengo B
Figura 32
ˆ XAY = 62 , YAB = 54 , ABX = 46 e XBY = 74 . Q u a l e´ a dist ancia e n t r e X e Y ? ◦
◦
◦
◦
Medir o raio da Terr Terr a. P r o b l e m a 4 . Medir Desde a an tiguidade, este pr oblema est eve presen te n a cabec¸a ¸a dos matem a´ ticos. Divers a s s olu olu c¸o˜ es apar eceram eceram mas os rresul esultata¨ ˜ eram bons dos freq uentemente n ao bons pois pois se exig exigia ia a medida entr e
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˜ da Trigonometria Apli Ap lica cac c¸ oes
69
d e 0,85 com o plano horizontal. Encontre uma medida aproximada par a o raio da Terra Terr a . ◦
P r o b l e m a 5 . Ainda o ra io da Terra . U m a b e la t en e n t a ti t i v a d e m ed e d ir ir o r a io io d a Te r r a d ev ev e -s e a E r a t o´ sten es n o terceiro terceiro s eculo e´ culo ant es de Cr isto. isto. Medidas fora fora m feitas nas cidades de Assu a˜ e Alex Alexan an dria, n o Egito, Egito, que que est ao ˜ a proxiproximada mente no mesmo meridiano meridiano t errestr e, e por por ra ra felic elicidade idade,, ˆ cer. Assu a˜ est a´ qu ase sobre o tr opico o´ pico de C an er. Ist o quer quer fizer fizer que no ˜ , a o meio dia, ˜ perfeitaprimeiro dia do ver ao, ao dia, os ra ios ios solares solares s ao perfeitamente verticais. verticais. Na quele tempo, tempo, uma unidade comum comum par a m edir edir ˆ cias gran des era o estadio. ´ ´ dist an adio . O est adio era o co compr iment o da pista de corrida utilizada nos jogos ol´ ol´ımpicos ı mpicos da antiguidade (de 776 a 394 aC.) e era equivalente a 1/10 de milha, ou seja, aproximadamente 161 161 m. solst ıcio ´ıcio de ver a˜ o , E r a t o´ sten es verifico verificou Enunciado: No dia do solst que, a o meio dia, o sol sol brilhava diretament e dentr o de um poc poc¸o ¸o ´ profun profun do em Assu a˜ e, em Alexandria, a 5000 est adios ao norte de ˜ algu em ˆ Assu a, e´ m mediu o angulo que os raios solares faziam com a vertic vertical, al, enco encontr an do 1/50 1/50 do c´ c´ırculo ı rculo.. Com Com base n estes dados, calcule o ra io da Terr a. P r o b l e m a 6. 6 . O problema da corrida. ˜ os seOs dados e o objetivo deste interessante problema s ao
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70
Temas e Problemas
r
X
A
?
B Figura 33
2) Considere Considere BAX = 110 , velocidade de A igua igua l a 8 m/s, velo velocidade de B igua igua l a 8,1 8,1 m/s e AB = 50 m. Sendo B um corˆ cia ele perco redor redor intelige inteligente, nte, determ ine que dist an percorr eu a t e´ a lca n c¸a ¸a r A. ◦
P r o b l e m a 7 . Novamente a corrida, mas um fato muito estranho acontece. Enunciado: Considerando ainda a Figura 33, seja BAX = 60 . O corredor A tem velocidade 15% maior que a de B. P or or e´ m , o corredor B e´ int elig eligente, ente, planejou planejou cuidadosamen uidadosamen te sua tr ajet oria, o´ ria, ˆ e a lcan c¸ou ¸ ou o corr edor A no ponto C d a r e t a r. Calc Calcule o angulo ABC. ◦
ˆ ABC. Observa¸ca˜ o: voceˆ vai encontrar dois valores para o angulo ´ıveis? Ambos s ˜ ı veis? Por que ocorre isto?
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˜ da Trigonometria Apli Ap lica cac c¸ oes
71
constr u ´ıda ı da em um plano horizo horizonn3 . Uma estrada que est a´ sendo co tal e ser a´ formada pelos trechos retos XP, PQ e QY como mostra a Figura 34. No trecho PQ s er a´ constru ´ıdo ı do um t unel ´ para atravessar a montan montan ha. Os engenhei engenheiro ross devem devem saber t anto em P q u a n t o ˜ devem ´ AB de forem Q, q u e dir ec¸ ao devem tomar para constr uir o t unel ˜ fixaram ma que o tr echo echo PABQ seja reto. Eles ent ao fixaram um ponto C d o ´ıvel ta nto de P quanto de Q e det plano h orizont rizont al, vis´ vis ıvel deteerminara rminara m as seguintes medidas: CP = 1,2 k m , CQ = 1,8 k m e PCQ = 27 . ˆ Calcule os angulos CPQ e CQP. ◦
Q B x
P
A y
C Figura 34
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72
Temas e Problemas
´ cume do Corcovado e p e´ d a e s t atua do Cristo Cristo Redento Redentorr. Sendo ˜ da perpen dicular dicular tr ac¸ada por C ao plano horizonP a in t er se c¸ ao tal que cont cont em e´ m A e B, considere os seguintes dados: AB = 660 m , CAP = 29,7 , CBP = 30,6 , PAB = 70,5 , PBA = 77,9 . Calcule Calcule a altu ra do morro do Corco Corcovado. vado. ◦
◦
◦
CRI RIST STO O R ED EDEN ENTO TOR R
P
◦
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C a p ´ıtu ı tu lo 5
U m a In t r odu c¸ a˜ o a o ´ C alculo de Volumes Para introduzir o conceito de volume, o professor deve, antes de ˜ fo qua lquer t ent at iva iva de um a defin ic¸ao form al, apresentar um a idei e´ i a intuitiva e fornecer diversos exemplos para que os alunos possam compreender mpreender do que que vai se falar. falar. E qual e´ a primeira coisa que ˜ nos ocorre nenhuma outra frase melhor que devemos dizer? N ao a seguinte: Volume de um s´ olido e´ a quantidade de espa¸co c o por ele
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74 Temas e Problemas ˜ das saber o volume volume de concreto concreto que ser a´ ut ilizado ilizado n a cons cons tr uc¸ ao ´ıcio.. A fo colunas, vigas vigas e lajes de u m edif edif ıcio forma e a s dimens oes o˜ es de cada um destes objetos objetos est ao ˜ na plant a e o c´ calculo ´ do volum volum e deve ˜ pequenos ser feito eito an tes qu e o edif ´ıcio ı cio exista. Alguns objetos s ao ˜ inacess demais, ou grandes demais, ou s ao inacess´´ıveis ı veis ou, simplesmen˜ existem ˜ a necessi te, n ao existem concretamen te. Sentimos Sentimos en t ao necessidade dade de ´ obter m etodos e´ todos para o calculo de volumes, pelo menos de objetos simples, conhecendo sua forma e suas dimens oes. o˜ es. Para medir esta grandeza chamada volume, devemos compar a´ la com uma unidade e, tradicionalmente, a unidade de volume e´ o cubo cuja aresta mede uma unidade de comprimento, denominado de cubo unit ar ´ io. io. P or exemplo, exemplo, se um cubo tem 1 cm cm de ar esta , seu volume e´ a u nidade cham cham ada de cent´ ımetro cubico ´ (cm ). ¿
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˜ ao Calculo ´ Uma Um a Intro In trodu duc c¸ ao de Volumes
1
75
O v o lu l u m e d o b lo l o c o r e ta t a n g u la la r
Imaginemos inicialmente umm bloco retangular com dimens oes o˜ es 4 cm, 3 cm cm e 2 cm. cm. Qua l e´ o seu volume? volume?
2
3 4 Figura 37
Observando o desenho, n ˜
h ´ d uvida ´ que este bloco pode ser
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76 Temas e Problemas
2
4,7 5,6 Figura 38
1
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˜ ao Calculo ´ Uma Um a Intro In trodu duc c¸ ao de Volumes
77
´ıtulo). final deste cap´ cap ıtulo). Para o caso caso gera gera l, onde onde a s m edidas edidas das ar estas do blo blocco retan ´ gular s a˜ o n umeros reais positivos quaisquer, o volume e´ a in in d a o produto dessas dessas medidas e, para demonstr demonstr ar, usar emos emos o teorema teorema fun dam ent al da proporcio proporciona na lidade. lidade. O roteiro par a a demonstr ac¸a˜ o es t a´ no Problema Problema 2. Considerem Considerem os port an to esta belec belecido que o volume volume de u m bloc bloco retangular cujas arestas medem medem x, y e z, e´ da do por por V = xyz.
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78 Temas e Problemas Seja V o volume de S e seja v(P) o volume de um poliedro retangular P cont cont ido em S. O n u´ m e r o V n a˜ o e´ ainda conhecido mas deve satisfazer a` con d ic¸ a˜ o v(P) ≤ V para todo poliedro retangula r P cont cont ido em S. Par a cada poli poliedro edro retan gular P cont cont ido em S, ˜ igual a S, e´ poss´ m a s n ao poss ´ıvel ı vel sempre obter um poli poliedro edro reta ngula r P , ma ior ior que P e a inda contido em S. Basta B asta acrescentar a P novos blocos retangulares que ainda estejam dentro de S. Portan Portan t o, v(P) < v(P ) ≤ V , o que qu er dizer qu e v(P) e´ u m a a pr oxima oxima c¸ a˜ o
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˜ ao Calculo ´ Uma Um a Intro In trodu duc c¸ ao de Volumes
S’
S
z y
kz
79
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80 Temas e Problemas Esta id eia e´ ia inicial ja´ nos conduz a dois importantes resultados. a ) D o i s p r i s m a s d e m e s m a b a s e e m e s m a a l t u r a t em eˆ m mesmo volume volume (Figura 43). 43).
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˜ ao Calculo ´ Uma Um a Intro In trodu duc c¸ ao de Volumes
B
A
S
S A h
B
81
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82 Temas e Problemas
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122 Temas e Problemas O valor valor de um a perpetuid perpetuid ade d e term term os iguais a P, um tem tem po antes antes P do prim eiro pagamento, pagamento, e, ´ sendo i a taxa de juros, A = . i
E x e m p l o 4 . Se o dinh eiro eiro vale 1% a o m es, eˆ s, por por quan to deve ser alugado um im ovel o´ vel qu e va le R$40 00,00? 00,00?
Qua ndo voc voceˆ aluga um imovel, o´ vel, vocˆ voceˆ cede a posse do im ovel o´ vel em ˜ igua troca troca de u ma r enda perpetua e´ tua cujos termos s ao igua is ao valor valor do ˜ o valor aluguel aluguel.. Ent ao, valor do im o´ vel deve deve ser igua igua l ao valor valor da serie e´ rie de alugu eis. e´ is. P Logo, com com o A = , tem os P = Ai = 4000 × 0,01 = 400. i Deve ser alugado por R$400,00.
E x e m p l o 5 . Helena Helena tem dua s alternat ivas ivas pa ra obter uma copiadora:
´ a) Aluga-la por R$480,00 por m es. eˆ s. N esse caso, o locador locador se responsa biliza biliza pela s despesa s de man ut enc¸a˜ o. ´ b) Compr a-la por R$8 000,00. Nesse caso, ja´ que a vida econ ooˆ mica mica da copiadora e´ de 2 an os, Helena Helena vender vender a´ a copiadora a p os o´ s 2 an os, os, por R$1 000,00. 000,00. As despesa s de m an ut enc¸a˜ o s a˜ o ˜ de R$100,00 por m es de responsabilidade responsabilidade de Helena e sao eˆ s no primeiro an o e de R$150,00 R$150,00 por m es, eˆ s, no ano seguinte:
˜ para Helena? Se o dinheiro vale 1% ao m es, eˆ s, qu a l a m elhor opc¸ao Helena?
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123
˜ de Matem atica ´ Noc¸ oes Financeira
i) um a p ar cela de R$8 000,00; 000,00; ii) ii) o valo valorr atua l de uma s erie e´ rie de 12 pa gamentos de R$100, R$100,00, 00, 1 − 1,01 igual a 100 125,51 51; = R$ 1 125, 0,01 iii) o valor, na epoc e´ poca da compr compr a, dos gastos no segun segun do ano. Pa ra determ in a-lo, a-lo ´ , calcula calcula mos o valor at ua l dos dos gast os no segun do 1 − 1,01 688,26, e dividimos esse valor por = 1 688,26 ano, 150 0,01 ´ obtendo finalmente eˆ -lo um ano para tr as, 1,01 , p a r a t r a ze-lo R$1 498,25; ¹
½
¾
¹
½
½
¾
¾
iv) o valor, valor, na epoca e´ poca da compra, da receita auferida com a venda, ´ isto e, R$1 000,00 trazidos dois anos para tr as, e´ , 1000 ÷1,01 = 787,57. ¾
125,51 + 1 498, 498,25 25 − 787,57 = Portanto, os gastos s a˜ o d e 8000 + 1 125,51 9 836,1 836,19 9. Na alternativa a), o valor dos gastos na epoca e´ poca da compra e´ o v a l o r a t u a l d e u m a s erie e´ rie de 24 pagamentos iguais a R$480,00, 1 − 1,01 480 196,83. = R$ 10 196,83 0,01 A melhor melhor alterna tiva e´ a compra. ¹
¾
Problemas Propostos∗ 1 . Um t elevis or, or, cu jo pr ec¸o a` vista e´ R$900,00, e´ vendido em dez ˜ pa gos pr es t a c¸ o˜ es men sais igua igua is. Se s ao gos jur os de 4% ao m es, eˆ s, det er˜ paga: min e o valor da s p rest ac¸oes, o˜ es, su pondo a pr imeira pr esta c¸ao
a) no ato da compra compra . b ) u m m es eˆ s ap o´ s a compr a. c) dois dois meses ap o´ s a compr a.
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124 Temas e Problemas 2 . Se a taxa de juros juros e´ de 0,6% 0,6% ao m es, eˆ s, por quanto se aluga um im ovel o´ ve l cu jo pr ec¸o a` vista e´ R$80 000, 000,00, 00, supondo o aluguel m ensa l pago venci vencido do?? E se fo fosse pago adiant ada men te? 3 . Supondo juros juros de 1% ao m es, eˆ s, quan to voc voceˆ deve investir mensalmente salmente durant e 10 a nos nos par a obter a o fim desse desse pra zo, zo, por por 30 an os, uma renda mensal de R$500, R$500,00? 00? 4 . Su pondo jur jur os de 1% ao m es, eˆ s, quan to voc voceˆ deve investir men salment e duran te 35 an os par a obter, bter, ao fim fim desse prazo, prazo, uma renda perp etu e´ tu a de R$1 000,0 000,00? 0? 5 . Considere uma renda perp etua e´ tua cujos termos crescem a uma taxa constante j e cujo primeiro ter mo e´ igua igua l a P . Supondo juros de taxa i (i > j), determine o valo valorr da renda na epoca e´ poca do primeiro pagamento. 6 . Minha mulher acha que devemos vender o carro novo que compr am os por R$18 000,00 000,00 qu an do ele estiver com com dois dois a nos de ´ıamos vend e-lo ´ıamos uso. uso. Co Consegui nseguirr ıamos eˆ -lo por R$14 000,00 000,00 e compr compr ar´ ar ıamos outr o igua igua l, zero quiloˆ metro. Eu acho que seria melhor melhor esperar quatro anos para vender o carro, caso em que s o´ cons cons eguir´ eguir ´ıamos ı amos R$10 000,00 000,00 na venda, m esmo levando levando em cont cont a qu e gasta r´ıamos ı amos em consertos cerca de R$1 000,00 no terceiro ano e de R$2 000,00 no quart o ano. Supondo que o dinheiro dinheiro valha 15% ao ano, quem quem tem raz a˜ o?
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˜ de Matem atica ´ Noc¸ oes Financeira
125
ˆ AP ENDICE C o m o c a l c u l a r a t a x a d e j u r o s u t i li li z a n d o o E x c e l
Para calcular a taxa de juros em s´eries e ries unifo uniform es de pagamen tos, inicialmente, deve-se clicar na tecla do menu f . Ü
˜ a parecer Com e st a oper a c¸ao parecer a´ na tela: tela:
Figura 51
Role o cursor no quadro a` esquerda e clique em Financeira, como mostra a Figura 52.
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126 Temas e Problemas
Figura 52
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˜ de Matem atica ´ Noc¸ oes Financeira
127
´ ´ Aparecer a´ u m a c a i x a d e d ialogo e ser a´ necess ario preencher preencher algumas janelas: N p e r coloque nesta lacuna o n u´ m e r o t ot ot a l d e t e r m os os d a s erie e´ rie uniforme. P g t o coloque nesta lacuna o n u´ m e r o t ot ot a l d e t e r m os os d a s erie e´ rie uniforme. VP preen cha este qu adr o co com o valor valor presen te (valor (valor at ua l), l), co co m ´ sinal contr ario ao pagamento pagamento.. Se o VF VF e´ preenchido esta c´ celula e´ lula deve ficar ficar em br an co.
´ Vf preencha este qu adr o co com o valor valor fut fut ur o, co com sinal cont cont r ario a o p a ga ga m e n t o. o. S e o Vp Vp e´ preenc preenchido hido esta esta celula e´ lula deve ficar em branco. ´ ero 0 ou 1, confo Tipo e´ o n um conform rm e os paga men tos sejam sejam postecipostecipados ou antecipados. Se for deixado em branco, o Excel assumira´ 0, considera considera ndo os pa gamen tos postecipados. postecipados. E s t i m a t i v a e´ a sua estimativa estimativa pa ra a t axa. Deixe Deixe em branco. branco.
o´ gica do contador”, na Obs e rv ac¸ a˜ o . O Excel trabalha com a “logica ´ qua l os os paga men tos e os os recebimen recebimen tos devem devem t er sina is contr contr arios. Logo Logo, se o valor valor pr esent e e´ um valor p ositivo, ositivo, o va va lor da s pr est a c¸oes o˜ e s e´ obrigatoriam obrigatoriam ent e nega tivo. tivo. E x e m p l o 1 . Qual e´ a t axa de juros juros na compra de um ve´ ve´ıculo ı culo cujo p r e c¸ o a` vista e´ de R$8 000,00 e e´ pago em 24 pagament os men sais
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128 Temas e Problemas E x e m p l o 3 . O Excel tamb em e´ m calcula taxas de juros em s eries e´ ries ˜ n ao-uniformes. Vejamos como calcular a taxa de juros ao ano do finan ciamento a seguir: seguir:
An o Valor
0 80
1 50
2 50
3 0
4 − 40
5 − 40
6 −60
7 −70
Os valores est ao ˜ em milhares de reais, as entradas de capital fora m considera considera das positiv positivas as e as sa´ sa ´ıdas, ı das, negativas. Inicialmente devemos colocar os valores do fluxo em celulas e´ lulas adjacentes de uma mesma coluna da planilha, por exemplo, nas celulas e´ lulas de B1 a B8. Procedendo Procedendo co como an teriorment teriorment e, usamos os comandos f , Finan ceira eira e TIR (enc (enco ontr a-se a-se imediatam ente a pos o´ s TAXA). ´ go.. Par preench ˜ digite Aparecer a´ uma caixa de dialogo alo preench e-la, eˆ -la, n ao ˜ esquerdo do mouse aperta do, nada. Com o bot ao do, cubra cubra as celulas e´ lulas na s qu ais foi colo coloccado o fluxo de caixa, n o caso caso a s celulas e´ lulas de B1 a ˜ dentro de um ret angulo ˆ B8. Elas ficar ao com efeito de movimento ´ go se p reencher a´ sozinha, aparecendo: na borda e a caixa de di´ dialogo alo VALORES B1:B8 TIR(B1:B8) = 0,031826856 A taxa e´ de 3,18% 3,18% ao an o. Ü
Problemas Propostos∗
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˜ de Matem atica ´ Noc¸ oes Financeira
129
´ s de certa 3 . Uma caixa de funcion arios ario certa empresa empr esta dinheiro a seus a ssoc ssociados iados e calcula alcula os juros de modo pec peculi uliar. ar. Para um empr estimo e´ stimo de R$1 000,00, para pagamento em 5 vezes, os juros ˜ de “3% ao m es”, s ao eˆ s”, isto e, e´ , 15% em 5 meses. Portanto, o total a ser pago e´ de R$1 150,00, ou seja, 5 p re st a c¸oes o˜ es de R$230,00 cada. Qual e´ na realidade realidade a ta xa de juros juros co com que t raba lha a caixa?
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 1
133
S o l u c¸ o˜ e s d o s P r o b le l e m a s d o C a p ı´t u l o 1 1 . Caso particular do Exemplo 3. ˆ o´ rmula da a´ r e a d e u m t r iangulo, o resultado e´ 2 . Admitindo a f ormula ˆ imediato. O fat fat or de pr oporcio porciona na lidade lidade e´ a m e t a d e da d a d is is t ancia ˆ d e P a r, que e´ a medida co comu m das a ltur as de todos todos esses triangulos. Deste modo, o exerc´ exerc´ıcio ı cio se torna uma conseq u¨ encia eˆ ncia (indireta) ´ ˆ gulo resulta da area ´ do Exemp lo 2, pois pois a f o´ rmu la da area do tr triian ˆ do ret angulo. ´ ea do par alelo alelogramo que t em OX e OY como la3 . Seja f(x, y) a ar dos. dos. A prova de qu e f(x, y) e´ proporcional a x e y se faz faz exatam enˆ gulo. te como n o caso do ret an gulo. A const an te de proporcio proporciona na lidade lidade ´ ea do lo ˆ gulo e´ a = f(1, 1) = (ar losan go de lado lado 1 e an gulos internos α e 180 − α) = s en α. Portanto P or tanto f(x, y) = (s en α)x · y. N a˜ o e´ preciso tabela bela de senos senos n em calc calculadora uladora par a responder responder a ultima ulti ´ ma pergunta pergunta do exerc´ exerc´ıcio ı cio.. Saben do que a · 6 · 7 = 29, obtemos a = 29/42. Logo f(2, 3) = (29/42) · 2 · 3 = 29/7. ve r ifica ifi ca c¸ ao ˜ de que o volume f(x,y,z ) do pa ra lele lelep p ´ıpedo ı pedo que 4 . A ver t em OX, OY e OZ como a resta s e´ proporcional a x, y e z, se faz do mesmo modo que que no caso caso do bloc bloco reta ngular. Tem-se port port an to f(x,y,z) = axyz, onde o fat or de pr oporcio porciona na lidad lidad e a = f(1,1,1) e´ ˜ sobre o volume do paralelep´ paralelep ´ıpedo ı pedo cujas arestas medem 1 e est ao OB = α, AOC OC = β e BOC OC = γ as semi-retas OA, OB e OC. S e AOB
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134 Temas e Problemas significa dizer que f(t + h ) − f(t), espa c¸o ¸ o per corr corr ido n o int er valo de ˜ de t. A co tempo [t, t + h ], depende depende apenas de h , m a s n ao constant e v = f(t + 1) − f(t), espac¸o percorr percorr ido na un idade de t empo, cha cha ma s e a velocidade do ponto m ovel. o´ vel. Pelo Teorema Teorema de Ca ra cteriza c¸a˜ o da s F u n c¸oes o˜ es Afins, se pusermos b = f(0) = abscissa abscissa do ponto ponto de partida, teremos, para todo t, f(t) = vt + b, desde que saibamos qu e f e´ u m a fun c¸ oes o˜ es crescente (ou decrescente). Do ponto de vista f ´ısico ı sico, isto significa significa que o ponto se m ove ove sem pr e no mesm o sentido. ˜ pode, sem pr eju E st a cond ic¸ ao eju ´ızo ı zo algum, ser inclu ´ıda ı da n a d efin ic¸a˜ o ´ ´ 101 do de movimento uniforme. (Veja tamb em e´ m o coment coment ario a` p ag. ´ ica livro “A Matem at ica do Ensino M edio”, e´ dio”, vol. 1.) Deve-se notar que ˜ afim f(t) = vt + b a posi¸ca˜ o d o m ovel o´ vel no eixo e´ da da pela fun c¸ao p or em e´ m o espa¸co ) que ele ele perco percorr eu e´ d a do p ela fun c¸a˜ o co (dist ancia) ancia ˆ linear e = vt. e´ brica do fato fato de que por por dois dois pontos pontos dis6 . A contrapartida algebrica t in i n t os os d o p la la n o p a s s a u m a , e s om o m e n t e u m a , r e t a e´ a pr opos ic¸a˜ o segundo a qual, dados x , x , y , y em R, com x = x , existe ˜ afim f(x) = ax + b, tal que f(x ) = y um a, e somen te u ma , fun fun c¸ ao ˜ afim fica determinada e f(x ) = y . Em palavras: palavras: uma func¸ ao quando se conhecem seus valores (tomados arbitrariamente) em ˜ que resulta imediatamendois dois p ontos dist int os. E st a pr oposic oposic¸ ao, ˜ geom te da sua vers vers ao geom etrica e´ trica acima mencionada, pode ser provada algebric algebricamen amen te observandobservando-se que, dados a rbitrar iamente x , x , y , y , com x = x , o sistema linear ½
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 1
135
˜ e alem ´ ma is vacas, men os du ra a ra c¸ao e´ m disso, disso, se o n umero de va´ ca s e´ multiplicado por um n u´ m e r o n a t u r a l n, o n umero de dias que d ur a a ra c¸a˜ o e´ d ividido ividido por n. Passados os primeiros 14 dias, ˜ suficie o fazen fazen deiro tin ha ain da r ac¸ao suficiente nte pa ra alimentar 16 vacas vacas d u r a n t e 62 − 14 = 48 dias. Par a saber durante quantos dias esta ˜ poderia r a c¸ ao poderia alimentar as 12 (= 16 − 4) vacas vacas resta ntes, observaobserva´ mos que 12 = 16 × (3/4) logo logo esse n umer o de de dias e´ 48 ÷ 3/4 = 64. ˜ que Depois de m a is 15 dia s, a r a c¸ao que res resta ta d a´ para alimentar alimentar a s 12 vacas durante 64 − 15 = 49 dias. Essa mesm a r ac¸a˜ o e´ suficiente para a limentar limentar 21(= 12 + 9) vacas durante 49 ÷ (21/12) = 28 dias, 21 ˆ pois 21 = 12 × . Tota lizand lizand o, na s circun circun st ancias do problema, a 12 ˜ dur r a c¸ ao duraa 14 + 15 + 28 = 57 dias. ´ıcio e´ a n alogo ´ ˜ sse8 . Este exerc´ exercıcio alog o a o a nt erior, erior, porta nt o su a soluc soluc¸ao egue as m esmas linhas. Ao final de 12 dias de viagem, viagem, a caravan a ´ t em agua suficiente par a ser vi r 7 pessoas durante 42 − 12 = 30 dias. E 10(= 7 + 3) pessoas? Como 10 7 (10/7) a´ g u a d u r a r ´
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136 Temas e Problemas ´ du a. A soluc ˜ mais simples consiste em e´ u m a t a r e f a m u i t o ar soluc¸ ao calcular alcular o tempo. tempo. Os dois dois tr ens se chocam hocam no momento t em que 46t + 58t = 260, ou seja, ap os o´ s t = 260/104 = 2 horas e 30 minutos ´ E s t e e´ exatamente o tempo em em que o p assaro voou. Portanto ele percorreu 60 × 2,5 = 150 quilometros. oˆ metros.
1 1 . Ap os o´ s x meses de uso, quem comprou o aparelho na loja A gastou f(x) = 3800 + 20x reais, enquan to quem comprou comprou na loj loja B gastou g(x) = 2500 + 50x. Tem-se g(x) − f(x) = 30x − 1300. Logo, 1 para todo x ≥ 1300/30 = 43 , tem-se g(x) − f(x) ≥ 0. N ou ou t r a s 3 palavr palavras, as, ap os o´ s 43 m eses eˆ ses e 10 dias de u so, so, o aparelho co comprado na loja B, que inicialmente era mais barato, torna-se mais caro do que o compr compr ado n a loja loja A.
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 1
137
S o lu c¸ a˜ o d o p r o b l e m a d a l e b r e e d o c a c h o r r o Um pulo de lebre lebre e´ igua igua l a 2/3 de um pulo de cachorro. Portanto a dianteira da lebre e´ d e 100/3 pulos de cachorro. No momento em que alcanc¸ar a lebre, o cachorr cachorr o t era´ d a d o x pulos e a lebre ter a´ dado 4/3 pulos (de lebre), ou seja, ( 4/3) × (2/3)x = (8/9)x pulos ˆ cia per corr ida pelo cachorro de cachorr cachorr o. Nesse m omen to, a dist an cachorro ` (medida em termos de pulos) e´ igua igua l aquela percorrida pela lebre mais a dianteira que ela levava levava no princ´ princ´ıpio. ı pio. Assim:
x=
8 100 x+ , donde 9x = 8x + 300 e x = 300. 9 3
Port an to, dan do 300 pu los, los, o cachorr cachorr o alcan c¸a ¸ a a lebre.
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138 Temas e Problemas
S o l u c¸ o˜ e s d o s P r o b le l e m a s d o C a p´ p ı´t u l o 2 1 1 e´ o d ob ob r o d a m edia e´ dia aritm etica e´ tica de x e , lox x go e´ maior do que ou igual ao dobro da m edia e´ dia geom etrica e´ trica desses 1 ´ 2 para todo x > 0. A n umeros, que e´ igual igual a 1. Assim, ssim, x + x 1 igua igua ldade oc ocorr e apena s qua ndo x = , ou ou seja, seja, qua ndo x = 1. x 1 . Note que x +
≥
2 . A soma ax + by e´ o d o b r o d a m edia e´ dia aritm etica e´ tica de ax e by, logo e´ maior do que ou igual ao dobro da m edia e´ dia geom etrica e´ trica des´ ses n umeros, umeros , sendo s endo igual igual apenas quando ax = by, caso em que ax + by = 2 ax by = 2 abc. E st s t e e, e´ , portanto, o menor valor d e ax + by quando xy = c. Como ax = by, cada um destes dois ´ n umeros e´ igual a abc.
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 2
139
˜ pode As ra´ r a ´ızes ı zes dest a equa c¸a˜ o s a˜ o 20 e −6. C om o x n ao pode ser negativo, deve ser x = 20. Logo Logo, uma das torneiras enche o tanque em 20 hora hora s e a outra em 20 + 10 = 30 horas. ´ o de hora hora s que as duas t orn eira eira s jun jun tas levar levar iam 5 . Seja z o n umer para encher encher o tanque. Em uma hora, hora, a frac¸a˜ o d o t a n qu qu e q u e a s 1 1 1 1 x+y duas torneiras juntas enchem e´ . Logo = + = e d a ´ı z z x y xy xy z= . (Note que z e´ a metade da m edia e´ dia h arm onica oˆ nica de x e y.) x+y ´ ´ de horas horas necess necessarias 6 . Sejam x e y respectivamente o n umero para que cada guindaste guindaste desc descarr egue egue sozi sozinho nho o na vio vio. Temos emos xy x(x + 5) = 6 e y = x + 5, logo = 6 e d a ´ı x − 7x − 30 = 0. x+y 2x + 5 As ra ´ızes ı zes d est a equ a c¸a˜ o s a˜ o 10 e −3, logo x = 10 e y = 15. ¾
7
O primeiro primeiro comercian mercian te vendeu x met ros ao pr ec¸o ¸ o de p reais
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140 Temas e Problemas
√
˜ dada sob a form form a x = x − m e elevando8 . E scre ven do a equ a c¸ao a a o qua qua dra do, do, vemos vemos que ela e´ equivalent equivalent e a` s segu in t es condic¸oes o˜ es simult aneas: ˆ
x − ( 2m + 1)x + m = 0, ¾
¾
x
≥ 0,
x
≥ m.
˜ do segundo O discrimin an te da equa c¸ ao segundo grau acima acima e´ ∆ = 4m + 1. Vam os sepa ra r os valores poss´ poss´ıveis ı veis de m em cinco cinco casos:
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 2
Y
Y
y = x
y = x y
y
=
m + x
141
=
m + x
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142 Temas e Problemas ´ de diagonais de um pol´ pol ´ıgono ı gono convexo de n lados e´ 1 0 . O n umero i g u a l a n(n − 3)/2. I gu gu a la la n do d o-o a 405, obtemos btemos a equac¸a˜ o n − 3n − 810 = 0, cuja unica ´ raiz positiva e´ n = 30. P or or t a n t o o pol´ pol´ıgono ı gono tem 30 lados. ¾
´ 1 1 . O n umer o de jogo jogoss nu m cam peona peona to disputado por por n clubes em dois turnos e´ n(n − 1). Logo Logo n − n = 306. Resolv Resolvendo endo esta ˜ encontramos a un ´ ica e q u a c¸ao, ica ra iz positiva positiva n = 18. ¾
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 2
143
i o em em k m p or or h or a . N o 1 4 . Seja x a velocidade das a´ g u a s d o r io movimento uniforme, tem-se tempo = es p a c¸ o/veloci velocida da de. Logo Logo, soma nd o o tem po de ida com o de volta volta r esu lta a equa c¸a˜ o
12 8 + = 2, 12 + x 12 − x
ou s ej e ja ,
x − 2x − 24 = 0. ¾
´ ica A un ica ra iz positiva positiva dest a equ ac¸a˜ o e´ x = 6. Port an to a veloci velocidade dade do rio e´ de 6 km por hora. O remador levou 40 minutos para ir e
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144 Temas e Problemas ´ diagona is do losango s a˜ o x e 8 − x logo a area do mesmo mesmo e´ 1 8 . As diagona 1 ´ x(8 − x)/2 = x 4 − x e seu valor m aximo e´ obtido obtido par a x = 4. 2 ˜ as duas diagonais ˜ iguais e o losango e´ u m q u a dr E n t ao di agonais s ao dr a d o cujo lado m ede 2 2.
√
´ ent a˜ o x + 8 e´ o outro outro e seu 1 9 . S e x e´ o menor dos dois n umeros produto f(x) = x(x + 8) assum e o valo valorr m ´ınimo ı nimo f(− 4) = −16, logo os valores poss´ poss ´ıveis ı veis desse produto formam o intervalo [−16, )
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146 Temas e Problemas
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156 Temas e Problemas
P
9
x
A 119
1 52
B Figura 58
Y
6
X 18
74
62 54
A
46
B
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 4
157
o que fornece XA = 1,328. Sendo ABY = 120 , novam novam ent e a lei dos dos senos fornece: YA 1 , = s en 120 s en 6 ˆ AXY usamos agora a lei dos o q u e d a´ YA = 8,285. N o t r iangulo cossenos: ◦
◦
◦
XY = XA + YA − 2 · XA · YA · cos (XAY ) ¾
¾
¾
e os calculos ´ indicam XY = 7,48.
Problema 4 plano horizontal
h horizonte
R α
R
O
Figura 60
ˆ α entre a hor Observando a Figura 60, vemos que o angulo horiizontal e a linha de visada ao horizonte, aparece tamb em e´ m no centr centr o da Terra. Da´ Da ´ı, ı, R cos α = h + R e, portanto, h cos α R= . 1 − cos α P a r a h = 0,703 k m e α = 0,85 encontra-se R = 6633 k m . ◦
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158 Temas e Problemas Problema 5
A
O
S
Figura 61
360 ˜ o comprimento da circunfer encia Se α = en t ao eˆ ncia da Terra 50 ´ e´ 50 vezes o compr compr iment o do arco SA, ou seja 250 000 est adios ou 40250 40 250 250 km. Da´ Da ´ı, ı, R = = 6409 km, um resultado mu ito bom. bom. 2π ◦
Problema 6 ˜ proporcio 1) Os compr compr iment os de AC e BC s ao proporciona na is respectivament e a 8 e 9 (Fig (Figur ur a 62). 62). Da´ Da ´ı, ı , pela lei dos senos, 9k 8k = s en 110 s en θ ◦
Encontramos sen θ = 0,835 e como θ e´ u m an ˆ gulo agudo, agudo, tem-se θ = 56,6 . ◦
2) Vej Vejaa a Figura 63.
8,1 8 = s en 110 s en θ ◦
◦
⇒
θ = 68,14
⇒
BC
Logo, ACB = 1,86 .
◦
BC
50
1448
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 4
r
C
8k
A
159
110
9k
θ
B Figura 62
C
8k
A 110
50 8,1k 8,1 k
θ B Figura 63
Problema 7 ˜ os lados Como a veloci velocida da de d e A e´ 15% ma ior ior qu e a de B, ent ao BC e AC do tr ian ˆ gulo s ao ˜ respectivamen te pr oporcio porciona na is a 1 e 1,15 (Figura 64). Da´ Da ´ı, ı, 1 1,15 = s e n 60 s en θ ◦
⇒
s en θ = 0,99593.
Mas, isto fornece θ = 84,8 ou θ = 180 − 84,8 = 95,2 . ◦
◦
◦
◦
Por que h a´ d u a s r e s p o s t a s ? ˜ Os v ertices ˆ ABC Im agin e a segu int e situ ac¸ao. e´ rtices A e B do tr iangulo ˜ fixos ˜ entr e os s ao fixos e a ra zao os lados CA e CB e´ cco onst an te (Figur (Figur a 65). 65).
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160 Temas e Problemas
1,15k 1,15 k
A
C
60
k θ
B Figura 64
C
A Figura 65.
CA CB
B
e´ tr ico ico do v ertice e´ rtice C ? = r, constante. Qual e´ o lugar geom etr
Divi Dividamos damos o segmento AB harmonicamente na raz a˜ o r. I st st o significa encontrar os pontos M e N d a r e t a AB, um inter interior ior ao segmento AB e outr o exter exter ior, ior, ta is que
MA NA = = r. MB NB MA CA ˆ gulo interno C d o = , ent a˜ o CM e´ bissetriz do an MB CB ˆ ABC (reco t r i angulo (recorde o teorem teorem a da s bissetrizes e sua rec´ rec´ıproca). ı proca). NA CA Como
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 4
C
161
β
β
α α
A
M
B
N
Figura 66
˜ fixos e o angulo ˆ MCN e´ reto. Logo, C Ora , os os pont os M e N s ao es t a´ sobre sobre a circunferencia eˆ ncia de diaˆ m e t r o MN (Figura 67). 67).
C
A
M
B
N
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162 Temas e Problemas No n osso problema, problema, h a´ dois pontos poss´ poss ´ıveis ı veis para o encontro: ˆ C ou C . Os angulos calculados foram ABC = 84,8 e ABC = 95,2 . (Figura 68). 68).
◦
r
A
C
◦
C
’
B
Figura 68
Problemas suplementares ˜ os corr velocidades forem igua igua is ent ao corr edores edores percorr percorr er a˜ o 1 Se a s veloc
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˜ do Cap´ıtulo Soluc¸ oes ıtulo 4
A
r
C
163
x
α
α B Figura 69
3 . Veja a Figura 70. Pela lei dos cossenos, ◦
PQ = 1,2 + 1,8 − 2 · 1, 2 · 1, 8 · cos 27 , donde PQ = 911 m . ¾
¾
¾
Aplicando a lei dos senos,
1,8 0,911 = s en α s en 27
◦
⇒
s en α = 0,897.
β = 89,2 . Temos ent a˜ o α = 63,8 e, conseq uentemente, ¨ ◦
◦
ˆ PAB e PBA foram medidos, eja a Figura 71. Como Como os os angulos 4 . Veja encontramos APB = 31,6 . ◦
PA 660 = s en 77,9 s en 31,6 ◦
◦
⇒
PA = 1231,6
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164 Temas e Problemas
Q
β x
P
α
y 1,8
1,2
27° C Figura 70
C