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[SPAM] Re: [obm-l] FORMIGA

To: obm-l@xxxxxxxxxxxxxx
Subject: [SPAM] Re: [obm-l] FORMIGA
From: alexmay nunes soares <alexmaynunes@xxxxxxxxxxxx>
Date: Sat, 22 Mar 2008 21:41:11 -0300 (ART)
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  Tente acompanhar com a figura abaixo:
   
   
  Qdo vc planificar o cilindro vc terá um retângulo de dimensões 30cm por 24cm (H por 2pi*raio). Dos dados do exercícios constatamos q a planificação pode ser dada com a Formiga F no "meio" do retângulo e o mel no ponto M. Para a formiga entrar no menor tempo possível ela terá que sair de F e caminhar ate um ponto P, na borda, e desse ponto P entrar no cilindro e caminhar sobre a superfície interna até um ponto T, onde ela deverá encontrar o Mel, a distância mínima implicará um tempo mínimo, logo MP + PT deve ser mínimo, usando alinhamento dos pontos, podemos prolongar PQ (veja figura abaixo) de modo a termos FQ = QF', daí F'P = FP, mas temos F', P e T alinhados! Trace um reta ligando os pontos F' , P e T, se vc prolongar TM até um ponto R, com RF' ortogonal a MR, teremos um triângulo retângulo TRF', sabemos que MQ = 24/2, isto é: MQ = 12 cm, como RF' = MQ, temos RF' = 12 cm, por outro lado F'T é a distância percorrida pela formiga, se t for o tempo necessário para o
 encontro temos (distância = velocidade * tempo)
  F'T = 3*t e MT = 1*t, note que MT é a distãncia percorrida pela gota de mel, como TR = TM + MR vem que TR = t+4, pronto agora é só o teorema de pitágoras, com: hipotenusa F'T = 3t e catetos TR = t+4 e RF' = 12
   
  (3t)^2 = (t+4)^2+12^2
  9t^2=t^2+8t+16+144
  8t^2-8t-160=0
  t^2-t-20=0
  t=-4(não serve) ou t=5segundos
  como t deve ser expresso em décimos de segundos vem que t = 50décimos de segundos! Fiz uma tentativa de figura abaixo:                                     
   
                       R                  F'
                                           I                     
                                           I4cm              
                      M______P__ I Q____________ Borda
                        I                  I                    I
                        I                  I4cm              I
                        I                  I                    I
                        I                 F                    I
                        I                                       I
                        I                                       I
                     T I                                       I
                        I                                       I
                        I                                       I
   
   
  

arkon <arkon@xxxxxxxxxx> escreveu:
        PESSOAL ALGUÉM PODE RESOLVER, POR FAVOR, ESSA:
   
  Num vaso aberto, com formato de um cilindro circular reto, de altura igual a 30 cm e raio da base igual a 4 cm, existe uma formiga e uma gota de mel.
  A Formiga localiza-se num ponto da superfície externa do vaso a 4 cm da sua borda. A gota de mel, por sua vez, localiza-se na superfície interna do vaso, junto à borda, o mais afastado possível da formiga.
  Sabe-se que a gota de mel escorre pela superfície interna do vaso a uma velocidade constante de 1 cm/seg e a formiga é capaz de caminhar, no máximo, a uma velocidade de 3 cm/seg. Calcular, em décimos de segundo, o tempo mínimo que a formiga gastará para alcançar a gota de mel. Despreze a espessura do vaso e adote para cálculos pi = 3.
   
  GABARITO 50.
   
  DESDE JÁ AGRADEÇO




       
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<div><?xml:namespace prefix = v ns = "urn:schemas-microsoft-com:vml" /><v:line id=_x0000_s1028 style="Z-INDEX: 1; POSITION: absolute" coordsize="21600,21600" to="54.45pt,63.45pt" from=".45pt,.45pt"></v:line></div>  <div><v:line id=_x0000_s1028 style="Z-INDEX: 1; POSITION: absolute" coordsize="21600,21600" to="54.45pt,63.45pt" from=".45pt,.45pt"></v:line><v:line id=_x0000_s1028 style="Z-INDEX: 1; POSITION: absolute" coordsize="21600,21600" to="54.45pt,63.45pt" from=".45pt,.45pt"></v:line><v:line id=_x0000_s1028 style="Z-INDEX: 1; POSITION: absolute" coordsize="21600,21600" to="54.45pt,63.45pt" from=".45pt,.45pt"></v:line></div>  <div><v:line id=_x0000_s1028 style="Z-INDEX: 1; POSITION: absolute" coordsize="21600,21600" to="54.45pt,63.45pt" from=".45pt,.45pt"></v:line>Tente acompanhar com a figura abaixo:<v:line id=_x0000_s1028 style="Z-INDEX: 1; POSITION: absolute" coordsize="21600,21600" to="54.45pt,63.45pt" from=".45pt,.45pt"></v:line></div>  <div>&nbsp;</div> 
 <div>&nbsp;</div>  <div>Qdo vc planificar o cilindro vc terá um retângulo de dimensões 30cm por 24cm (H por 2pi*raio). Dos dados do exercícios constatamos q&nbsp;a planificação pode ser dada com a Formiga F no "meio" do retângulo e o mel no ponto M. Para a formiga entrar no menor tempo possível ela terá que sair de F e caminhar ate um ponto P, na borda, e desse ponto P entrar no cilindro e caminhar sobre a superfície interna até um ponto T, onde ela deverá encontrar o Mel, a distância mínima implicará um tempo mínimo, logo MP + PT deve ser mínimo, usando alinhamento dos pontos, podemos prolongar PQ (veja figura abaixo) de modo a termos FQ = QF', daí F'P = FP, mas temos F', P e T alinhados! Trace um reta ligando os pontos F' , P e T, se vc prolongar TM até um ponto R, com RF' ortogonal a MR, teremos um triângulo retângulo TRF', sabemos que MQ = 24/2, isto é: MQ = 12 cm, como RF' = MQ, temos RF' = 12 cm, por outro lado F'T é a distância percorrida pela formiga, se t for o
 tempo necessário para o encontro temos (distância = velocidade * tempo)</div>  <div>F'T = 3*t e MT = 1*t, note que MT é a distãncia percorrida pela gota de mel, como TR = TM + MR vem que TR = t+4, pronto agora é só o teorema de pitágoras, com: hipotenusa F'T = 3t e catetos TR = t+4 e RF' = 12</div>  <div>&nbsp;</div>  <div>(3t)^2 = (t+4)^2+12^2</div>  <div>9t^2=t^2+8t+16+144</div>  <div>8t^2-8t-160=0</div>  <div>t^2-t-20=0</div>  <div>t=-4(não serve)&nbsp;ou t=5segundos</div>  <div>como t deve ser expresso em décimos de segundos vem que t = 50décimos de segundos! Fiz uma tentativa de figura abaixo:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </div>  <div>&nbsp;</div> 
 <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;R&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <STRONG>F'</STRONG></div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I4cm&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div> 
 <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<STRONG>M</STRONG>______<STRONG><U>P</U></STRONG>__ I<U> Q</U>____________ Borda</div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;I</div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I4cm&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;I</div>  <div>&nbsp;
 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;I</div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <STRONG>F</STRONG>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;I</div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I</div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;I</div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;T I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;I</div> 
 <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;I</div>  <div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;I</div>  <div>&nbsp;</div>  <div>&nbsp;</div>  <div><BR><BR><B><I>arkon &lt;arkon@xxxxxxxxxx&gt;</I></B> escreveu:</div>  <BLOCKQUOTE class=replbq style="PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #1010ff 2px solid">  <DIV style="FONT-SIZE:
 12px; FONT-FAMILY: verdana, arial">  <DIV>  <div class=MsoNormal style="MARGIN: 0cm 56.2pt 0pt 0cm; TEXT-ALIGN: justify; tab-stops: 369.0pt"><B style="mso-bidi-font-weight: normal"><SPAN style="COLOR: black"><FONT size=3><FONT face="Times New Roman">PESSOAL ALGUÉM PODE RESOLVER, POR FAVOR, ESSA:<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p></FONT></FONT></SPAN></B></div>  <div class=MsoNormal style="MARGIN: 0cm 56.2pt 0pt 0cm; TEXT-ALIGN: justify; tab-stops: 369.0pt"><B style="mso-bidi-font-weight: normal"><SPAN style="COLOR: black"><o:p><FONT face="Times New Roman" size=3>&nbsp;</FONT></o:p></SPAN></B></div>  <div class=MsoNormal style="MARGIN: 0cm 56.2pt 0pt 0cm; TEXT-ALIGN: justify; tab-stops: 369.0pt"><B style="mso-bidi-font-weight: normal"><SPAN style="COLOR: black"><FONT size=3><FONT face="Times New Roman">Num vaso aberto, com formato de um cilindro circular reto, de altura igual a <?xml:namespace prefix = st1 ns =
 "urn:schemas-microsoft-com:office:smarttags" /><st1:metricconverter ProductID="30 cm" w:st="on">30 cm</st1:metricconverter> e raio da base igual a <st1:metricconverter ProductID="4 cm" w:st="on">4 cm</st1:metricconverter>, existe uma formiga e uma gota de mel.<o:p></o:p></FONT></FONT></SPAN></B></div>  <div class=MsoNormal style="MARGIN: 0cm 56.2pt 0pt 0cm; TEXT-ALIGN: justify; tab-stops: 369.0pt"><B style="mso-bidi-font-weight: normal"><SPAN style="COLOR: black"><FONT size=3><FONT face="Times New Roman">A Formiga localiza-se num ponto da superfície externa do vaso a <st1:metricconverter ProductID="4 cm" w:st="on">4 cm</st1:metricconverter> da sua borda. A gota de mel, por sua vez, localiza-se na superfície interna do vaso, junto à borda, o mais afastado possível da formiga.<o:p></o:p></FONT></FONT></SPAN></B></div>  <div class=MsoNormal style="MARGIN: 0cm 56.2pt 0pt 0cm; TEXT-ALIGN: justify; tab-stops: 369.0pt"><B style="mso-bidi-font-weight: normal"><SPAN style="COLOR:
 black"><FONT size=3><FONT face="Times New Roman">Sabe-se que a gota de mel escorre pela superfície interna do vaso a uma velocidade constante de 1 cm/seg e a formiga é capaz de caminhar, no máximo, a uma velocidade de 3 cm/seg. Calcular, em décimos de segundo, o tempo mínimo que a formiga gastará para alcançar a gota de mel. Despreze a espessura do vaso e adote para cálculos pi = 3.<o:p></o:p></FONT></FONT></SPAN></B></div>  <div class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><B style="mso-bidi-font-weight: normal"><SPAN style="COLOR: black"><o:p><FONT face="Times New Roman" size=3>&nbsp;</FONT></o:p></SPAN></B></div>  <div class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><B style="mso-bidi-font-weight: normal"><SPAN style="COLOR: blue"><FONT size=3><FONT face="Times New Roman">GABARITO 50.<o:p></o:p></FONT></FONT></SPAN></B></div>  <div class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><B style="mso-bidi-font-weight:
 normal"><SPAN style="COLOR: black"><o:p><FONT face="Times New Roman" size=3>&nbsp;</FONT></o:p></SPAN></B></div>  <div class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><B style="mso-bidi-font-weight: normal"><SPAN style="COLOR: black"><FONT size=3><FONT face="Times New Roman">DESDE JÁ AGRADEÇO<o:p></o:p></FONT></FONT></SPAN></B></div></DIV></DIV></BLOCKQUOTE><BR><p>&#32;


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