domingo, 20 de janeiro de 2008

A Formiga e a Escada

PROBLEMA

DA AULA



Um artista construiu um monumento com 10 escadarias.

Concebeu a sua obra de maneira simples: para passar de uma escadaria para a seguinte, substituiu cada degrau por dois degraus com metade das dimensões, como mostra o projecto das três primeiras.



Uma sobe a décima escadaria, seguindo as setas, de A até B.


Qual é o comprimento do trajecto da formiga?


Aqui ficam algumas resoluções!






3 comentários:

Tânia Aires disse...

Queria só deixar um beijinho e saber se está tudo bem...enviei-lhe um e-mail, mas se calhar não recebeu...
Beijinhos

Anónimo disse...

Just a guess for the ant problem.

The distance covered in every staircase should be the same, since we multiply the number of steps with a factor of 2 but the ant has to cover twice the distance (based on the Pythagorean theorem).E.g. in staircase #1 (with two steps) the distance would be =sqrt(2)*10, thus independent of the number of steps. So forth the total distance would be ten times this = 100*sqrt(2).

Does it make sense?

thanks for making me take a break off work.

Camila disse...

Thank you of having answered to a problem that I put to my pupils.

I believe that it drove the drawing to a wrong reading of the problem, because you can´t read Portuguese!
Notice: An artist built a monument with 10 staircases.

It conceived his work of simple way: to pass of a staircase for the next one, it substituted each step for two steps with half of the dimensions, how it shows the project of the first three.
Does the ant rise even to the tenth staircase, all the meters it went?
Now, as the Phitagorean Theorem the hypotenuse in fig.1 has 10*squared( 2), what it does that each cathetus is the same as 5 m. In the next staircase each cathetus is divided in 2, and the same thing happens in each cathetus that is divided of staircase in staircase! So the ant goes through 20 m in each staircase, since it rises even to the 10th one it will go 200m.

I have posted other fun problems, you may solve them!

NOv,27, Dec 9,16, 29, for instance!

But...

I am going to post other ones, in English, even I make a lot of mistakes! :(

Which is your language? I can put in French and in Spanish.

Thank you!
:) Camila

H2=10^2 +10^2
H=10sqrt(2)
h=5sqrt(2)
2c^2=25*2
c^2=25
c=5
In Fig 1. Each step = 10 m
2 Steps = 20 m
In Fig 2. 2 steps is the same that 1 step in fig 1 ---- 20m
In Fig 3. 4 steps is the same that 1 step in Fig 1 ------20 m
And so on....
As the unt rise till the tenth staircase, the total distance is 10*20 m, 200 m