I have a box of 100 cubes, 1 or 2 of the cube faces are painted, lets call this

Question

I have a box of 100 cubes, 1 or 2 of the cube faces are painted, lets call this 'X-face' it will represent an atom going under radioactive decay. The procedure is as follows: i have to shake the box ONCE, record and remove the number of X-face up, then shake the box TWICE, record and remove the number of X-face up... i have to continue this up to TEN shakes.

Q1)If you have a very large number of cubes, what sort of relationship is there between the number of 'undecayed' cubes and the number of shakes? Write an equation that relates the number of cubes remaining (N), the initial number of cubes (N0) and the number of shakes (n).

Q2)Use your equation to predict the number of shakes needed so that there are only 50 of the initial 100 cubes remaining- ie, what is the half life?

Explanation / Answer

Q1) supposed 1 face painted, each time you shake you will get 1/6 of the cubes being painted face up.
so each time you shake you lose 1/6 or in other words keep 5/6
N=N0 (5/6) after one shake
after two shakes N=N0(5/6)(5/6)
so with n shakes N=N0 (5/6)n

(if it was two faces painted it would be 4/6 instead of 5/6)

Q2) so we have N=50 and N0=100 so

50=100(5/6)n so 1/2=(5/6)n

take the log of both sides and use that log xn=nlogx so we have log(1/2)=nlog(5/6)

so n= log(1/2)/log(5/6)=3.8

so the half life is 3.8 shakes.

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