When comparing the half-lives of sample A and sample B, we need to consider that the half-life is a constant characteristic of a radioactive nuclide and does not depend on its initial activity. Therefore, the half-life of A is the same as the half-life of B.
After each sample has passed through five half-lives, we can calculate the ratio of their activities. The activity of a radioactive substance is proportional to the number of radioactive atoms present.
Let's assume the initial activity of sample A is A0, and the initial activity of sample B is B0. Since sample A has twice the initial activity of sample B, we can write A0 = 2B0.
After one half-life, the activity of a radioactive substance decreases to half of its initial value. Therefore, after five half-lives, the activity decreases by a factor of (1/2)^5, or 1/32.
To calculate the ratio of their activities, we divide the final activity of sample A (A5) by the final activity of sample B (B5):
A5/B5 = (A0/32)/(B0/32)
= A0/B0
Since A0 = 2B0, the ratio of their activities after five half-lives is:
A5/B5 = 2B0/B0
= 2
Therefore, after passing through five half-lives, the ratio of the activities of sample A to sample B is 2.