Answer:
The half-life of a radioactive substance is the time it takes for half of the initial amount of the substance to decay. We can use the fact that 8% of the original amount remains after 100 days to determine the half-life of the isotope.
Let's assume that the initial amount of the substance is 1 unit (it could be any amount, but we're assuming 1 unit for simplicity). After one half-life, half of the original amount remains, or 0.5 units. After two half-lives, half of the remaining amount remains, or 0.25 units. After three half-lives, half of the remaining amount remains, or 0.125 units. We can see that the amount of substance remaining after each half-life is half of the previous amount.
We can use this information to set up the following equation:
0.08 = (1/2)^n
where n is the number of half-lives that have elapsed. We want to solve for n.
Taking the logarithm of both sides, we get:
log(0.08) = n*log(1/2)
Solving for n, we get:
n = log(0.08) / log(1/2) = 3.42
So the number of half-lives that have elapsed is approximately 3.42. Since we know that 100 days is the time for three half-lives (from the previous calculation), we can find the half-life by dividing 100 days by 3.42:
Half-life = 100 days / 3.42 = 29.2 days (rounded to one decimal place)
Therefore, the half-life of the radioactive isotope that leaked into the stream is approximately 29.2 days.