the isotope has a very short life of 13 half-life days.

estimate the percentage of the original amount of the isotope released by the explosion that remains 3 days after the explosion?

after t days, the amount left is

(1/2)^(t/13)

So, after 3 days, the amount left is

(1/2)^(3/13) = .852 = 85.2%

To estimate the percentage of the original amount of the isotope that remains 3 days after the explosion, we need to understand how half-life works.

The half-life of an isotope is the time it takes for half of the original amount of the isotope to decay or disintegrate. In this case, you mentioned that the isotope has a very short life of 13 half-life days.

To estimate the percentage remaining after a certain amount of time, we can calculate the number of half-lives that have occurred within that time period.

In this case, since each half-life is 13 days, we need to divide the number of days that have passed by 13 to determine the number of half-lives. So, if 3 days have passed since the explosion, we divide 3 by 13.

Number of half-lives = 3 days / 13 days = 0.2308

Now, we need to calculate the remaining percentage after 0.2308 half-lives.

To do this, we can use the formula:

Remaining Percentage = (1/2) ^ (Number of half-lives) * 100

Substituting the value of the number of half-lives, we have:

Remaining Percentage = (1/2) ^ 0.2308 * 100

Using a calculator, we find:

Remaining Percentage ≈ 93.4 %

Therefore, approximately 93.4% of the original amount of the isotope remains 3 days after the explosion.