Alright, so let's figure out what it means to have a half life of 3.8 days.
It means that after each set of 3.8 days, half of the previous amount of Radon-222 will remain.
To put this into equation form, we recognize that after the 2nd set of 3.8 days, 1/4 of our original amount will remain. After the 3rd set of days, 1/8 will remain. This leads us to the equation:
m(t) = m_0 * (1/2)^(t/3.8)
Where t is the time in days, m_0 is the initial mass, and m(t) is the mass after time (t).
So, let's fill in the numbers we got from the relation above:
m(15.3) = 4.5 * (1/2)^(15.3/3.8)
Well, we don't have to solve for anything. We just need to evaluate the expression on the right! Putting it into the calculator:
m(15.3) = 0.276 g
Therefore, after 15.3 days, we'll have 0.276 grams.
I hope that helps!