In order to calculate the amount of radon-222 left after 45 days, we can use the formula:
![P=P_0\cdot0.5^{(t)/(n)}](https://img.qammunity.org/2023/formulas/physics/college/qrcivkmx2zrr5snbnvog708s6mfuqo71jx.png)
Where P is the final amount after t days, P0 is the initial amount, t is the number of days and n is the period of half-life.
So, using P0 = 100, t = 45 and n = 3.8, we have:
![\begin{gathered} P=100\cdot0.5^{(45)/(3.8)} \\ P=100\cdot0.5^(11.842) \\ P=0.0272 \end{gathered}](https://img.qammunity.org/2023/formulas/physics/college/yh6kmev91z5ce9ctksjrknj8p2qxmq10tf.png)
So the amount left after 45 days is 0.0272 grams.