Explanation
Part A
The formula for an exponential decay model is given as:
[tex]y=C(1-r)^t[/tex]
Where C is the initial amount and r is the decay rate and t is the time.
Given that the amount of radon has a half-life of 3.8 days, it implies that the original amount will decay to half of itself in 3.8 days. Therefore, we will have
[tex]\begin{gathered} \frac{1}{2}R_0=R_0(1-r)^{3.8} \\ \frac{1}{2}=(1-r)^{3.8} \\ \sqrt[3.8]{0.5}=1-r \\ r=1-\sqrt[3.8]{0.5} \\ r=0.1667 \\ r=16.7\text{ \%} \end{gathered}[/tex]
Answer: The daily decay rate is 16.7% per day
Part B
We can find the formula below;
[tex]\begin{gathered} R=R_0(1-0.1667)^t \\ R=R_0(0.833)^t \end{gathered}[/tex]
Answer: Option A
Part C
The percentage of radon after 13 days will be gotten with the formula below.
[tex]\frac{Amount\text{ after 13 days}}{Original\text{ amount}}\times100[/tex]
Therefore,
[tex]\begin{gathered} \frac{R_0(0.833)^{13}}{R_0}\times100 \\ =9.30\text{\%} \end{gathered}[/tex]
Answer: 9.30%
