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(A) The activity $A$ of a radioactive sample is given by $A = \lambda N = \frac{0.693}{T_{1/2}} N$,where $N$ is the number of nuclei and $T_{1/2}$ is

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$4$

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(A) The activity $A$ of a radioactive sample is given by $A = \lambda N = \frac{0.693}{T_{1/2}} N$,where $N$ is the number of nuclei and $T_{1/2}$ is the half-life.From the given information,$N_{1} = 2 N_{2}$ and $A_{2} = 2 A_{1}$.We can write the ratio of activities as:$\frac{A_{1}}{A_{2}} = \frac{N_{1} / T_{1}}{N_{2} / T_{2}} = \frac{N_{1}}{N_{2}} 	imes \frac{T_{2}}{T_{1}}$Rearranging for the ratio of half-lives $\frac{T_{1}}{T_{2}}$:$\frac{T_{1}}{T_{2}} = \frac{A_{2}}{A_{1}} 	imes \frac{N_{1}}{N_{2}}$Substituting the given values:$\frac{T_{1}}{T_{2}} = \frac{2 A_{1}}{A_{1}} 	imes \frac{2 N_{2}}{N_{2}} = 2 	imes 2 = 4$Thus,the ratio of the half-life of $S_{1}$ to the half-life of $S_{2}$ is $4$.

$A$ radioactive sample $S_{1}$ having the activity $A_{1}$ has twice the number of nuclei as another sample $S_{2}$ of activity $A_{2}$. If $A_{2} = 2 A_{1}$,then the ratio of half-life of $S_{1}$ to the half-life of $S_{2}$ is

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