In a radioactive decay process, the activity is defined as A=−dNdt where N(t) is the number of radioactive nuclei at time ( t ). Two radioactive sources, S1 and S2 have same activity at time t = 0. At a later time, the activities of S1 and S2 are A1 and A2 , respectively. When S1 and S2 have just completed their 3rd and 7th half-lives, respectively, the ratio A1A2 is __________

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In a radioactive decay process, the activity is defined as $A = - \frac{dN}{dt}$ where N(t) is the number of radioactive nuclei at time ( t ). Two radioactive sources, S₁and S₂have same activity at time t = 0. At a later time, the activities of S₁ and S₂ are A₁ and A₂ , respectively. When S₁ and S₂ have just completed their 3^rd and 7^th half-lives, respectively, the ratio $\frac{A_{1}}{A_{2}}$ is ___________.

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answer is 16.

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Detailed Solution

$\frac{A_{1}}{A_{2}} = \frac{A_{0} e^{- x_{1} t}}{A_{0} e^{- x_{2} t}} = \frac{A_{0} / 2^{n}}{A_{0} / 2^{n}}$
$= \frac{A_{0} / 8}{A_{0} / 2^{7}} = \frac{2^{7}}{2^{3}} = 2^{4} = 16$