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(C) The law of radioactive decay is given by $A_t = A_0 e^{-\lambda t}$,where $A_0$ is the initial amount,$A_t$ is the amount remaining at time $t$,an

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$\frac{1}{e}$

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(C) The law of radioactive decay is given by $A_t = A_0 e^{-\lambda t}$,where $A_0$ is the initial amount,$A_t$ is the amount remaining at time $t$,and $\lambda$ is the decay constant.The mean lifetime $(	au)$ of a radioactive element is defined as $	au = \frac{1}{\lambda}$.We need to find the fraction of the initial amount remaining after time $t = 	au = \frac{1}{\lambda}$.Substituting $t = \frac{1}{\lambda}$ into the decay equation:$A_t = A_0 e^{-\lambda 	imes (\frac{1}{\lambda})}$$A_t = A_0 e^{-1}$$A_t = \frac{A_0}{e}$The fraction of the initial amount remaining is $\frac{A_t}{A_0} = \frac{1}{e}$.

In a radioactive element,the fraction of the initial amount remaining after its mean lifetime is:

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