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(A) The number of radioactive nuclei remaining at time $t$ is given by $N(t) = N_0 e^{-\lambda t}$.Average life $	au$ is defined as $	au = 1/\lambda

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more than half the active nuclei decay

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(A) The number of radioactive nuclei remaining at time $t$ is given by $N(t) = N_0 e^{-\lambda t}$.Average life $	au$ is defined as $	au = 1/\lambda$.At time $t = 	au$, the number of remaining nuclei is $N(	au) = N_0 e^{-\lambda(1/\lambda)} = N_0 e^{-1} \approx N_0 / 2.718 \approx 0.368 N_0$.The number of nuclei that have decayed is $N_{decayed} = N_0 - N(	au) = N_0 - 0.368 N_0 = 0.632 N_0$.Since $0.632 N_0 > 0.5 N_0$, more than half of the active nuclei decay in one average lifetime.

In one average lifetime of a radioactive nucleus,

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