Discussion
The half-life periods of two radioactive materials A and B are 1500 years and 1200 years respectively. If their mean life periods are \(\tau_{A}\) and \(\tau_{B}\) respectively, then the value of the ratio \(\frac{\tau_{A}}{\tau_{B}}\)
Answer: A
The ratio is \(\frac{5}{4}\).
Mean life and half-life are proportional: \(\tau = \frac{1}{\lambda}\) and \(T_{1/2} = \frac{\ln 2}{\lambda}\), so \(\tau = \frac{T_{1/2}}{\ln 2}\).
\[ \frac{\tau_{A}}{\tau_{B}} = \frac{1500}{1200} = \frac{5}{4} \]
The factor \(\ln 2\) cancels in the ratio. The other fractions, all less than 1, would make A shorter-lived even though its half-life is longer.
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