Polonium-210 is a radioactive element whose time rate of decay is proportional to the quantity present at any time. A nuclear accident, confined to a single room of a nuclear research laboratory has left the room contaminated with polonium-210 which has a half-life of 140 days. If the initial contamination of the room is 5 times the amount safe for long term human exposure, how long should laboratory workers wait to re-enter the room to decontaminate it?
2026-04-05 17:46:11.1775411171
Rate of change optimisation
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If you look at elemental definitions, as for example at
http://en.wikipedia.org/wiki/Radioactive_decay,
you will see that the half-time of a radiaactive substance is the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value and that it obeys an exponential law, that is to say Q = Q0 Exp[- k t].
According to your numbers, using Q = Q0 /2 and t = 140, this gives k = log(2) / 140.
The same equation will be now used with Q0 = 5 and Q = 1 to compute the time. This leads to t = 140 log(5) / log(2) which is close to 325 days (so almost a year).