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g100num [7]
3 years ago
6

The core melt- down and explosions at the nuclear reactor in Chernobyl in 1986 released large amounts of strontium-90, which dec

ays exponentially at the rate of 2.5% per year. Areas downwind of the reactor will be uninhabitable for 100 years. What percent of the original strontium-90 contamination will still be present after 50 years?
Mathematics
1 answer:
elena-14-01-66 [18.8K]3 years ago
5 0

If S(t) is the amount of strontium-90 present in the area in year t, and it decays at a rate of 2.5% per year, then

S(t+1)=(1-0.025)S(t)=0.975S(t)

Let S(0)=s be the starting amount immediately after the nuclear reactor explodes. Then

S(t+1)=0.975S(t)=0.975^2S(t-1)=0.975^3S(t-2)=\cdots=0.975^{t+1}S(0)

or simply

S(t)=0.975^ts

So that after 50 years, the amount of strontium-90 that remains is approximately

S(50)=0.975^{50}s\approx0.282s

or about 28% of the original amount.

We can confirm this another way; recall the exponential decay formula,

S(t)=se^{kt}

where t is measured in years. We're told that 2.5% of the starting amount s decays after 1 year, so that

0.975s=se^k\implies k=\ln0.975

Then after 50 years, we have

S(50)=se^{50k}\approx0.282s

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