If the fraction of the atoms of a radioactive material left after 120years is 1/64, what is the half-life of the material?

Answer Details

The half-life of a radioactive material is defined as the time it takes for half of the original sample to decay. In this problem, we know that after 120 years, only 1/64th of the original sample is left. This means that the fraction of the original sample that has decayed is:
1 - 1/64 = 63/64
We want to know the half-life, which we can call "t". We know that after one half-life, half of the original sample will remain. So we can write:
(1/2) = (63/64)^(120/t)
To solve for t, we can take the natural logarithm of both sides and use the fact that ln(a^b) = b*ln(a):
ln(1/2) = ln[(63/64)^(120/t)]
ln(1/2) = (120/t)*ln(63/64)
t = (120/ln(63/64))*ln(2) ≈ 20.2 years
Therefore, the half-life of the radioactive material is approximately 20.2 years. The answer is closest to the option: 20 years.