Radioactive decay is a process where the nucleus of an unstable atom, such as Uranium-238 loses energy by emitting radiation.
The half life is the average time it take for half the nuclei in a sample to undergo radioactive decay.
Given an initial sample of x with mass N(0). After a time t the mass of x left in the sample N(t) is given by:
N(t) = N(0).2-t/t1/2 (1)
Where t1/2 is the halflife.
To answer the question we need to find t. Rearranging equation (1) we have:
- t1/2 log2[N(t)/N(0)] = t (2)
subbing the values from the question into (2)
-4.5x109 log2 [0.4/ 2] = 10.4 billion years