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 t_{1/2} is the halflife.

To answer the question we need to find t. Rearranging equation (1) we have:

- t_{1/2} * **log _{2}[N(t)/N(0)] = t (2)*

subbing the values from the question into (2)

*
-4.5x109 * log2 [0.4/ 2] = 10.4 billion years