Answer:
6 billion years.
Step-by-step explanation:
According to the decay law, the amount of the radioactive substance that decays is proportional to each instant to the amount of substance present. Let
be the amount of
and
be the amount of
after
years.
Then, we obtain two differential equations

where
and
are proportionality constants and the minus signs denotes decay.
Rearranging terms in the equations gives

Now, the variables are separated,
and
appear only on the left, and
appears only on the right, so that we can integrate both sides.

which yields
,
where
and
are constants of integration.
By taking exponents, we obtain

Hence,
,
where
and
.
Since the amounts of the uranium isotopes were the same initially, we obtain the initial condition

Substituting 0 for
in the general solution gives

Similarly, we obtain
and

The relation between the decay constant
and the half-life is given by

We can use this fact to determine the numeric values of the decay constants
and
. Thus,

and

Therefore,

We have that

Hence,

Solving for
yields
, which means that the age of the universe is about 6 billion years.