According to one cosmological theory, there were equal amounts of the two uranium isotopes 235u and 238u at the creation of the universe in the "big bang." at present there are 137.7 atoms of 238u for each atom of 235u. using the half-lives 4.51×109 years for 238u and 7.10×108 years for 235u, calculate the age of the universe.

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.

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