Carbon-14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the...

It is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant in time. Therefore, given a sample of a particular radioisotope, the number of decay events  expected to occur in a small interval of time  is proportional to the number of atoms present  i.e.

For different atoms different decay constants apply.

The above differential equation is easily solved by separation of variables.

 

where  is the number of undecayed atoms at time 

We can now calculate decay constant  for carbon-14 using the given half-life.

Note that the above constant is usually measured in seconds rather than years.

Now we can return to the problem at hand. Since the charcoal contains only 15% ( ) of the original carbon-14, we have

Now we solve for

According to our calculation the tree was burned approximately 15679 years ago.