c) 22920 years
I am not 100% on this solution, so someone more knowledgeable than I should definitely feel free to jump in and correct if and where necessary.
Assumption : The question doesn't make sense if
Given the above assumption we have this:
Where subscript s denotes the sample's ratio and c the current ratio.
The question then is: how many half lives are required to reduce the ratio by that factor. Mathematically that can be expressed like this:
The question provides the half life of carbon-14, so we multiply the number of half lives by the time for each half life to determine the total length of time:
This is not exactly equal to either of the four options (not ideal!), but it is closest to option (c) 22,920 years.
(c) 22,920 years
If the ratio of
Then use this equation:
Remember that tthe decay constant, λ is given by:
Then the equation becomes:
Substitute in the values:
Option (c) is closest to this value, so that is the solution.