Examine the graph showing the half-life of the radioactive isotope substance x. based on this graph, what might be the best use of this radioactive isotope?

 T_{1/2} = 5,776  10³ years

We see this life time as it is very close to the life time of carbon 14, so it could be used for dating ancient objects

Explanation:

The radioactive decay of described by an equation of the form

        N = N₀  

where N is the number of atoms present, N₀ is the number of initial atoms λ is the activity of the material.

The average life time is defined as the time for which the number of remainng atoms is N = N₀ / 2

          = ln 2 /λ

With these expressions, the best method to determine the average life time is to find the activity of the first equation.

For this we look for a point on the graph as accurate as possible,

  N₀ = 100,  N = 30 and  t = 10,000 years

we substitute in the equation

         30 = 100 e^{-\lambda 10000}

        ln 0.3 = - λ 10000

        λ = - (ln 0.3) / 10000

        λ = 1.2 10-4

now we can find the average life time

          = ln 2 / 1,2 10-4

         T_{1/2} = 5,776  10³ years

We see this life time as it is very close to the life time of carbon 14, so it could be used for dating ancient objects