Carbon-14 half life question

Question

Radioactive Carbon-14 has a half-life of approximately 5730 years. It collects in living plants and plant-eating organisms and will start to decay once the organism dies.
Assume an animal bone is discovered in an archaeological dig and is Carbon-14 dated to be roughly 9,000 years old.

 

What fraction of the initial amount of Carbon-14 is remaining? (round to three decimal places)

Answer 1

Let A(t) = amount of Carbon 14 remaining after t years.

 

A(t) = A₀e^kt, where A₀ is the initial amount.  

 

Since A(5730) = (1/2)A₀, we have (1/2)A₀ = A₀e^5730k

 

                                                      0.5 = e^5730k

 

                                                 ln(0.5) = 5730k

 

                                                       k ≈ -0.00012

 

So, A(t) = A₀e^-0.00012t

 

When t = 9000, A(9000) = A₀e^{-0.00012(9000)}

 

                                   ≈ 0.340A₀