Ancient paintings were found on cave walls in South America. The Carbon-14 in the paintings was measured and was found to be 19% of the original weight. How old were the paintings?A. 3,839B. 9,239C. 13,839D. 19,239

Answer: C. 13,839 (the answer is not among the given options, however the result is near this value)Step-by-step explanation:The exponential decay model for Carbon- 14 is given by the followig formula:[tex]A=A_{o}e^{-0.0001211.t}[/tex]  (1)Where:[tex]A[/tex] is the final amount of Carbon- 14  [tex]A_{o}=[/tex] is the initial amount of Carbon- 14[tex]t[/tex] is the time elapsed (the value we want to find)On the other hand, we are told the current amount of Carbon-14 [tex]A[/tex]  is [tex]19\%=0.19[/tex], assuming the initial amount of Carbon-14 [tex]A_{o}=[/tex] is  [tex]100\%[/tex]:[tex]A=0.19A_{o}[/tex] (2)This means: [tex]\frac{A}{A_{o}}=0.19[/tex] (2)Now,finding [tex]t[/tex] from (1):[tex]\frac{A}{A_{o}}=e^{-0.0001211.t}[/tex]  (3)Applying natural logarithm on both sides:[tex]ln(\frac{A}{A_{o}})=ln(e^{-0.0001211.t})[/tex]  (4)[tex]ln(0.19)=-0.0001211.t[/tex]  (5)[tex]t=\frac{ln(0.19)}{-0.0001211}[/tex]  (6)Finally:[tex]t=13713.717years[/tex]  This is the age of the paintings and the option that is nearest to this value is C. 13839 years