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
