Question

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. 3,839
B. 9,239
C. 13,839
D. 19,239

Answer 1

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:

$A=A_{o}e^{-0.0001211.t}$  (1)

Where:

$A$ is the final amount of Carbon- 14  

$A_{o}=$ is the initial amount of Carbon- 14

$t$ is the time elapsed (the value we want to find)

On the other hand, we are told the current amount of Carbon-14 $A$  is $19\%=0.19$, assuming the initial amount of Carbon-14 $A_{o}=$ is  $100\%$:

$A=0.19A_{o}$ (2)

This means: $\frac{A}{A_{o}}=0.19$ (2)

Now,finding $t$ from (1):

$\frac{A}{A_{o}}=e^{-0.0001211.t}$  (3)

Applying natural logarithm on both sides:

$ln(\frac{A}{A_{o}})=ln(e^{-0.0001211.t})$  (4)

$ln(0.19)=-0.0001211.t$  (5)

$t=\frac{ln(0.19)}{-0.0001211}$  (6)

Finally:

$t=13713.717years$  This is the age of the paintings and the option that is nearest to this value is C. 13839 years