Respuesta :
Answer:
Painting is 13725 years old.
Step-by-step explanation:
C-14 dating is the process through which age of fossils or dead tissues are determined.
One important thing about the C-14 dating is half life life period of C-14 that is 5730 years.
Now we know for C-14 decay we use the formula
[tex]A_{t}=A_{0}e^{-kt}[/tex]
where [tex]A_{t}[/tex] = present quantity of C-14
[tex]A_{0}[/tex] = Initial quantity of C-14
k = Decay constant
t = Duration or time
We will calculate the decay constant first to use this formula for age determination further.
For Half life of C-14
A_{t}=A_{0}e^{-kt}
[tex]\frac{1}{2}=1.e^{-k(5730)}[/tex]
Taking ln on both the sides
[tex]ln(\frac{1}{2})=ln(1.e^{-k(5730)})[/tex]
[tex]-ln2=-5730k(lne)[/tex]
k = [tex]\frac{ln2}{5730}=(1.21)(10)^{-4}[/tex]
Now we have been given in the question that C-14 found to be 19% and we have to determine the age of the painting.
[tex]A_{t}=A_{0}e^{-kt}[/tex]
[tex]0.19=1\times (e^{-(1.21\times 10^{-4})t})[/tex]
We will take ln on both the sides
[tex]ln(0.19)=ln[1\times (e^{-(1.21\times 10^{-4})t})][/tex]
[tex]-1.6607=-(1.21\times 10^{-4})t\times lne[/tex]
[tex]t=\frac{1.6607}{1.21\times 10^{-4}}[/tex]
t = 1.3725×[tex]10^{4}[/tex]
t = 13725 years
Therefore, painting is 13725 years old.
Answer:
13713.717 years
Step-by-step explanation:
The exponential decay model for Carbon-14 is modeled by the following formula:
[tex] A=A_{0}\;e^{-0.0001211\;t} [/tex]
where:
[tex]A_0[/tex] is the initial amount of Carbon-14
A is the final amount of Carbon-14, [tex]A = 0.19 \times A_0[/tex]
t is the elapsed time in years since the painting was made
Replacing in the formula:
[tex] 0.19 \times A_0=A_{0}\;e^{-0.0001211\;t} [/tex]
[tex] 0.19=e^{-0.0001211\;t} [/tex]
[tex] ln(0.19)=-0.0001211\;t [/tex]
[tex] \frac{ln(0.19)}{-0.0001211}=t [/tex]
[tex] 13713.717 years=t [/tex]
