The half-life is the time used by the concentration to get half of its initial value. The cotton cloth is 5097 years old.
Radioactive decay is the reduction of the initial concentration to the half value. The exponential decay can be given as:
[tex]\rm N(t) = N_{0}e ^{-\lamda t}[/tex]
The decay constant is calculated as:
[tex]\begin{aligned} \lamda &= \rm \dfrac {ln (2)}{t\frac{1}{2}}\\\\&= \dfrac{\rm ln(2)} {5730}\\\\&= 0.0001209\end{aligned}[/tex]
Solving further:
[tex]\begin{aligned}0.22 &= \rm e^{-0.0001209 \times t}\\\\\rm ln 0.22 &= -0.0001209 \times t\\\\\rm t &= \rm \dfrac{log 0.22}{-0.0001209}\\\\&= 5097 \;\rm years\end{aligned}[/tex]
Therefore, the cotton cloth is 5097 years.
Learn more about decay here:
https://brainly.com/question/16254926
#SPJ1
