Scientist can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon,14C, with a halflife of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14C begins to decrease through radioactive decay. Therefore, the level of radioactivity must also decay exponentially.
Given that a parchment fragment was discovered that had about 61% as much 14C radioactivity as does plant material on Earth today.
This means that the parchment has 61% of its fragment remaining.
The amount of substance [tex]N(t)[/tex] remaining of a radioadtive substance with initial amount of substance [tex]N_0[/tex] and half life [tex]t_{\frac{1}{2}}[/tex] after t years is given by:
[tex]N(t)=N_0e^{ \frac{t}{t_{\frac{1}{2}}} [/tex]
Thus the number of years it will take the parchment to have 61% of its fragment is obtained as follows:
[tex]61=100e^{ -\frac{t}{5,730}} \\ \\ e^{ -\frac{t}{5,730}}= 0.61 \\ \\ -\frac{t}{5,730}=\ln0.61=-0.4943 \\ \\ t=5,730(0.4943)=2,832[/tex]
Therefore, the age of the parchment is 2,800 years to the nearest hundred years.
