If 13 % of the carbon-14 in a sample of cotton cloth remains, what's the approximate age of the cloth?
To solve a half-life problem we usually use this formula:
amount remaining = initial amount (1/2)^n
Where n is the number of half-lives. We can replace n for T/Thalf:
amount remaining = initial amount (1/2)^(T/Thalf)
Where t is the age or time elapsed and Thalf is the half-life or our isotope. The half life of C-14 is 5730 years. Then:
Thalf = 5730 years
Let's supose that the cotton cloth originally had 100 g of C-14. If 13 % remains, the amount remaining is 13 g. So:
amount remaining = 13 g initial amount 100 g
Rearranging the formula we get:
amount remaining = initial amount (1/2)^(T/Thalf)
amount remaining/initial amount = (1/2)^(T/Thalf)
ln (amount remaining/initial amount) = ln [(1/2)^(T/Thalf)]
ln (amount remaining/initial amount) = (T/Thalf) *ln (1/2)
T = Thalf * ln (amount remaining/initial amount) /ln (1/2)
Now we can replace the given values:
T = 5730 years * ln(13 g/100g) / ln(1/2)
T = 5730 years * ln(0.13)/ln(0.5)
T = 5730 years * 2.94
T = 16825 years
Answer: the approximate age of the cloth is 16825 years.