If 13 percent of the carbon-14 in a sample of cotton cloth remains, what's the aproximate age of the cloth?
When we are dating something using the carbon-14 decay we usually use this formula:
N = N0 * (1/2)^(t/thl)
Where N is the quantity remaining after the time t. N0 is the initial amount. t is the elapsed time and thl is the half-life of the substance.
We know that 13 % of the carbon-14 in the sample remains. So let's supose that we originally had 100 g of carbon-14, the amount that remains is 13 g.
N = quantity remaining = 13 g
N0 = initial quantity = 100 g
The half-life of carbon-14 is a constant. And t is our unkwnon.
thl = 5730 years t = ?
Now we can replace the values that we know and try to solve the equation for t that is the age of the cloth.
N = N0 * (1/2)^(t/thl)
13 g = 100 g * (0.5)^(t/5730 years)
13/100 = (0.5)^(t/5730 years)
0.13 = (0.5)^(t/5730 years)
We can apply on both sides ln.
ln 0.13 = ln [(0.5)^(t/5730 years)]
ln 0.13 = (t/5730 years) * ln 0.5
ln 0.13/ln 0.5 = t/5730 years
t = (ln 0.13/ln 0.5) * 5730 years
t = -2.04/(-0.69) * 5730 years
t = 16940 years
Answer = the aproximate age of the cloth is 16940 years