Question

An archaeologist in Turkey discovers a spear head that contains 32% of it's original amount of C-14.
Find the age of the spear head to the nearest year

Answer 1

Assuming that the amount of C-14 decreases by a rate constant on 1...
\[x = N \times e ^{-t}\]
x = current amount of C-14
N = original amount of C-14
t = time in years
It is stated that x = 0.32 x N
Therefore,
\[0.32 \times N = N \times e ^{-t}\]
Divide both sides by N,
\[0.32 = e ^{-t}\]
using natural logarithms,
\[\ln 0.32 = \ln e ^{-t}\]
\[\ln 0.32 = -t \times \ln e, where \ln e=1\]
So t = - ln 0.32 = 1.139.. = 1 (to the nearest year)

I'm not sure if the answer is correct because i have assumed that t is already in years and that the rate constant of this exponential decay is 1.