The half-life of carbon-14 is 5600 years if a piece of charcoal made from the word of a tree shows only 77% of the carbon 14 expected in living matter when did the tree die?

Question

The half-life of carbon-14 is 5600 years if a piece of charcoal made from the word of a tree shows only 77% of the carbon 14 expected in living matter when did the tree die?

Answer 1

We know that the amount of carbon-14 present in the tree t years after it dies is given by

f(t)=A_{0}e^{-kt}

where A_0 is the amount of carbon-14 present in the tree when it originally dies and k is a positive constant. Since we also know that the half-life of carbon-14 is 5600 years, we have that

(1/2)*A_{0}=A_{0}e^{-k(5600)}

1/2=e^{-5600k} (divided both sides by A_{0})

\ln(1/2)=-5600k (took natural logarithm of both sides)

-\ln(2)=-5600k

\ln(2)/5600=k (divided both sides by -5600)

k=\ln(2)/5600

Finally, since we are told that a piece of charcoal made from the wood of a tree contains only 77% of the carbon-14 that we would expect to find in living matter we have

0.77*A_{0}=A_{0}e^{-(\ln(2)/5600)t}

0.77=e^{-(\ln(2)/5600)t} (divided both sides by A_{0})

\ln(0.77)=-(\ln(2)/5600)t (took natural logarithm of both sides)

\ln(0.77)/[-(\ln(2)/5600)]=t (divided both sides by -(\ln(2)/5600))

-5600*\ln(0.77)/\ln(2)=t

t=-5600*\ln(0.77)/\ln(2)≈2112 years

This tells us that the tree died approximately 2112 years ago.