The age of an ancient tree trunk is estimated using radiocarbon dating. If the trunk has a C-14 decay rate that is 34% of what it is in living plants, how old is the trunk?

The half-life of C-14 is 5730 years.

1 Answer
Feb 1, 2018

I got 8939 years old.

Explanation:

I'm assuming that you are saying that there is only 34% of the total carbon-14 in the trunk left. If that's the case, then we have to do the following steps to solve the problem.

We have to use the half-lives of carbon to calculate the age, as well as the fraction remaining.

Know that, after n half-lives of the substance have passed, there is only going to be 100/(2^n)% of the substance left.

(Source: https://en.wikipedia.org/wiki/Half-life)

So, we can setup the following equation:

100/(2^n)%=34%

Removing percent from both sides, we get

100/(2^n)=34

100*2^-n=34

From here, cross multiplication gives us

2^-n=34/100=17/50

Now, we have to use logarithms. We can take the natural log of both sides, which yields

ln(2^-n)=ln(17/50)

Using the power rule for logarithms, log_c(a^b)=blog_c(a), so we get

-nln(2)=ln(17/50)

Now, we can divide by ln(2) in both sides to get

-n=(ln(17/50))/(ln(2))

n=-(ln(17/50))/(ln(2))

From here, we need to use a calculator to figure out the answer. Plugging this into a calculator, it gives us

n~~1.56 to 3 significant figures

So, the carbon-14 in the trunk has elapsed 1.56 half-lives. From the start, the problem tells us that the half-life of carbon-14 is 5730 years, so the carbon in the tree trunk is

5730*1.56=8938.8~~8939 years old

I hope that my explanation was clear! Feel free to ask me anything in the comments or send me a message.