Question

The ratio of carbon-14 to carbon-12 in a piece of charcoal from a fire pit in an archaeological excavation is found to be 12.5% of that in a sample of modern wood. Approximately how old is the site? (Carbon-14 half-life is 5730 years.) Answer as a whole number with no units.

Answer 1

17190 years

Step-by-step explanation:

The exponential decay equation is:

`N_(t) = N_(0)e^(-\lambda t)`

`(N_(t))/(N_(0)) = e^(-\lambda t)`

Where:

N(t) is the quantity at time t

N₀ is the initial amount

λ is the decay constant = ln(2)/t(1/2)

t(1/2) is the half-life

Since the ratio of carbon-14 to carbon-12 is 12.5%, we have that:

`(N_(t))/(N_(0)) = e^(-\lambda t)`

`(0.125N_(0))/(N_(0)) = e^(-\lambda t)`

`ln(0.125) = -\lambda t`

By solving the above equation for t:

`t = (ln(0.125))/(-\lambda) = (ln(0.125))/(-ln(2)/t_(1/2)) = (5730 y* ln(0.125))/(-ln(2)) = 17190 y`

Therefore, the site is 17190 years old.

I hope it helps you!