Question

why would a radioactive element need to have a known rate of decay in order for it to be useful as a dating tool?

Answer 1

Because the amount of radioactive sample left at time t, N(t), is related to the initial amount of the radioactive sample at time t=0 by the law

`N(t) = N_0 e^(-\lambda t)`
where
`\lambda` is the decay rate of the sample. If we are able to measure the fraction of sample left at time t,
`N(t) / N_0`, we can estimate its age t, but only if we know the decay rate of the sample,
`\lambda`. Without knowing this value, we cannot estimate t from the equation.