Question

I need a clear explanation please bc i don't get how I'm supposed to do this. :,)

Answer 1

The half-life $t_{1/2}$ is the amount of time it takes for some quantity $N_0$ of carbon-14 to decay to half the original amount, or $N=\frac{N_0}2$.

In terms of the formula, it's the time such that

$\dfrac{N_0}2=N_0e^{-kt_{1/2}}$

and we can divide both sides by the original amount to get

$\dfrac12=e^{-kt_{1/2}}$

We want to find the time $t$ it takes for 57%, or 0.57, of the original amount to remain. This means we solve for $t$ in

$0.57N_0=N_0e^{-kt}$

or

$0.57=e^{-kt}$

We're given $k=0.0001$; plug this in and solve for $t$:

$0.57=e^{-kt}\implies\ln0.57=-kt\implies t=-\dfrac{\ln0.57}k\approx\boxed{5621\,\mathrm{years}}$