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

`\frac{N_0}2=N_0e^{-kt_(1/2)}`

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

`\frac12=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=-\frac{\ln0.57}k\approx\boxed{5621\,\mathrm{years}}`