Radiocarbon dating, or carbon-14 dating, is a scientific method that can accurately determine the age of organic materials as old as approximately 60,000 years. The technique is based on the decay of the carbon-14 isotope. The time in which half of the original number of atom decay is defined as the half-life. It is modelled by the equation $N = N_0 e^{-kt}$, where $k$ is a constant and $t$ is the time in years. $N_0$ is the initial amount of material in an object. Carbon-14 has a half life of 5730 years, meaning that 5730 years after an organism dies, half of its carbon-14 atoms have decayed.

1. Substitute the half-life conditions into the given model

When $t = 5730$, $N = \frac{1}{2}N_0$.
$\frac{1}{2}N_0 = N_0 e^{-5730k}$

2. Solve for $k$ using natural logarithms

$0.5 = e^{-5730k}$
$\ln 0.5 = -5730k$
$k = \frac{\ln 0.5}{-5730}$
$k \approx 0.000120968 \approx 0.000121 \text{ (to 3 s.f.)}$

1. Substitute the given percentage and the value of $k$ from part (a) into the model

From part (a), $k = \frac{\ln 2}{5730}$.
Given $N = 0.08N_0$,
$0.08N_0 = N_0 e^{-\left(\frac{\ln 2}{5730}\right)t}$

2. Solve for $t$ using logarithms

$0.08 = e^{-\left(\frac{\ln 2}{5730}\right)t}$
$\ln 0.08 = -\left(\frac{\ln 2}{5730}\right)t$
$t = -\frac{5730 \ln 0.08}{\ln 2}$
$t \approx 20879.43$

3. State the final answer rounded to the nearest whole number

$t = 20879 \text{ (to nearest whole number)}$