Question $2$: $(10$ points$)$

The decay of radioactive materials can be used to date archaeological remains. If the amount present at time $\backslash ($ t $\backslash )($in years$)$ is given by $\backslash (\backslash$mathrm$\{$N$\}(\backslash$mathrm$\{$t$\})\backslash )($in grammes$),$ then the decay satisfies the equation

$\backslash [$

$\backslash$frac$\{\backslash$mathrm$\{$d$\}\}\{\backslash$mathrm$\{$~d$\}$ t$\}$ N$($t$)=-$a N$($t$)$

$\backslash ]$

where where $\backslash (\backslash$mathrm$\{$a$\}>0\backslash )$ is a positive constant. Given that the half$-$life $($the amount of time after which the mass of the material has halved$)$ of some radioactive material is $5,782$ years, find a value for the constant a$.$ If $1$ gramme of the material was present in a sample from a pot when it was created and only $27/100$ grammes remain now, then how old is the pot?

The pot is years old $($give an approximation with at least $3$ decimal places AFTER the decimal point$)$