A sample has a $14$C activity of $0\mathrm{.}0082$ Bq per gram of carbon. $($a$)$ Find the age of the sample,

assuming that the activity per gram of carbon in a living organism has been constant at a value of

$0\mathrm{.}23$ Bq$.($b$)$ Evidence suggests that the value of $0\mathrm{.}23$ Bq might have been as much as $46\%$ larger.

Repeat part $($a$),$ taking into account this $46\%$ increase.