$(5)$ It is known from experiment that an initial mass $A_0$ of carbon$-14$ decays accoeding to the function

$A(t)=A_0{e}^{-kt}$

with $k=0\mathrm{.}000120.$ This coestant $k$ is the particular value acsociated with carboe$-14.$ tional to the amount present. $($This means the rate of deciy shorld be equal to some constant times the amount present at time t$.)$

$($b$)$ Suppose a famous painting was carefully analyzed and found to have $94\mathrm{.}0\%$ to $94\mathrm{.}5\%$ of its original carbon$-14$ remaining from the time it was painted. Estimate an age range for this putating.