Answer
d) $3.30\times10^{9}$ years.
Work Step by Step
Mass of $^{238}U$ decayed to $^{206}Pb$=
$2.47\,mg\,\,^{206}Pb\times\frac{238\,mg\,\,^{238}U}{206\,mg\,\,^{206}Pb}=2.85\,mg$
Original mass of $\,\,^{238}U=4.31\,mg+2.85\,mg=7.16\,mg$
Decay constant $k=\frac{0.693}{t_{1/2}}=\frac{0.693}{4.51\times10^{9}\,y}=1.54\times10^{-10}\,y^{-1}$
$\ln(\frac{original\,mass}{present\,mass})=kt$
$\implies \ln(\frac{7.16}{4.31})=0.5076$$=1.54\times10^{-10}\,y^{-1}(t)$
$\implies t=\frac{0.5076}{1.54\times10^{-10}\,y^{-1}}=3.30\times10^{9}\,y$
