c
(c) \({\lambda _\alpha } = \frac{1}{{1620}}\) per year and \({\lambda _\beta } = \frac{1}{{405}}\) per year and it is given that the fraction of the remained activity \(\frac{A}{{{A_0}}} = \frac{1}{4}\)

Total decay constant

\(\lambda = {\lambda _\alpha } + {\lambda _\beta } = \frac{1}{{1620}} + \frac{1}{{405}} = \frac{1}{{324}}per\,year\)

We know that \(A = {A_0}{e^{ - \lambda t}}\)==> \(t = \frac{1}{\lambda }{\log _e}\frac{{{A_0}}}{A}\)

==>\(t = \frac{1}{\lambda }{\log _e}4 = \frac{2}{\lambda }{\log _e}2\)

\(=324 × 2 × 0.693 = 449 \,years.\)