The activity of a sample of a radioactive material is ${\mathrm{A}}_{1}$, at time ${\mathrm{t}}_{1}$ and ${\mathrm{A}}_{2}$ at time ${\mathrm{t}}_{2}$ (${\mathrm{t}}_{2}$>${\mathrm{t}}_{1}$). If its mean life T, then
(a) ${\mathrm{A}}_{1}{\mathrm{t}}_{1}={\mathrm{A}}_{2}{\mathrm{t}}_{2}$            (b) ${\mathrm{A}}_{1}-{\mathrm{A}}_{2}={\mathrm{t}}_{2}-{\mathrm{t}}_{1}$
(c) ${\mathrm{A}}_{2}={\mathrm{A}}_{1}{\mathrm{e}}^{\left({\mathrm{t}}_{1}-{\mathrm{t}}_{2}\right)/\mathrm{T}}$    (d) ${\mathrm{A}}_{2}={\mathrm{A}}_{1}{\mathrm{e}}^{\left({\mathrm{t}}_{1}/{\mathrm{t}}_{2}\right)\mathrm{T}}$

(c) $\mathrm{A}={\mathrm{A}}_{0}{\mathrm{e}}^{-\mathrm{\lambda t}}={\mathrm{A}}_{0}{\mathrm{e}}^{-\mathrm{t}/\mathrm{\tau }}$ ;  where $\mathrm{\tau }$ = mean life

So ${\mathrm{A}}_{1}={\mathrm{A}}_{0}{\mathrm{e}}^{-{\mathrm{t}}_{1}/\mathrm{T}}⇒{\mathrm{A}}_{0}=\frac{{\mathrm{A}}_{1}}{{\mathrm{e}}^{-{\mathrm{t}}_{1}/\mathrm{T}}}={\mathrm{A}}_{1}{\mathrm{e}}^{{\mathrm{t}}_{1}/\mathrm{T}}$

$\therefore {\mathrm{A}}_{2}={\mathrm{A}}_{0}{\mathrm{e}}^{-\mathrm{t}/\mathrm{T}}=\left({\mathrm{A}}_{1}{\mathrm{e}}^{{\mathrm{t}}_{1}/\mathrm{T}}\right){\mathrm{e}}^{-{\mathrm{t}}_{2}/\mathrm{T}}⇒{\mathrm{A}}_{2}={\mathrm{A}}_{1}{\mathrm{e}}^{\left({\mathrm{t}}_{1}-{\mathrm{t}}_{2}\right)/\mathrm{T}}$

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