The decay equation can be written for 100 years.

\(\displaystyle{N}={N}_{{{0}}}{e}^{{-\lambda{t}}}\)

\(\displaystyle{96}={100}{e}^{{-{100}\lambda}}\)

\(\displaystyle{\left({96}\right)}^{{{2}}}={\left({100}{e}^{{-{100}\lambda}}\right)}^{{{2}}}\)

\(\displaystyle{96}^{{{2}}}={10}^{{{4}}}{e}^{{-{200}\lambda}}\) (A)

Let \(\displaystyle\lambda\) be the decay constant.

The decay equation can be written for 100 years.

\(\displaystyle{N}'={N}_{{{0}}}{e}^{{-{200}\lambda}}\)

\(\displaystyle{N}'={100}{e}^{{-{200}\lambda}}\) (B)

Divide A by B.

\(\displaystyle{\frac{{{96}^{{{2}}}}}{{{N}'}}}={\frac{{{10}^{{{4}}}{e}^{{-{200}\lambda}}}}{{{10}{e}^{{-{200}\lambda}}}}}\)

\(\displaystyle{N}'={92.16}\)

\(\displaystyle{N}={N}_{{{0}}}{e}^{{-\lambda{t}}}\)

\(\displaystyle{96}={100}{e}^{{-{100}\lambda}}\)

\(\displaystyle{0.96}={e}^{{-{100}\lambda}}\)

\(\displaystyle{\ln{{\left({0.96}\right)}}}={\ln{{\left({e}^{{-{100}\lambda}}\right)}}}\)

\(\displaystyle-{100}\lambda={\ln{{\left({0.96}\right)}}}\)

\(\displaystyle\lambda={0.0004}\)

The expression for half-life \(\displaystyle{\left({T}_{{{\frac{{{1}}}{{{2}}}}}}\right)}\) is,

\(\displaystyle{T}_{{{\frac{{{1}}}{{{2}}}}}}={\frac{{{0.693}}}{{\lambda}}}\)

\(\displaystyle={\frac{{{0.693}}}{{{0.0004}}}}\)

\(\displaystyle={1732.5}\) years