# Question #bbcf9

Mar 2, 2016

#### Answer:

$45 \text{min}$

#### Explanation:

The equation for radioactive decay is:

${N}_{t} = {N}_{0} {e}^{- \lambda \text{t}}$

${N}_{0}$ is the initial number of undecayed atoms.

${N}_{t}$ is the number of undecayed atoms after time $t$.

$\lambda$ is the decay constant.

We can say that ${N}_{0}$ represents 100% of undecayed atoms.

If 34% decay then there must be 100 - 34 = 66% atoms remaining.

$\therefore 66 = 100 {e}^{- \lambda \text{t}}$

$\therefore \ln 66 = \ln 100 - \lambda \text{t}$

$\therefore \lambda \text{t} = \ln 100 - \ln 66$

$\therefore \lambda = \frac{4.605 - 4.19}{27} = 0.0514 {\text{min}}^{- 1}$

${t}_{\frac{1}{2}} = \frac{0.693}{\lambda}$

$\therefore {t}_{\frac{1}{2}} = \frac{0.693}{0.0514} = 45 \text{min}$