Question

Some amount of a radioactive substance (half-life $=10days)$ is spread inside a room and consequently the level of radiation becomes 50 times the permissible level for normal occupancy of the room. After how many days will the room be safe for occupation?

Answer 1

We have, $\frac{N}{N_0}={\left(\frac{1}{2}\right)}^{t/T}$

Given, half-life $T=10days$
$\frac{N}{N_0}=\frac{1}{50}\Rightarrow \frac{1}{50}={\left(\frac{1}{2}\right)}^{t/T}$

Taking log:
$\Rightarrow log1-log50=\frac{t}{T}(log1-log2)$

$\Rightarrow 0-log\frac{100}{2}=\frac{t}{T}(0-log2)$

$\Rightarrow log100-log2=\frac{t}{T}(0-log2)$

$\Rightarrow log100-log2=\frac{t}{T}log2$

As $lo{g}_{10}2=0.3010$, we have
$\Rightarrow 2-0.3010=\frac{t}{T}\times 0.3010$

$t=\frac{1.6990}{0.3010}T=\frac{1699}{301}\times \left(10days\right)=56.45days$