A radioactive elements get spilled over the floor of a room. Its half-liife period is 30  day's. If the initial activity is ten times the permissible value, after how many days will it be safe to enter the room?

The fraction of radioactive material remaining undecayed after each successive half-life is as follows. $1⟶\underset{t_{1/2}}{2}\underset{t_{1/2}}{⟶}\frac{1}{4}\underset{t_{1/2}}{⟶}\frac{1}{8}\underset{t_{1/2}}{⟶}\frac{1}{16}$1/10th of the radioactive material will be present in between third and fourth half-lives, i.e. in between 90 days and 120 days. The minimum time after which it is safer to enter the room may be 100 days. This may be quantitatively verified by using the rate constant 

In $\frac{N_0}{N}=kt$ For a half-life; $t_{1/2}=\frac{\ln 2}{k}$ i.e. $k=\frac{\ln 2}{t_{1/2}}$

For $N_0/N=10$ time required will be

$t=\frac{\ln N_0/N}{k}=\left(\frac{\ln 10}{\ln 2}\right)t_{1/2}=\frac{t_{1/2}}{\log 2}=\frac{30}{0.3010}=99.67\mathrm{d}$