Minimum Days Required for Radioactive Decay in Laboratory

Let $A_0$ be the initial radiation level, $A(t)$ be the radiation level at time $t$, and $A_{\text{safe}}$ be the safe threshold radiation level required for normal operation.

According to the radioactive decay law, the radiation level $A(t)$ after time $t$ is given by: $A(t) = A_0 \left(\frac{1}{2}\right)^{t / T_{1/2}}$

where:

• $T_{1/2} = 18 \text{ days}$ is the half-life of the radioactive isotope.
• $A_0 = 64 A_{\text{safe}}$ is the initial radiation level.

For the laboratory to be declared safe for work, the radiation level must reduce to $A_{\text{safe}}$, so we set $A(t) = A_{\text{safe}}$: $A_{\text{safe}} = 64 A_{\text{safe}} \left(\frac{1}{2}\right)^{t / 18}$

Dividing both sides by $A_{\text{safe}}$: $1 = 64 \left(\frac{1}{2}\right)^{t / 18}$

$\left(\frac{1}{2}\right)^{t / 18} = \frac{1}{64}$

Expressing $64$ as a power of $2$, we have $64 = 2^6$, so $\frac{1}{64} = \left(\frac{1}{2}\right)^6$: $\left(\frac{1}{2}\right)^{t / 18} = \left(\frac{1}{2}\right)^6$

Equating the powers on both sides: $\frac{t}{18} = 6$

$t = 6 \times 18 = 108 \text{ days}$

Thus, the minimum number of days after which the laboratory can be declared safe for work is $108$ days.

Hence, the correct option is (C).