A freshly prepared radioactive sample of half life 1 hour emits radiations that are 128 times as intense as the permissible safe limit. What is the minimum time after which the sample can be safely used?

7 h

The correct answer is Option (4) → 7 h

The intensity of radiation from a radioactive sample decays over time -

$I(t)=I_0\left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$

where,

$I(t)$ = Intensity at time t

$I_0$ = Initial Intensity

$T_{1/2}$ = half-life of substance = 1

t = time passed since the sample was prepared

$I_0=128×I_{safe}$ [given]

$I(t)=I_0\left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$

$I_{safe}=128×I_{safe}\left(\frac{1}{2}\right)^{t/1}$

$2^{-7}=\left(\frac{1}{2}\right)^t$

$t=7hours$