At time $t=0$ some radioactive gas is injected into a sealed vessel. At time $T$ some more of the gas is injected into the vessel. Which one of the following graphs best represents the logarithm of the activity $A$ of the gas with time $t$?

$A$ radioactive sample of $U^{238}$ decays to $Pb$ through a process for which the half-life is $4.5 \times 10^9$ years. Find the ratio of the number of nuclei of $Pb$ to $U^{238}$ after a time of $1.5 \times 10^9$ years (given $2^{1/3} = 1.26$).

$A$ radioactive sample has an activity $A$ in air. If the sample is kept inside water,then its activity $A^{\prime}$

The half-life of $Bi^{210}$ is $5$ days. If we start with $50,000$ atoms of this isotope,the number of atoms left over after $10$ days is:

The decay constant of a radioactive element is $0.01 \ s^{-1}$. Its half-life period is ....... $s$.

$A$ radioactive sample decays $\frac{7}{8}$ times its original quantity in $15$ minutes. The half-life of the sample is $......$ minutes.