The number of undecayed nuclei $N$ in a sample of radioactive material as a function of time $t$ is shown in the figure. Which of the following graphs correctly shows the relationship between $N$ and the activity $A$?

The fossil bone has a ${}^{14}C:{}^{12}C$ ratio,which is $\frac{1}{16}$ of that in a living animal bone. If the half-life of ${}^{14}C$ is $5730 \, years$,then the age of the fossil bone is .......... $years$.

Half-life of a radioactive element depends upon:

$A$ radioactive material of half-life $T$ was kept in a nuclear reactor at two different instants. The quantity kept the second time was twice that kept the first time. If their present activities are $A_1$ (first) and $A_2$ (second) respectively,then their age difference is equal to:

At time $t=0$, a material is composed of two radioactive atoms $A$ and $B$, where $N_{A}(0)=2 N_{B}(0)$. The decay constant of both kinds of radioactive atoms is $\lambda$. However, $A$ disintegrates to $B$ and $B$ disintegrates to $C$. Which of the following figures represents the evolution of $N_{B}(t) / N_{B}(0)$ with respect to time $t$?
$N_{A}(0) = \text{Number of } A \text{ atoms at } t=0$
$N_{B}(0) = \text{Number of } B \text{ atoms at } t=0$

The radiation emitted by a radioactive substance with a half-life of $30 \ min$ is measured by a Geiger-Muller counter. If the count rate drops to $5 \ \text{disintegrations/sec}$ in $2 \ \text{hours}$, find the initial disintegration rate.