The half-life of a radioactive substance is $20 \ min$. The approximate time interval $(t_{2}-t_{1})$ between the time $t_{2}$,when $2/3$ of it has decayed,and time $t_{1}$,when $1/3$ of it has decayed,is (in $min$):

$1 \, mg$ of gold undergoes decay with a $2.7 \, \text{days}$ half-life period. The amount left after $8.1 \, \text{days}$ is ......... $mg$.

The half-life of radium is about $1600$ years. Of $100 \, g$ of radium existing now, $25 \, g$ will remain unchanged after .......... $years$.

Write a formula showing the relation between half-life and average life of a radioactive substance.

$A$ radioactive nucleus $n_2$ has $3$ times the decay constant as compared to the decay constant of another radioactive nucleus $n_1$. If initial number of both nuclei are the same,what is the ratio of number of nuclei of $n_2$ to the number of nuclei of $n_1$,after one half-life of $n_1$?

The activity $R$ of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:

$t (h)$	$0$	$1$	$2$	$3$	$4$
$R (MBq)$	$100$	$35.36$	$12.51$	$4.42$	$1.56$

$(i)$ Plot the graph of $R$ versus $t$ and calculate the half-life from the graph.
$(ii)$ Plot the graph of $\ln \left( \frac{R}{R_0} \right)$ versus $t$ and obtain the value of the half-life from the graph.