The radioactivity of a sample is $R_1$ at time $T_1$ and $R_2$ at time $T_2.$ If the half-life of the specimen is $T,$ the number of atoms that have disintegrated in time $(T_2 - T_1)$ is proportional to

If the measurement errors in all the independent quantities are known,then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example,consider the relation $z = x / y$. If the errors in $x, y$ and $z$ are $\Delta x, \Delta y$ and $\Delta z$,respectively,then $z \pm \Delta z = \frac{x \pm \Delta x}{y \pm \Delta y} = \frac{x}{y}(1 \pm \frac{\Delta x}{x})(1 \pm \frac{\Delta y}{y})^{-1}$. The series expansion for $(1 \pm \frac{\Delta y}{y})^{-1}$,to first power in $\Delta y / y$,is $1 \mp(\Delta y / y)$. The relative errors in independent variables are always added. So the error in $z$ will be $\Delta z = z(\frac{\Delta x}{x} + \frac{\Delta y}{y})$. The above derivation makes the assumption that $\Delta x / x \ll 1, \Delta y / y \ll 1$. Therefore,the higher powers of these quantities are neglected.
$(1)$ Consider the ratio $r = \frac{(1-a)}{(1+a)}$ to be determined by measuring a dimensionless quantity $a$. If the error in the measurement of $a$ is $\Delta a$ $(\Delta a / a \ll 1)$,then what is the error $\Delta r$?
$(2)$ In an experiment,the initial number of radioactive nuclei is $3000$. It is found that $1000 \pm 40$ nuclei decayed in the first $1.0 \ s$. For $|x| \ll 1$,$\ln(1+x) \approx x$ up to the first power in $x$. The error $\Delta \lambda$,in the determination of the decay constant $\lambda$,in $s^{-1}$,is:

The half-life of $^{131}I$ is $8 \, days$. Given a sample of $^{131}I$ at time $t = 0$,we can assert that

$A$ radioactive element of mass $1 \,kg$ after $N$ years is left with only $125 \,g$. If the half-life of the element is $12.5 \,y$, then the value of $N$ is:

$A$ radioactive substance has a half-life of $3.8 \, days$. If the initial mass is $10.38 \, g$, how much mass (in grams) will remain after $19 \, days$?

The nuclide $^{131}I$ is radioactive,with a half-life of $8.04$ days. At noon on January $1$,the activity of a certain sample is $600 \, Bq$. The activity at noon on January $24$ will be