In a nuclear reactor,the activity of a radioactive substance is $2000 / s$. If the mean life of the products is $50 \text{ minutes}$,then in the steady power generation,the number of radionuclides is:

The activity of a sample is $64 \times 10^{-5} \, Ci$. Its half-life is $3 \, days$. The activity will become $5 \times 10^{-6} \, Ci$ after ......... $days$.

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 $|x| < 1$,$\ln(1 + x) = x$ up to first power in $x$. The error $\Delta \lambda$,in the determination of the decay constant $\lambda$,in $s^{-1}$,is:

The count rate of $10\,g$ of radioactive material was measured at different times and this has been shown in the figure. The half-life of the material and the total counts (approximately) in the first half-life period,respectively,are:

$A$ radioactive substance is being produced at a constant rate of $10 \text{ nuclei/s}$. The decay constant of the substance is $0.5 \text{ s}^{-1}$. After what time will the number of radioactive nuclei become $10$? Initially,there are no nuclei present. Assume the decay law holds for the sample.

The half-life of a sample of a radioactive substance is $1 \text{ hour}$. If $8 \times 10^{10}$ atoms are present at $t = 0$,then the number of atoms decayed in the duration $t = 2 \text{ hours}$ to $t = 4 \text{ hours}$ will be