Carbon Dating The Carbon isotope C is widely used for dating of ancient artifacts containing biological material. It undergoes decay with a half-life time of Ti/2 = 5700 ycars. Suppose an ancient artifact originally contained 10-12 kg of C (recall hat 1 mol of a substance, which has a mass corresponding to the atomic mass number in grams, contains NA -6.022 102 particles). a) Derive analytically the relation between half life time, Tia and decay constant T as defined in class. [2 points b) write a computer code to numerically calculate the activity of the sample, defined as R(t)-dN/dt, over a duration of 20,000 years. Use numerical time-step widths of 10, 100 and 1000 years. Plot the results in appropriate units together with the exact (analytical) solution in the same graph. (Use the log scale for the y-axis for a better display.) [12 points for the python code and 6 points for plotting and documentation