$($LSN$4)$ A notional spherical weapon assembly composed of pure Pu$-239$ has an initial radius of $7\mathrm{.}75cm$ and a density of $27\mathrm{.}6\frac{g}{c}{m}^3.$ The fission process is initiated with one thousand neutrons $($which will be considered as neutron population in generation zero$),$ and the constant non$-$leakage probability of the assembly is $0\mathrm{.}435.$ Assume the multiplication factor and non$-$leakage probability remain unchanged through the life of the weapon assembly.

a$.$ Calculate the total number of neutrons present at the end of the $5^{th}$ neutron generation.

b$.$ Calculate the total number of neutrons present at the end of the ${65}^{th}$ generation.

c$.$ Approximate the yield at the end of $65$ generations. State your assumptions