$($LSN$4)$ A notional spherical weapon assembly composed of pure Pu$-239$ has a radius of $6\mathrm{.}00cm$ and a density of $19\mathrm{.}6\frac{g}{c}m3.$ The fission process is initiated with ${10}^6$ neutrons, and the non$-$leakage probability of the assembly is $0\mathrm{.}650.$ Assume the neutron lifetime, multiplication factor, and non$-$leakage probability remain unchanged through the life of the weapon assembly.

a$.$ Calculate the neutron lifetime of the first generation and show all work.

b$.$ Calculate the multiplication factor and show all work.