A building in can be modeled as a one degree of freedom system with a weight W $=1000$ tons and a damping fraction of Z$\%=0\mathrm{.}25.$ If the seismic force is simulated as a concentrated lateral load $($and applied instantaneously$)$ that generates a lateral displacement of A $=8\mathrm{.}8$ cm and an initial velocity of B $=72$ cm$/$s$,$ a$)$ what should be the maximum period of the system if by regulation it can only move $18$ cm$?$ b$)$ If dampers capable of tripling the original damping fraction are installed, how much additional weight could the building have to continue complying with the $15$ cm lateral deformation limit assuming that the stiffness is the same as in item a$)?$