A weather balloon filled with ideal gas is released from earth and floats upward through the atmosphere. Its initial pressure and volume at the surface are Pi $=$ Patm $($standard atmospheric pressure$),$ and Vi $=1\mathrm{.}0$m$3.$ At some later time the pressure in the balloon has decreased to Pf $=0\mathrm{.}4$Patm and the volume has increased to Vf $=2\mathrm{.}2$m$3.$ The internal energy of the gas in the ballon is given by U $($T $)=506\mathrm{.}5$ KJ $$ T $.$ Assume that during its ascent, the pressure and volume are always related by P $=$ AV $+$ B$.($a$)$ Determine the values of A and B$,$ include the appropriate units. $($b$)$ If the initial temperature is $300$K$,$ what is the final temperature?$($c$)$ How much work is done by the gas in the balloon during this time? $($d$)$ How much heat was absorbed by the gas during this process?