During an expansion process, the pressure of a gas changes from $18$ to $120$ psia according to the relation $P=aV+b,$ where $a=6\mathrm{.}25$ psia$/$ft $f{t}^3$ and $b$ is a constant. If the initial volume of the gas is $7\mathrm{.}5f{t}^3,$ calculate the work done during the process.