Answer
$W = 180.89 BTU$
Work Step by Step
For the first point:
$P = aV+b$
$15\ psia = 5 \frac{psia}{ft³}\times 7\ ft³ + b$
$b = -20\ psia $
For the second point:
$P = aV+b$
$100\ psia = 5 \frac{psia}{ft³}\times V_2 -20\ psia$
$V_2 = 24\ ft³$
The work done:
$W = \int_{V_1}^{V_2} P dV$
$W = \int_{V_1}^{V_2} (aV+b) dV$
$W = (\frac{a}{2}V^2+bV)|_{V_1}^{V_2}$
$W = 977.5\ psi.ft^3 \times \frac{1\ BTU}{5.40395\ psi.ft³}$
$W = 180.89 BTU$
