Answer
a) $\sum\tau=1280N.m$
b) $I=711kg.m^2$
Work Step by Step
a) There are 2 forces acting on the ladder that produce torque: its weight $W$ and the painter's pulling force $P$. The torques produced are in the opposite direction: $$\sum\tau=\tau_P-\tau_W$$
The lever arm of $W$ is half the ladder's length, while that of $P$ is the whole ladder's length.
$$\sum\tau=(9.75m)P-(0.5\times9.75m)W$$
The ladder's weight $W=23.2kg\times9.8m/s^2=227.36N$ and $P=245N$ $$\sum\tau=1280N.m$$
b) We have $$I=\frac{\sum\tau}{\alpha}=\frac{1280N.m}{1.8rad/s^2}=711kg.m^2$$
