Question 2.12: A barge has a uniform rectangular cross section of width 2L ...

If the barge has length b into the paper, the waterline area, relative to tilt axis O, has a base b and a height 2L; therefore, I_{o} = b(2L)^3/12. Meanwhile,\nu_{sub}= 2LbH. Equation (2.37) predicts
\overline{MG}= \frac{I_{o}}{\nu_{sub} }-\overline{GB}=\frac{8bL^3/12}{2LbH} -\frac{H}{2} =\frac{L^2}{3H}-\frac{H}{2}
The barge can thus be stable only if
L^2 \gt 3H^2/2   or   2L\gt 2.45H
The wider the barge relative to its draft, the more stable it is. Lowering G would help also.