Treat the gate as weightless with a frictionless hinge, as implicit in the source. For unit width, the vertical leg has triangular water-pressure force $F_H=\gamma h^2/2$ acting h/3 above the hinge, with moment $\gamma h^3/6$. The2 m horizontal leg has upward pressure $\gamma h$,resultant $F_V=\gamma hL$ at L/2,with opposing moment $\gamma hL^2/2$. At opening,$h^3/6=hL^2/2$,so $h=\sqrt3L=2\sqrt3=\mathbf{3.46}$ m.
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