Problem $1$

Two tanks are separated by a massless, Lshaped gate as shown. The left side of the gate contains water of density ${}_w=1000k\frac{g}{m^3}$ while the right side of the gate contains oil of density ${}_o=600k\frac{g}{m^3}.$ The tops of both sides of the tank are open to air at atmospheric pressure, $P_{\text{a}\text{t}\text{m}}.$ The known lengths are $a=1m,b=1m,$ and $c=0\mathrm{.}25m.$ The oil surface is at height $h$ above the hinge, as shown. The gate can swing freely to the right if not forced closed by the weight of the oil. Determine the minimum height, $h,$ to hold the gate closed.