I. The self$-$regulating floodgate ABC $,$ pinned at B $,$ is pressed against the lip of the spillway at C by the action of the $36$ kN weight at A $.$ The gate is to open when the water level reaches a height $\backslash (\backslash$mathrm$\{$h$\}=6\backslash$mathrm$\{$~m$\}\backslash )($Hint: i$.$e$.$ the force at C is zero when $\backslash (\backslash$mathrm$\{$h$\}=6\backslash$mathrm$\{$~m$\}\backslash )).$ Assume the width of the gate into the page is $0\mathrm{.}5$ m $,$ and the height $\backslash (\backslash$mathrm$\{$h$\}=6\backslash$mathrm$\{$~m$\}\backslash ).$

a$)$ Given that $\backslash (\backslash$mathrm$\{$h$\}=6\backslash$mathrm$\{$~m$\}\backslash ),$ determine the equivalent force acting on the gate BC due to the water pressure. Assume the density of water, $\backslash (\backslash$rho $\backslash ),$ is $\backslash (1000\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$ Specify the equivalent force magnitude and location from $\backslash ($ B $\backslash ).$ b$)$ If the gate is to open when the water level reaches a height $\backslash (\backslash$mathrm$\{$h$\}=6\backslash$mathrm$\{$~m$\}\backslash ),$ determine the distance $\backslash ($ x $\backslash )$ locating the weight $\backslash ($ A $\backslash ).$ Neglect the weight of the gate.

c$)$ Determine the reactions at pin B$.$