Problem $2$: Fluid Statics $[10$ points$]8$

The figure below shows a container of water connected to a mercury manometer. The right side of the container is slanted and has a rectangular gate installed. The slant makes an angle of $40$ with the vertical axis. The rectangular gate has a width of $1m$ into the page.

Note: The moment of inertia about the $x-$axis for a rectangular shape is given by

$I_{}^{(c)}=\frac{1}{12}b{h}^3$

where $b$ is the base of the rectangle and $h$ is the height of the rectangle.

a$)[2$ points$]$ What is the value of the height, $h,$ of the fluid in the mercury manometer?

Ob$)[2$ points$]$ What is the resultant force due to hydrostatic pressure acting on the gate?

c$)[2$ points$]$ What is the depth at which the resultant hydrostatic pressure acts on the gate?

d$)[2$ points$]$ What is the force, $F_x$ required to keep the gate from opening? $F_x$ acts in the horizontal direction. $($If you were unable to find a value for part $c),$ assume the resultant acts at a depth of $9\mathrm{.}600m)$

C$)[2$ points$]$ If the rectangular gate were swapped for the trapezoidal gate of the same area as shown below $($on the next page$),$ would you expect $F_x$ to increase or decrease? Provide a physical explanation?