$6.$ a$)$ Find the hydrostatic force on a circular gate $4$ m in diameter in a vertical dam wall where the gate is i$)50$ m below the water surface from its center.

i$)50$ m below the water surface from its top.

iii$)50$ m below the water surface from its bottom.

b$)$ A trough is $6$ m long and $1$ m high. Vertical cross sections are equilateral triangles with the top size horizontal. Find the force on one end if the trough is filled with water.

c$)$ The cubical metal tank has a parabolic gate held in place by bolts and designed to withstand a fluid force of $160$ lb without rupturing. The liguid you plan to store has a weight$-$density of $\backslash (50\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$ft$\}^\{3\}\backslash ).$

i$)$ What is the fluid force on the gate when the liquid is $2$ ft deep?

i$)$ What is the maximum beight to which the container can be filled without exceeding the gate's design limitation$?$