I'm 95% certain the answer is 3.528 ft.
The water pressure on the side of the trough is equal to the density of water multiplied by the depth at the point. So p = (d - y) * 62.5. To calculate the the force on the side of the trough, we need to multiply the pressure by the area. To do this we need an integral because the pressure on the side varies with depth. The area of a tiny of the side is 2 * sqrt(y) * dy. dy represents an infinitesimal vertical sliver of the trough, and 2 * sqrt(y) is the width of that sliver. To solve for the force we multiply the area by the force and take the integral from 0 to d (d is depth).
Your final equation should look like this: integral(62.5 * (d - y) * 2 * sqrt(y) * dy) from 0 to d. Setting this equal to 779.423 and solving for d, we get the depth to be 3.528 feet.
Tough guy online. Non-confrontational in person.