4 is the maximum depth of the trough so we expect the answer to be less than 4. As the cross section of the trough has the shape y = x², then the width of the water level at any depth y is 2x.
Let a to b be the water depth, σ is the weight density of the fluid (water), y is the depth of water at any level and the width of water at that level is w(x).
The fluid force against the walls of the trough equals :-
b
∫σyw(x) dx. . . . a = 0 and b is the desired depth, y = x², w(x) = 2x a
b
∫62.5 * x² *2x dx = 62.5 * 2 ∫x³ dx = 125 * ¼x⁴ = 31.25x⁴ 0
As the lower level as zero, we now have :-
31.25b⁴ = 779.423
b⁴ = 24.941536
b = 2.23476 ft . . . which is the required depth of water.
