Question

The ends of a water trough have the shape of the region bounded by the graphs of y = x^2 and y = 4 with x and y both measured in feet. To what depth must the trough be filled with water so that the force exerted by the water on either end of the trough is 779.423 lb? (Density of water is 62.5 lb per cubic foot).

Answer 1

please help if possible !

Answer 2

What'd you get?

Answer 3

3.528 feet

Answer 4

Let h be the water level
Each small slice of area from one end of parabola to the other end = 2*sqrt(y) dy
Pressure exerting on small slice = density * (h-y) ; h-y is the depth of water from the surface.
So, F = 62.5 integral of (h-y) 2*sqrt(y) dy from 0 to h
F = 125 (2/3*h*y^3/2 - 2/5*y^5/2) from 0 to h
Substitue F with 779.423
h = 3.528 ft.