Submerge planes

Mar 2018
I'm having trouble with the following problem, can anyone help? Refer to attached picture.

A 25-mm thick steel-gate of 15-mx8-m size, specific gravity of 7.8 is hinged, as shown in the figure below, to store flash-flooding water. The density of water, at standard temperature condition, is 990 kg/m3. Compute the water level h for which the gate starts to fall.


Aug 2010
The gate "starts to fail" when the total weight of water on the gate is 12500g Newtons. To find the total weight imagine the gate divided into many horizontal strips of with "dx" so area 8dx. At height x, measured along the gate, we are at height \(\displaystyle y= x cos(30)= \frac{x\sqrt{3}}{2}\) above the bottom so at depth \(\displaystyle h- x cos(30)= h- \frac{x\sqrt{3}}{2}\). The water pressure at that depth, so the force on a unit square on the gate, is the weight of a column of water a unit square of that height- the density of water (1 in grams per cubic cm) times that depth. Multiply that by the area of the strip, 8 dx, and integrate with x from 0 to 15. Set that equal to 1250000g and solve for h.