find the force in newtons

Question

A tank full of water is designed with ends in the shape of an isosceles triangle with height 13 meters and width, at the top, of 14 meters. Find the hydrostatic force in newtons on one end of the tank.

Answer 1

height, h = 13 m

base, b = 14 m

Force, F = ρgh_cA

where ρg = the weight of water = 9800

h_c = the depth from surface of the water to the vertical centroid of the inverted triangle = h/3

A = the area of the triangle = bh/2

F = 9800(13/3)(13•14/2) = 3864466.67 Newtons

Answer 2

Let W=width of base = 14m and h=height =13m Let w= width of the water in the tank and x= the height of the water from the bottom of the tank. Use similar triangles to establish a relationship between the parameters of the end of the tank. So w/x= W/h =14/13 and the area of the rectangular strip is14xdx/13. The pressure is P=ρg(13-x) where ρ is the density of water in km/m³. Since force is pressure times area the integral to find force becomes ∫₀¹³ρg(13-x)14xdx/13 =14ρg/13∫₀¹³(13-x)dx. Doing the integration 14ρg(13x²/2 -x³/3)₀¹³. Solving this expression gives F= 7ρg169/3 and using ρ=1000 km/m³ gives F= 3.864x10⁶N.