hydrostatic force problem

[SUhottie10]

A gate in an irrigation canal is in the form of a trapezoid 3 feet wide at the bottom, 5 feet wide at the top, with the height equal to 2 feet. It is placed vertically in the canal with the water extending to its top. For simplicity, take the density of water to be 60 lb/ft cubed. Find the hydrostatic force in pounds on the gate.

I am having problems setting this problem up. It looks like its really easy but I am just not sure how to start it.

 

[galactus]

Hello SUHottie. Hope I'm on the right track with this one.

Let \(\displaystyle \L\\w(x) = 3+2u(x)\)

By similar triangles:

\(\displaystyle \L\\\frac{u(x)}{1}=\frac{2-x}{2}\)

So, \(\displaystyle \L\\u(x)=\frac{2-x}{2}\)

and \(\displaystyle w(x)=3+2(\frac{2-x}{2})=5-x\)

\(\displaystyle \L\\60\int_{0}^{2}x(5-x)dx\)

\(\displaystyle \L\\60\int_{0}^{2}(5x-x^{2})dx\)