Moment produced by a fluid of varying density

[wahaj]

Homework Statement

Find F, the force acting at the top of the gate.
The gate is 2m high and 2m into the page. It is hinged at the bottom (red dot in the diagram)
Density of fluid varies linearly. 1000 kg/m³ at the top and 1600 kg/m³ at the bottom (depth of 4m)

Homework Equations

M_s=-\int \int \vec{r} \times \hat{n} \ P dA

The Attempt at a Solution

My origin is at the red dot. z is positive upwards and x is positive towards the left into the fluids
\vec{r} \times \hat{n} = z \hat{k} \times \hat{i} = z \hat{j}
I'm going to ignore the vector part and also the negative sign of this since I only need the magnitude.
P= -\rho g z + C \ (negative\ because\ g\ is\ negative) \\<br /> P = 0\ when\ z = 4 \\<br /> 0= -4 \rho g + C \\ C = 4 \rho g \\<br /> P = \rho g (4 - z) \\<br /> \rho = 1000 \ when\ z = 4 \ \ \ \rho = 1600 \ when\ z = 0 \\<br /> \rho = mz +b \\<br /> 1000 = 4m +b \ \ \ \ \ \ \ \ \ \ \ 1600 = 0m+b \\<br /> \rho = 1600 - 150z \\ <br /> dA = dz dy , \ limits: \ [0,2],[0,2] \\<br /> M = g \int_0^2 \int_0^2 z(1600 - 150z)(4 - z) \ dzdy \\<br /> M = 147.804 kN \\<br /> 147.804kN = 2F \\<br /> F = 73.902 kN
The actual answer is 63.1 kN. What am I doing wrong?

 

[SteamKing]

Is there supposed to be a diagram attached?

 

[wahaj]

I did attach the diagram. I don't know where it went. Let me try again
Edit: There we go

 

[SteamKing]

Your calculations check. I can't see how F = 63.1 kN.

 

[wahaj]

That's the answer my professor gave. If I am right that means the professor must have given the answer for a different question by mistake. I will ask him about this when classes start again. Thanks for checking my work, I've been losing my mind trying to find the mistake in my work.