Pressure and heights of a manometer -- Find the fluid flow rate

[Jenny Physics]

Homework Statement

Find the flow rate of water. The fluid in the manometer is mercury. $d_{1},d_{2}$ are the diameters in the figure

Homework Equations

Bernoulli, continuity equations

The Attempt at a Solution

We know that

$p_{1}=p_{1Top}+\rho_{water}gh_{2}$, $p_{2}=p_{2Top}+\rho_{water}gh_{3}$

which means

$p_{1Top}-p_{2Top}=p_{1}-p_{2}+\rho_{water}gh$

on the other hand

$p_{1}-p_{2}=\rho_{Hg}gh$

so

$p_{1Top}-p_{2Top}=gh(\rho_{water}+\rho_{Hg})$

But in the solution it is

$p_{1Top}-p_{2Top}=gh\rho_{Hg}$

where did I go wrong?

 

[TSny]

We know that

$p_{1}=p_{1Top}+\rho_{water}gh_{2}$, $p_{2}=p_{2Top}+\rho_{water}gh_{3}$

which means

$p_{1Top}-p_{2Top}=p_{1}-p_{2}+\rho_{water}gh$

Check the sign of the last term on the right side of the second equation. This will modify your result.
Otherwise, I don't see anything wrong with your work. It does appear that the solution you were given is incorrect.