Flow Through a Large Orifice

For large orifice, the flow rate varies with the head $h$ . We have to consider the flow rate $dQ$ of every elementary horizontal strip of depth $dh$ . We have $dQ = C_d v\ dA$ , where $C_d$ is the coefficient of discharge (assume $C_d=1$ in this problem), $v$ is the flow velocity at $h$ , and $dA-b\ dh$ is the elementary area of the orifice. By Bernoulli's theorem we have $\frac 12 \rho v^2 = \rho gh \implies v = \sqrt{2gh}$ . Then the flow rate $Q$ through the orifice is given by:

$\begin{aligned} dQ & = \sqrt{2gh}b \ dh \\ Q & = \int_{H_1}^{H_2} b\sqrt{2gh}\ dh \\ & = \frac {2b\sqrt{2g}h^\frac 32}3 \ \bigg|_{H_1}^{H_2} \\ & = \frac {2b\sqrt{2g}\left(H_2^\frac 32 - H_1^\frac 32\right)}3 \end{aligned}$

Therefore $A+B+C = 2+\dfrac 32 + 3 = \boxed{6.5}$ .