A cubical box is completely filled with mass m of a liquid and is given horizontal acceleration a as shown. Match the force due to fluid pressure on the faces of the cube with their appropriate values (assume zero pressure at minimum pressure).

Pressure varies with height $\Rightarrow \mathrm{P}=\rho gl$

and in horizontal with acceleration $\Rightarrow \mathrm{P}=\rho l\mathrm{a}$

so on (a) $\rho gl$ part is zero while average of $\rho l\mathrm{a}$ is

$\left[\frac{0+\rho l\mathrm{a}}{2}\right]\left(l^2\right)=\frac{\rho \mathrm{a}}{2}l\left(l^2\right)=\frac{\left(\rho l^3\right)}{2}\mathrm{a}=\frac{\mathrm{ma}}{2}$

In (B) $\rho l\mathrm{a}$ part is zero while average of $\rho gl$ is 

$\left[\frac{0+\rho \mathrm{g}l}{2}\right]l^2=\frac{\rho g}{2}\left(l^3\right)=\frac{\rho \left(l^3\right)}{2}\left(\mathrm{g}\right)=\frac{\mathrm{mg}}{2}$

In (C) Average of  $\rho gl$ and constant pressure of $\rho l\mathrm{a}$ is 

$\left[\rho al+\frac{\rho \mathrm{g}l}{2}\right]l^2=\rho a+\frac{\left(\rho g\right)}{2}\left(l^3\right)=\rho \left(l^3\right)(a+\frac{\mathrm{g}}{2})=\frac{mg}{2}+ma$

In (D) Average of $\rho l\mathrm{a}$ and constant pressure of $\rho gl$ is 

$\left[\rho gl+\frac{\rho al}{2}\right]l^2=\rho g+\frac{\left(\rho a\right)}{2}\left(l^3\right)=\rho \left(l^3\right)(g+\frac{a}{2})=\frac{ma}{2}+mg$