A rectangular concrete block (specific gravity = 2.5) is used as a retaining wall in a reservoir of water. The height and width of the block are x and y respectively. The height of water in the reservoir is $\mathrm{z}=\frac{3}{4}\mathrm{x}$. The concrete block cannot slide on the horizontal base but can rotate about an axis perpendicular to the plane of the figure and passing through point A

 Calculate the minimum value of the ratio y/x for which the block will not begin to overturn about A.
 

 Let the dimension of the block perpendicular to the given figure be b. Consider a strip of width dh on the wall of the block in contact with water at a depth h below the free surface of water.
Hydrostatic force on the strip is dF = $\mathrm{\rho }$gh(b dh)
Torque of this force about rotation axis through A is $\mathrm{d\tau }=(\mathrm{z}-\mathrm{h})\mathrm{dF}$
$\mathrm{\tau }=\mathrm{\rho gb}\left[\mathrm{z}{\int }_0^{\mathrm{z}} \mathrm{hdh}-{\int }_0^{\mathrm{z}} {\mathrm{h}}^2\mathrm{dh}\right]=\mathrm{\rho gb}\left[\frac{{\mathrm{z}}^3}{2}-\frac{{\mathrm{z}}^3}{3}\right]=\mathrm{\rho gb}\frac{{\mathrm{z}}^3}{6}=\frac{\mathrm{\rho gb}}{6}{\left(\frac{3}{4}\mathrm{x}\right)}^3=\frac{9}{128}{\mathrm{\rho gbx}}^3$
Torque of weight of concrete block is
$$\begin{gathered} {\mathrm{\tau }}_{\mathrm{g}}=\mathrm{Mg}\cdot \frac{\mathrm{y}}{2}=\left(\mathrm{xyb}\right)(2.5\mathrm{\rho })\mathrm{g}\frac{\mathrm{y}}{2}=\frac{5}{4}{\mathrm{\rho gbxy}}^2 \\ \text{ In critical case }{\mathrm{\tau }}_{\mathrm{g}}=\mathrm{\tau } \\ \therefore \frac{5}{4}{\mathrm{\rho gbxy}}^2=\frac{9}{128}{\mathrm{\rho gbx}}^3\Rightarrow {\left(\frac{\mathrm{y}}{\mathrm{x}}\right)}^2=\frac{9}{32\times 5}\Rightarrow \frac{\mathrm{y}}{\mathrm{x}}=\frac{3}{4\sqrt{10}} \end{gathered}$$