A large block of ice $10 \mathrm{m}$ thick with a vertical hole drilled through it is floating in a lake. The minimum length of the rope required to scoop out a bucket full of water through the hole is ($0.2x$)m. Find the value of $x$. Take density of ice $= 0.9 g/c{m}^3$

In equilibrium, weight=Upthrust

i.e., ${\mathrm{V\rho }}_i\mathrm{g}={\mathrm{V}}_{\mathrm{i}}{\rho }_{\mathrm{w}}\mathrm{g}$

or  $\frac{{\mathrm{V}}_{\mathrm{i}}}{\mathrm{V}}=\frac{{\rho }_{\mathrm{i}}}{{\rho }_{\mathrm{w}}}=\frac{0.9}{1.0}=0.9$

i.e., $90\%$ or 9 m of ice block is inside the water. 

Therefore, minimum length l required to scoop out a bucket of water is nothing but the length of ice block which is above the surface of water = 10–9 =1m.