A cube of side 10 cm is made of a material of density $$\frac{d}{2}$$. It is floating in a large vessel containing two immiscible liquids whose densities are $$d$$ and $$\frac{3d}{2}$$ (i) Find the depth of immersion of the cube in the denser liquid (ii) Two opposite faces of the cube are pulled apart such that they are 12 cm apart and the resulting cuboid floats vertically in the same vessel, find its depth of immersion in the denser liquid?
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PLACEHOLDER ANSWER - the question has two answers. From the figure the lighter liquid forms a 4 cm layer. The cube's weight equals the weight of $$1000 \times \frac{d}{2} = 500d$$; with depth $$x$$ in the denser liquid, $$100 \times 4 \times d + 100 \times x \times \frac{3d}{2} = 500d$$, so $$x = \frac{2}{3}\text{ cm}$$. For the cuboid, the volume stays $$1000\text{ cm}^3$$, so the cross section is $$\frac{1000}{12}\text{ cm}^2$$ and $$\frac{1000}{12}\left(4d + \frac{3d}{2}x\right) = 500d$$, giving $$x = \frac{4}{3}\text{ cm}$$.