Question

A solid block of wood is floating in water such that $\frac{1}{2}$ of its volume is submerged. When the same block is put in alcohol, $\frac{1}{3}$ of its volume is submerged. Now a mixture of water and alcohol is made taking equal volume of both and the wooden block is put it. Percentage (%) of its volume that is now submerged equals

• A. 5
• B. 10
• C. 15
• D. None of these

Answer 1

The correct option is D None of these

Weight of block = $\frac{1}{2}$ weight of water for that volume

i.e. ${\varrho }_b{V}_b={\varrho }_w{V}_w=\varrho w\frac{V_b}{2}$

& ${\varrho }_b{V}_b={\varrho }_a\frac{V_b}{3}$

so ${\varrho }_w=2{\varrho }_b\&{\varrho }_a=3{\varrho }_b$

now if average density of mixture is $\frac{{\varrho }_a{V}_a+{\varrho }_w{V}_w}{2}$

${\varrho }_m=\frac{2{\varrho }_b\frac{V}{2}+3{\varrho }_b\frac{V}{2}}{V}$

= $\frac{5}{2}{\varrho }_b$

so ${\varrho }_b{V}_b={\varrho }_m{V}_m$

${\varrho }_b{V}_b=\frac{5}{2}{\varrho }_b{V}_m$

$V_m=\frac{2}{5}{V}_b$

So % Volume submerged = $\frac{V_m}{V_b}\times 100$

= 40%