A bottle has an opening of radius $a$ and length $b$. A cork of length $b$ and radius $(a + \Delta a)$ where is $(\Delta << a)$ is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is $B$ and frictional coefficient between the bottle and cork is $\mu$ then the force needed to push the cork into the bottle is :

Question

$\begin{array}{1 1} (1) (\pi \mu B b) \Delta a \\ (2) (2\pi \mu B b ) \Delta a \\ (3) (\pi \mu B b) a \\ (4) (4 \pi \mu B b) \Delta a\end{array} $