Two small solid spheres, A and B, are dropped into a tall column of...

According to Stokes' law, the terminal velocity $v_t$ of a sphere of radius $R$ and density $\rho$ falling through a fluid of density $\sigma$ is given by:

$v_t = \dfrac{2}{9} \dfrac{R^2 (\rho - \sigma) g}{\eta}$

Thus, $v_t \propto R^2 (\rho - \sigma)$.

For sphere A:
$v_A \propto r^2 (4\rho_0 - \rho_0)$
$v_A \propto 3r^2\rho_0$

For sphere B:
$v_B \propto (2r)^2 (1.5\rho_0 - \rho_0)$
$v_B \propto 4r^2 (0.5\rho_0)$
$v_B \propto 2r^2\rho_0$

Taking the ratio of their terminal velocities:

$\dfrac{v_A}{v_B} = \dfrac{3r^2\rho_0}{2r^2\rho_0} = \dfrac{3}{2}$

Therefore, the ratio $v_A : v_B$ is $3:2$.

Answer: $3:2$