The variation of gauge pressure ( P ) with depth ( h ) for two...

Gauge pressure at a depth $h$ in a liquid of density $\rho$ is given by $P = \rho gh$.

Rearranging this equation, we get $\dfrac{P}{h} = \rho g$.

On a graph of $P$ (y-axis) versus $h$ (x-axis), the slope of the line is equal to $\dfrac{P}{h}$. Therefore, the slope is directly proportional to the density of the liquid ($\text{Slope} = \rho g$).

For liquid A, the slope is $\tan(60^\circ) = \sqrt{3}$.

For liquid B, the slope is $\tan(30^\circ) = \dfrac{1}{\sqrt{3}}$.

The ratio of their densities is:
$\dfrac{\rho_A}{\rho_B} = \dfrac{\text{Slope}_A}{\text{Slope}_B} = \dfrac{\tan(60^\circ)}{\tan(30^\circ)} = \dfrac{\sqrt{3}}{\dfrac{1}{\sqrt{3}}} = 3$

Thus, the ratio of the density of liquid A to liquid B is $3:1$.

Answer: $3:1$