The height to which a cylindrical vessel must be filled with a homogeneous liquid,such that the average force with which the liquid presses the side of the vessel is equal to the force exerted by the liquid on the bottom of the vessel,is equal to:

$A$ tank of oil has a height of $4 \,m$ and a density of $850 \,kg \,m^{-3}$. The gauge pressure at the bottom of the tank is (Given: $1 \,atm = 10^5 \,Pa$,Acceleration due to gravity $g = 10 \,m \,s^{-2}$) (in $\,kPa$)

$(a)$ Pressure decreases as one ascends the atmosphere. If the density of air is $\rho$,what is the change in pressure $dp$ over differential height $dh$?
$(b)$ Considering the pressure $P$ to be proportional to the density,find the pressure $P$ at a height $h$ if the pressure on the surface of the earth is $P_{0}$.
$(c)$ If $P_{0} = 1.03 \times 10^5 \text{ N/m}^2$,$\rho_0 = 1.29 \text{ kg/m}^3$,and $g = 9.8 \text{ m/s}^2$,at what height will the pressure drop to $\frac{1}{10}$ of the value at the surface of the earth?
$(d)$ This model of the atmosphere works for relatively small distances. Identify the underlying assumption that limits the model.

Show that the pressure produced due to a fluid column depends on the height of the fluid column and the density of the fluid.

As the depth increases in a vessel filled with liquid, the pressure ..........

Why does food cook faster in a pressure cooker?