$A$ nurse measures the blood pressure of a seated patient to be $190 \,mm$ of $Hg$.

Consider a vessel filled with a liquid up to height $H$. The bottom of the vessel lies in the $X-Y$ plane passing through the origin. The density of the liquid varies with the $Z$-axis as $\rho(z) = \rho_0 \left[ 2 - \left( \frac{z}{H} \right)^2 \right]$. If $P_1$ and $P_2$ are the pressures at the bottom surface and top surface of the liquid respectively, the magnitude of $(P_1 - P_2)$ is:

Consider a mercury-filled tube as shown in the figure below. Which of the following options about the pressures at the lettered locations $(A, B, C, D)$ is true?

Explain why:
$(a)$ The blood pressure in humans is greater at the feet than at the brain.
$(b)$ Atmospheric pressure at a height of about $6 \; km$ decreases to nearly half of its value at sea level,though the height of the atmosphere is more than $100 \; km$.
$(c)$ Hydrostatic pressure is a scalar quantity even though pressure is force divided by area.

When an air bubble of radius $r$ rises from the bottom to the surface of a lake,its radius becomes $\frac{5r}{4}$. If the atmospheric pressure is equal to the pressure exerted by a $10 \ m$ height of water column,the temperature is constant,and surface tension is neglected,what is the depth of the lake (in $m$)?