At shallow depth h,the pressure in the ocean | vedclass

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(A) The pressure at a depth $h$ is given by the differential equation $dP = \rho(P) g dh$. Since water is compressible,its density $\rho$ increases as

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(A) The pressure at a depth $h$ is given by the differential equation $dP = \rho(P) g dh$. Since water is compressible,its density $\rho$ increases as the pressure $P$ increases with depth $h$. This means that as $h$ increases,the rate of change of pressure with respect to depth,$\frac{dP}{dh} = \rho g$,also increases because $\rho$ is an increasing function of $P$. Mathematically,$\frac{d^2P}{dh^2} = g \frac{d\rho}{dP} \frac{dP}{dh} > 0$. This indicates that the graph of $P$ versus $h$ must be concave upwards (i.e.,the slope increases as $h$ increases). Among the given options,graph $A$ shows an upward-curving (concave up) relationship,which correctly represents the increasing rate of pressure change due to the increasing density of water at greater depths.

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