The density ρ of water of bulk modulus B at a depth y in the ocean is related to the density at surface ρ₀ by the relation

To find the relation between the density of water $\rho$ at a depth $y$ in the ocean and the density at the surface ${\rho }_0$, we can proceed step-by-step using the definitions of bulk modulus and hydrostatic pressure.

Step 1: Understand the definition of Bulk Modulus
The Bulk Modulus ($B$) of a fluid is a measure of its resistance to uniform compression. It is defined as the ratio of the infinitesimal pressure increase ($dP$) to the resulting relative decrease in volume ($-dV/V$):
$B=-V\frac{dP}{dV}$

Step 2: Relate Volume change to Density change
Since mass ($m=\rho V$) remains constant, taking the differential on both sides gives:
$dm=d\left(\rho V\right)=\rho dV+Vd\rho =0$
This simplifies to:
$-\frac{dV}{V}=\frac{d\rho }{\rho }$
Substituting this into the bulk modulus formula, we get:
$B=\rho \frac{dP}{d\rho }$
Or, rearranging for the change in density:
$d\rho =\frac{\rho dP}{B}$

Step 3: Relate Pressure to Depth
The hydrostatic pressure increases with depth $y$ according to the relation:
$dP=\rho gdy$
For small density changes, we can approximate the density in the pressure expression using its surface value ${\rho }_0$ to find the pressure at depth $y$:
$P-P_0\approx {\rho }_{0}gy$
Thus, the change in pressure from the surface to depth $y$ is:
$\Delta P={\rho }_{0}gy$

Step 4: Solve for Density at Depth y
Using the approximation for small density changes ($d\rho \approx \rho -{\rho }_0$ and replacing $\rho$ with ${\rho }_0$ on the right-hand side of the differential equation):
$\rho -{\rho }_0\approx \frac{{\rho }_0\Delta P}{B}$
Substituting $\Delta P={\rho }_{0}gy$ into the equation:
$\rho -{\rho }_0=\frac{{\rho }_0\left({\rho }_{0}gy\right)}{B}$
Rearranging for $\rho$ (notated as $p$ in the options):
$p={\rho }_0(1+\frac{{\rho }_{0}gy}{B})$

Therefore, the correct relation is indeed:
p= ρ₀(1+ρ₀gy/B)