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## Homework Statement

"Suppose that a liquid has an appreciable compressibility. Its density therefore varies with depth and pressure. The density at the surface is ρ

_{0}

(a) Show that the density varies with pressure according to ρ=ρ

_{0}e

^{kp}where P is gauge pressure at any depth and k is the compressibility, a constant.

(b) Find P as a function of depth y.

## Homework Equations

ρ = m/V (Density definition)

k = (1/V)(ΔV/ΔP) (Compressiblity definition)

dP/dy = ρg (Pascal's Law)

My professor also hinted that I should replace any deltas with differentials, that I need to eliminate V, and to find dρ/dV (which is -m/V

^{2}

## The Attempt at a Solution

From the second equation, I multiplied both sides by dP (after changing ΔV to dV and ΔP to dP) to obtain the following:

k dP = (1/V) dV

I then integrated both sides:

∫k dP (from P

_{0}to P) = ∫(1/V) dV (from 0 to V)

kP(gauge)=ln(V)

e

^{kP(gauge)}= V

I get the feeling that I'm going in the correct direction, but haven't found a viable solution from this point. I fooled around with the density definition and Pascal's Law to no avail. Any assistance would be much appreciated.