Pressure is force over surface.
p=F/S.
In this case the force is given by the weight of the water G, so
p = G/S;
G=m*g;
m=ρ*V,
V=S*h
since we are interested only in the volume of water which presses on the surface of the submarine S, at depth h.
So:
p= ρ*S*h*g/S = ρ*g*h.
This is the pressure at a certain depth h, exerted only by the water so we must add the atmospheric pressure p0.
Therefore the total pressure is pt = p0 + ρ*g*h.
We can see that when h=0, pt=p0 which is correct since at the surface we have only atmospheric pressure.
The ratio pt/p0 = 1+ ρ*g*h/p0
p0=101325 Pa
h = 500 m
g= 9.8 m/s^2
ρ = 1.025*10^3 kg/m^3.
We substitute in the equation and get pt/p0 = 50.56 .
So the pressure at 500m depth is 50.56 times greater than at the surface.
