A cylindrical diving bell 5 m in diameter and 10 m tall with an open bottom is submerged to a depth of 150 m in the ocean. The temperature of the air at the surface is 20 degrees Celsius, and the air temp 150 m down is 2 degrees Celsius. The density of seawater is 1025 kg/m^3. How high does the sea water rise in the bell when the bell is submerged?

STEPS:
(1) Find volume of cylinder
(2) Find volume of air in cylinder when submerged
(3) Vcyl - Vair = Vwater

I think you use the formula P=Po + pgh to find the height of the air pocket, but I'm not sure what values to use for P, Po, and p. Please help.

h=(2*sigma*cos(theta))/(rho*g*r)

r= radius
sigma= 0.123(1-0.00139T)
theta=contact angle (for water=0)
rho=density

To find the height of the seawater rise in the bell when submerged, we can follow these steps:

Step 1: Find the volume of the cylinder
The volume of a cylinder can be calculated using the formula:
Vcyl = πr^2h
Given that the diameter (d) of the diving bell is 5m, the radius (r) can be calculated as r = d/2 = 5/2 = 2.5m. The height (h) of the bell is given as 10m. Hence, the volume of the cylinder is:
Vcyl = π(2.5)^2(10)

Step 2: Find the volume of air in the cylinder when submerged
Since the diving bell has an open bottom, the volume of air will decrease as the water rises inside. To find the volume of air in the bell when submerged, we need to calculate the volume of the cylinder up to the water level. This can be done by multiplying the cross-sectional area of the cylinder by the height of the air pocket.
The cross-sectional area can be calculated by multiplying the radius (r) by itself and then multiplying it by π. The height of the air pocket is the total height of the bell minus the height of the water in the bell. Thus, the volume of air is:
Vair = πr^2(h - h_water)

Step 3: Calculate the volume of water inside the bell
The difference between the volume of the cylinder and the volume of air will give us the volume of the water inside the bell:
Vwater = Vcyl - Vair

Now, let's proceed with finding the height of the seawater rise in the bell.

To calculate the height of the air pocket inside the bell, we can use the given temperatures and the ideal gas law. Since the densities are not given explicitly, we can use the ideal gas law equation in terms of density to relate the pressures and temperatures.

The ideal gas law equation in terms of density is:
P = ρRT

Where:
P is the pressure
ρ is the density
R is the specific gas constant (constant for air)
T is the temperature

Step 4: Calculate pressure at the surface and at 150m depth
Using the ideal gas law equation, we can calculate the pressures at the surface and at a depth of 150m.

For the surface temperature of 20 degrees Celsius (293.15 Kelvin) and assuming standard atmospheric pressure:
P_surface = Po (pressure at the surface) = ρoRT_surface

For the depth temperature of 2 degrees Celsius (275.15 Kelvin):
P_depth = Po + ρg(h - h_water)

Step 5: Calculate the density of air at the surface and at 150m depth
Since the density of air changes with temperature and pressure, we need to calculate the densities at the surface and at a depth of 150m.

Using the ideal gas law equation, we can rearrange the equation to solve for density:
ρ = P / (RT)

The density of air at the surface:
ρ_surface = P_surface / (RT_surface)

The density of air at the depth of 150m:
ρ_depth = P_depth / (RT_depth)

Step 6: Calculate the density difference between seawater and air
To find the difference in density between seawater and air, we can subtract the density of air from the density of seawater:
Δρ = ρ_seawater - ρ_air

Step 7: Calculate the volume of water inside the bell
Using the mass equation, we can calculate the volume of water:
Vwater = (mass of water) / (density of water)

Step 8: Calculate the height of water rise in the bell
To find the height of the seawater rise in the bell, divide the volume of water by the cross-sectional area of the bell:
h_water = Vwater / (πr^2)

Finally, calculate the value of h_water to get the answer.