Suppose you have a perfectly spherical water tank with an inside diameter of 8.6 metres. If the drain at the bottom of the tank can't handle a hydrostatic pressure of more than 50 kilopascals, what is the maximum volume of water, in litres, that can be contained in the tank? Assume that gravitational acceleration is exactly 9.81 m/s2.
Height of water makes the pressure. So solve for height.
50kpa/101.3=H/.76m
Now, solve for H. That would be the height of mercury in the tank, but the tank holds water, so multiply H by 13.2. Now you have the depth of the water, use geometry (or calculus) to find the volume.
The author probably had a different technique in mind..
To find the maximum volume of water that can be contained in the tank, we need to consider the hydrostatic pressure at the bottom of the tank.
The hydrostatic pressure at a point in a fluid is given by the formula:
P = ρgh
Where:
P is the pressure (in Pascals)
ρ is the density of the fluid (in kilograms per cubic meter)
g is the acceleration due to gravity (in meters per second squared)
h is the height or depth of the fluid (in meters)
In this case, the hydrostatic pressure at the bottom of the tank is limited to 50 kilopascals (50,000 Pascals). We need to find the corresponding depth of water that will result in this pressure.
Let's start by re-arranging the formula to solve for the depth of water:
h = P / (ρg)
Given:
Inside diameter of the tank = 8.6 meters
Radius of the tank (r) = diameter / 2 = 8.6 / 2 = 4.3 meters
Density of water (ρ) = 1000 kg/m^3
Acceleration due to gravity (g) = 9.81 m/s^2
Pressure limit (P) = 50,000 Pascals
Substituting the values into the formula:
h = 50,000 / (1000 * 9.81)
Now, we can calculate the depth of water:
h ≈ 5.108 meters
The depth of water that will result in a pressure of 50 kilopascals at the bottom of the tank is approximately 5.108 meters.
To find the maximum volume of water that can be contained in the tank, we can calculate the volume of a spherical cap (the portion of the sphere submerged in water) and subtract it from the volume of the full sphere.
The formula to calculate the volume of a spherical cap is given by:
V_cap = (πh^2/3) * (3R - h)
Where:
V_cap is the volume of the spherical cap
h is the depth of water
R is the radius of the tank
Substituting the values:
V_cap = (π * 5.108^2 / 3) * (3 * 4.3 - 5.108)
Now, we can calculate the volume of the spherical cap:
V_cap ≈ 311.855 cubic meters
To find the volume of the full sphere, we can use the formula:
V_sphere = (4/3) * π * R^3
Substituting the value:
V_sphere = (4/3) * π * 4.3^3
Now, we can calculate the volume of the full sphere:
V_sphere ≈ 333.532 cubic meters
Finally, let's subtract the volume of the spherical cap from the volume of the full sphere to find the maximum volume of water that can be contained in the tank.
Max volume = V_sphere - V_cap
≈ 333.532 - 311.855
Therefore, the maximum volume of water that can be contained in the tank is approximately 21.677 cubic meters, which is equivalent to 21,677 liters.