How Do You Calculate Hydrostatic Pressure in a Spherical Water Tank?

[teamgonuts]

I have this homework problem and I can't figure it out:

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. Please round to the nearest 10 litre increment, and please submit only a number for your answer. (For example, if you calculate the answer to be 16277 litres, submit 16280 as your answer)

All I can figure out is the volume of the sphere =/

Help?

 

[catfood]

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[Moetasim]

Hydrostatic pressure is the force exerted by a fluid at rest due to the weight of the fluid above it. In this case, the fluid is water and the pressure is being exerted on the bottom of the spherical tank. This pressure increases with depth and is directly proportional to the density of the fluid and the acceleration due to gravity.

To solve this problem, we can use the formula for hydrostatic pressure:

P = ρgh

Where P is the pressure, ρ is the density of water (1000 kg/m3), g is the acceleration due to gravity (9.81 m/s2), and h is the height of the water column.

We know that the maximum pressure the drain can handle is 50 kPa, so we can set up the equation:

50 kPa = (1000 kg/m3)(9.81 m/s2)h

Solving for h, we get:

h = 0.0051 m

This means that the maximum height of the water column is 0.0051 meters, or 5.1 centimeters. To find the maximum volume of water, we can use the formula for the volume of a sphere:

V = (4/3)πr3

Where V is the volume, π is approximately 3.14, and r is the radius of the sphere (which is half the diameter).

Plugging in the given diameter of 8.6 meters, we get a radius of 4.3 meters. Substituting this into the volume formula, we get:

V = (4/3)(3.14)(4.3)3 = 328.2 m3

To convert this to liters, we can multiply by 1000 (since 1 m3 = 1000 liters). This gives us a maximum volume of 328,200 liters.

However, we need to round to the nearest 10 liter increment, so the final answer is 328,200 liters.