Introduction: In this problem, we are given a cylindrical tank filled with water and we need to find out the force required to close the discharge tube at the bottom of the tank.

Approach: To solve this problem, we need to use the concept of hydrostatic pressure.

Hydrostatic Pressure: Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It is directly proportional to the depth and density of the fluid and can be calculated using the formula:

P = ρgh

where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity and h is the depth of the fluid.

Explanation: In this problem, the water in the cylindrical tank exerts a pressure on the bottom of the tank due to the force of gravity. The pressure at the bottom of the tank is equal to the hydrostatic pressure exerted by the water at that depth.

The force required to close the discharge tube at the bottom of the tank is equal to the pressure exerted by the water on the bottom of the tank.

We can calculate the pressure using the formula:

P = ρgh

where ρ is the density of water, g is the acceleration due to gravity and h is the height of the water in the tank.

The force required to close the discharge tube is equal to the pressure multiplied by the area of the bottom of the tank.

Conclusion: In conclusion, we can say that the force required to close the discharge tube at the bottom of the cylindrical tank filled with water can be calculated using the formula P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity and h is the depth of the fluid. The force required is equal to the pressure multiplied by the area of the bottom of the tank.