A tank consists of a right circular cylinder at height (h)surmounted by a hemisphere at a radius (r) with an outlet atthe top. The tank is full of water at density ρkg/m3. Find the work required to empty thetank.

To find the work required to empty the tank, we need to calculate the potential energy of the water in the tank.

Step 1: Find the volume of the tank.
The tank consists of a cylinder and a hemisphere. The volume of the cylinder can be calculated using the formula Vcylinder = πr^2h, where r is the radius of the cylinder and h is the height of the cylinder. The volume of the hemisphere can be calculated using the formula Vhemisphere = (2/3)πr^3. The total volume of the tank is Vtotal = Vcylinder + Vhemisphere.

Step 2: Find the mass of the water in the tank.
Mass (m) is equal to density (ρ) times volume (V). So, the mass of the water in the tank is m = ρ * Vtotal.

Step 3: Find the height of the water in the tank.
When the tank is full, the height of the water is equal to the height of the cylinder (h).

Step 4: Find the potential energy of the water.
Potential energy (U) is equal to the mass (m) times the acceleration due to gravity (g) times the height (h). So, U = m * g * h.

Step 5: Find the work required to empty the tank.
The work done to lift the water out of the tank is equal to the potential energy of the water. So, the work required to empty the tank is W = U.

Note: Make sure to convert all units to the same system (e.g., meters, kilograms, etc.) before performing any calculations.

I hope this helps! Let me know if you have any further questions.

To find the work required to empty the tank, we need to determine the potential energy of the water in the tank.

First, let's break down the tank into two parts: the cylindrical region and the hemispherical region.

1. Cylindrical region:
The potential energy of water in the cylindrical region is given by the formula:
PE_cylinder = m × g × h_cylinder
where m is the mass of the water in the cylindrical region, g is the acceleration due to gravity, and h_cylinder is the height of the cylindrical region.

To find the mass of the water in the cylindrical region, we need to calculate the volume of the cylindrical region and multiply it by the density of water.

The volume of the cylindrical region is given by the formula:
V_cylinder = π × r^2 × h_cylinder

So, the mass of the water in the cylindrical region is:
m_cylinder = V_cylinder × ρ

Therefore, the potential energy of water in the cylindrical region is:
PE_cylinder = m_cylinder × g × h_cylinder

2. Hemispherical region:
The potential energy of water in the hemispherical region is given by the formula:
PE_hemisphere = m × g × (h_cylinder + r)
where h_cylinder is the height of the cylindrical region and r is the radius of the hemisphere.

The volume of the hemispherical region is given by the formula:
V_hemisphere = (2/3) × π × r^3

So, the mass of the water in the hemispherical region is:
m_hemisphere = V_hemisphere × ρ

Therefore, the potential energy of water in the hemispherical region is:
PE_hemisphere = m_hemisphere × g × (h_cylinder + r)

Finally, the total potential energy of the water in the tank is the sum of the potential energies in both regions:
PE_total = PE_cylinder + PE_hemisphere

To calculate the work required to empty the tank, we need to multiply the potential energy by -1, since work is done against gravity:
Work = -PE_total

By substituting the equations for PE_cylinder and PE_hemisphere into the equation for PE_total, we can find the work required to empty the tank.

Note: Please ensure that all units are consistent (e.g., using meters for length and kilograms per cubic meter for density).