Calculating the Volume of a Tank Not Occupied by Water

Problem:

Final answer:

To find the volume of the tank not occupied by water at the start of the process, we need to use the given rates of water entering and withdrawn from the tank. Using the formulas for the volume of a cylinder and the rate of change of height, we can determine the height and volume not occupied by water.

Explanation:

To find the volume of the tank not occupied by water at the start of the process, we first need to determine the initial volume of the tank. Since the tank is initially two thirds full, we can calculate the volume using the formula for the volume of a cylinder: V = (πr^2h). Let's assume the radius of the tank is 'r' and the height of the tank is 'h'.

Let's say the water level in the tank is at a height of 'x' at the start of the process.

This means that the initial height of the tank occupied by water ='h - x'.

the volume of water entering the tank per second = 5.01 kg/s

therefore, density of water = 1000 kg/m^3,

the volume of water entering the tank per second is;

V_in = m_in / ρ = 5.01 / 1000 = 0.00501 m^3/s.

Similarly,

the volume of water withdrawn from the tank per second is V_out = m_out / ρ = 3.50 / 1000 = 0.0035 m^3/s.

Therefore, the change in height of the water level in the tank per second is given by the formula:

dH/dt = (V_in - V_out) / (πr^2).

Since the cross-sectional area of the tank is constant, the rate of change of height is determined solely by the difference between the inflow and outflow rates of water.

From the given information, we have: dH/dt = (0.00501 - 0.0035) / (πr^2).

Using this information, we can solve for the rate of change of the height and integrate it to find the height. Once we have the height, we can calculate the volume not occupied by water using the formula: V_not_occupied = V_total - V_water.