Question

A brine solution of salt flows at a constant rate of 9 L/min into a large tank that initially held 100 L of brine solution in which was dissolved 0.4 kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is 0.04 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.03 kg/L? Determine the mass of salt in the tank after t min. mass = kg When will the concentration of salt in the tank reach 0.03 kg/L? minutes. The concentration of salt in the tank will reach 0.03 kg/L after (Round to two decimal places as needed.)

Answer 1

The mass of salt in the tank after t minutes is determined by solving the differential equation which models the rate of change of the mass of salt. The mass increases until reaching a steady state. To find when the concentration reaches 0.03 kg/L, set the function representing the mass equal to 3 kg and solve for t.

Step-by-step explanation:

To determine the mass of salt in the tank after t minutes, we can use a differential equation based on the rate of change of the mass of salt in the tank. Let's denote M(t) as the mass of salt in the tank at any time t. The rate at which salt enters the tank is the product of the concentration of salt in the incoming brine and the flow rate: 0.04 kg/L × 9 L/min = 0.36 kg/min.

At the same time, the salt is leaving the tank due to the outgoing brine. The rate at which salt leaves the tank is the product of the concentration of salt in the tank and the flow rate. The concentration of the outgoing brine is M(t)/100, since the volume of the solution in the tank stays constant at 100 L. So, the rate at which salt leaves the tank is M(t)/100 × 9 L/min.

The rate of change of the mass of salt in the tank is given by the difference of these two rates, which gives us the differential equation: dM/dt = 0.36 - (M(t)/100) × 9. To solve this equation, we separate variables and integrate. The initial condition is M(0) = 0.4 kg. Solving the equilibrium condition when dM/dt = 0 gives the steady state mass of salt.

To find when the concentration of salt in the tank reaches 0.03 kg/L, we set M(t)/100 = 0.03 kg/L and solve for t. This gives us the specific time at which the concentration in the tank is equal to 0.03 kg/L.

Applying the initial condition and integrating, we would find an exponential equation representing the mass of salt over time. M(t) would be a function of t that increases towards a steady state concentration. To get the time when the concentration is exactly 0.03 kg/L, M(t) should be set equal to 3 kg (since the tank's volume is 100 L) and t solved accordingly.