Final answer:
To address the paper mill's issue, a continuous mixing problem is modeled and solved using differential equations. The task involves maintaining a specific concentration of a chemical in a tank with a constant volume and flow rate while enhancing the quality of paper produced. A first-order linear differential equation depicting the rate of change of the chemical's quantity in the tank is constructed and solved to find the concentration over time.
Step-by-step explanation:
To solve the paper mill chemical addition problem using differential equations, we should first establish the variables and constants. We are dealing with a tank that has a steady input and output flow rate of 20 L/min and contains a certain concentration of chemical initially. We are asked to maintain or enhance the whiteness of the paper by adding more chemical into the mixture as the mill runs, without stopping the mill, which makes it a continuous mixing problem.
Let's denote the amount of chemical in the tank at any time t as Q(t), measured in kilograms. The concentration of the chemical entering the tank, Cin, can vary depending on how much additional chemical we decide to add. The initial concentration of the chemical in the tank is given as 0.3 kg/L, so our initial condition is Q(0) = 0.3 kg/L × 4000 L. The flow rate is equal for both input and output, so the volume of the solution in the tank remains constant at 4000 L.
The differential equation that represents the situation is dQ/dt = ratein - rateout, where ratein = Cin × 20 L/min and rateout = (Q(t)/4000 L) × 20 L/min. Solving this first-order linear differential equation with the initial condition will provide the function Q(t), which represents the quantity of the chemical in the tank over time.