Differential Equations: Modeling with First Order Equations
Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 200 L of a dye solution with a concentration of 1 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 2 L/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value.
Re: Differential Equations: Modeling with First Order Equations
Let the amount of dye in the water at time t be Q(t) g. As long as it is "well stirred" the amount of die in each liter is Q/200 g/L. dQ/dL is the rate at which the amount of dye is changing: + is dye coming in, - is dye going out. Pure water is coming in at 2 L/min so there no dye coming in. Water containing Q/200 g/L is going out at 2 L/min so dye is going out at (Q/200)(2)= Q/100 g/min. Initially, there was 1 g/L in the 200 L so initially, there were 200 g of dye in the water. Solve the differential equation dQ/dt= -Q/100 with initial condition y(0)= 200 and then find t such that Q(t)= .01(200)= 20 g.