Consider a tank used in certain experiments After one experi

Consider a tank used in certain experiments. After one experiment the tank contains 250 liters of a dye solution with a concentration of 3 g/liter. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 2 liters/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in t he tank reaches 1% of its original value.

Solution

Let x be amount of dye in tank in grams at time t

So,

x(0)=250*3=750 g

The volume of solution remains unchanged because outflow rate is same as inflow rate

The differential equation is given by

dx=-(x/250)*2 dt

dx/x=-dt/125

Integrating gives

x=A e^{-t/125}

x(0)=750 gives A=750

Hence, x=750 e^{-t/125}

So concentration at time t is

x/250=3 e^{-t/125}

We want concentration to be 1 %

So final concentration should be

3/100

So the time elapsed is given by

3/100 =3 e^{-t/125}

Solving gives t~575.65 minutes