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

Volume of solution remains unchanged becuase inflow rate and outflow rate is the same

So,

dx= -(x/250)*2*dt

dx/x=-dt/125

INtegrating gives

x=A e^{-t/125}

Using x(0)=750 gives A=750 g

x=750 e^{-t/125}

So concentration at time t is

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

Initial concentration is 3 g/l

We need final concentration to be 1 % of this . SO final concentration is

3/100

So the required time is

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

Solving gives

t~575.65 min