Question

A tank contains 1000 L of a solution consisting of 100kg salt dissolved in water. Pure water is pumped into the tank at rate of 5L/s and the mixture is pumped out at the same rate. How long will it be until only 10 kg of salt remains in the tank?

Answer 1

dx
dt
= −5 x(t)
1000
The general solution to this equation is
x(t) = Ce− t
200 .
Since x(0) = 100 we have
x(t) = 100e− t
200 .
Denote by t0 the time when only 10 kg of salt left in the tank. Then
x(t0) = 10 = 100e− t0
200
From this equation we have
t0 = 200 ln 10.
3