3.

The volume of liquid after t minutes is 1 + t litres. The differential equation for the amount Q(t) of salt in the tank (in grams) at time tis $\displaystyle\frac{dQ}{dt} = 2 - \frac{Q}{1+t} $. The general solution of this linear equation is $\displaystyle Q(t) = (2 t + t^2 + C)/(1+t)$. From the initial condition Q(0) = 2we have C = 2. The concentration is Q(t)/(1+t) = (2 t + t2 + 2)/(1+t)2grams/litre after t minutes. The limit of this as $t \to \infty$ is 1 gram/litre.




Robert Israel
2002-02-07