A 2000 L tank initially contains 40 kg of salt dissolved in 1000 L of water. A brine solution containing 0.02 kg/L of salt flows into the tank at a rate of 50 L/min. The solution is kept thoroughly mixed, and the mixture flows out at a rate of 25 L/min. (a) Find the quantity of salt in the tank at any time t > 0) prior to overflow. (b) Find the time of overflow.
Solution;
(a)
Let S(t) be the amount of salt in the tank at any time t.
Then;
S(0)=0
Let V(t) be the volume in the tank at any time t
Then V(0)=1000
But ,net flow is 50L/min -25L/min.
Hence;
Rate of salt going in;
=
Rate of salt going out;
Hence the net rate of salt in tank is;
Rearrange as follows;
The integrating factor;
=
Hence;
Integrating both sides;
Resolve as follows;
But ;
Applying this initial conditions;
Hence ;
Hence,the quantity of salt in the tank at t>0 prior to overflow is described as;
(b)
Time for overflow;
At time of overflow,the tank is full hence;
minutes.
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