Question

A tank contains 50 gallons of water in which 2 pounds of salt is dissolved. A brine solution containing 1.5 pounds of salt per gallon of water is pumped into the tank at the rate of 4 gallons per minute, and the well-stirred mixture is pumped out at the same rate. Let A(t) represent the amount of salt in the tank at time t. Derive the initial value problem for A(t). Also how much salt will there be in the tank after a long period of time?

Answer 1

• A(0) = 2; A'(t) = 6 - 0.08A(t)
• A(∞) = 75

Explanation:

The initial value is said to be 2 pounds, so A(0) = 2.

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The influx is (1.5 #/gal)×(4 gal/min) = 6.0 #/min.

The outflow is (4 gal)/(50 gal) × A(t) = 0.08A(t).

The rate of change of A(t) is then ...

A'(t) = 6 - 0.08A(t)

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When the system reaches steady state, A'(t) = 0, so ...

0 = 6 - 0.08A(∞)

A(∞) = 6/0.08 = 75

75 pounds of salt will be in the tank after a long period.