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?

Question

Mathematics

Barrett Coleman

11 November, 07:16

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?

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Anaya · Answer 1

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

Step-by-step 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.