A vat of volume 10001000 gallons initially contains 44 lbs of salt. For t > 0 pure water is pumped into the vat at the rate of 66 gallons per minute; the perfectly stirred mixture is pumped out at the same flow rate. Derive a formula for the concentration of salt in the tank at any time t.

Step-by-step explanation:

The quantity of salt inside the tank is modelled after the Principle of Mass Conservation:

Salt

Water

Given that water is an incompressible fluid, the expression can be simplified into the following expression:

Both flows have the same rate and tank can be modelled as a steady state system.

The expression for salt concentration in the tank is:

-

After some handling, the following homogeneous first-order linear differential equation is found:

Where . The solution is obtained by using Laplace transforms:

Where .

The formula for the concentration of salt in the tank is:

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step-by-step explanation: