Problem 10: A tank initially contains a solution of 10 pounds of salt in 60 gallons of water. Water with 1/2 pound of salt per gallon is added to the tank at 6 gal/min, and the resulting solution leaves at the same rate. Find the quantity Q(t) of salt in the tank at time t > 0.

The quantity of salt at time t is , where t is measured in minutes.

Step-by-step explanation:

The law of mass conservation for control volume indicates that:

Where mass flow is the product of salt concentration and water volume flow.

The model of the tank according to the statement is:

Where:

- The salt concentration in the tank, as well at the exit of the tank, measured in .

- Concentration rate of change in the tank, measured in .

- Volume of the tank, measured in gallons.

The following first-order linear non-homogeneous differential equation is found:

This equation is solved as follows:

The initial concentration in the tank is:

Now, the integration constant is:

The solution of the differential equation is:

Now, the quantity of salt at time t is:

Where t is measured in minutes.

step-by-step explanation:

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