**Question: **A tank originally contains 100 gallons of fresh water. Then water containing 1/2 pound of salt per gallon is poured into the tank at a rate of 2 gallons per minute, and the mixture is allowed to leave at the same rate. After 10 minutes the process is stopped, and fresh water is poured into the tank at a rate of 2 gallons per minute, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 minutes.

Answer the following questions in regard to the situation described above.

(a) (5 points) Write down the initial-value problem corresponding to the first part of the procedure, whereby water containing 1/2 pound of salt per gallon is poured into the tank containing 100 gallons of fresh water at a rate of 2 gallons per minute, and the mixture is allowed to leave at the same rate.

Use descriptive variables whenever it is appropriate, always explain what the variables stand for, and write the differential equation in standard form.

(b) (20 points) Solve the initial-value problem.

(c) (5 points) How much salt is in the tank at the end of 10 minutes?

(d) (5 points) What is the limiting value \({{Q}_{L}}=\underset{t\to \infty }{\mathop{\lim }}\,Q\left( t \right)\) ? Is this value reasonable? – explain why it is or is not reasonable.

(e) (5 points) Now set up the initial-value problem corresponding to the second part of the procedure, whereby after 10 minutes the inflow of water containing 1/2 pound of salt per gallon and at the rate of 2 gallons per minute is stopped, and fresh water is poured into the tank at the same rate of 2 gallons per minute, with the mixture again leaving at the same rate.

As in part (a), use descriptive variables whenever it is appropriate, always explain what the variables stand for, and write the differential equation in standard form. Furthermore, modify slightly the notation in respect of the dependent variable in this part so that it is very similar to the notation used in respect of the dependent variable used in the preceding parts.

(f) (20 points) Solve the initial-value problem of part (e).

(g) (5 points) How much salt is in the tank at the end of the additional 10 minutes?

(h) (15 points) At what time, if any, will there be exactly 1 pound of salt remaining in the tank.