Assignment:
Oscillating Inflow Concentration
A tank initially contains 10 lb of salt dissolved in 200 gallons of water. Assume that a salt solution flows into the tank at a rate of 3 gal/min and the well-stirred mixture flows out at the same rate. Assume that the inflow concentration oscillates I n time, however, and is given by ci(t) = 0.2(1 + sin t) lb of salt per gallon. Thus, as time evolves, the concentration oscillates back and forth between 0 and 0.4 lb of salt per gallon.
(a) Make a conjecture, on the basis of physical reasoning, as to whether you expect the amount of salt in the tank to reach a constant equilibrium value as time increases. In other words, will lim Q(t) exist?
(b) Formulate the corresponding initial value problem.
(c) Solve the initial value problem.
(d) Plot Q(t) versus t. How does the amount of salt in the tank vary as time becomes increasingly large? Is this behavior consistent with your intuition?
Provide complete and step by step solution for the question and show calculations and use formulas.