A A simple 2 tank system. The diagram shows two tanks, Tank X and Tank Y. Tank X

Question

A A simple 2 tank system. The diagram shows two tanks, Tank X and Tank Y. Tank X is the tank from problem 2. Tank Y initially contains 15 gallons of pure water. Liquid flows from tank X into tank Y at a rate of 3 gallons per minute and the well stirred mixture flows out at the same rate. Let Y(t) be the amount of salt in tank Y at time t. Find the initial value problem satisfied by Y(t). Find the amount of salt Y (t) in tank Y at time t. What is the largest amount of salt in tank Y in the time interval t ge 0? Plot and label (on the same set of axes) the graphs of X(t) and Y(t) for t ge 0. When, if ever, do the two tanks contain the same amount of salt? (They both have no salt at time t = 0. Are there other times where they have the same amount of salt? You may estimate the answer(s) by using the graphs of X(t) and Y(t) from part (d). Your estimate may be done "visually.") Also describe in a few sentences what the graphs tell you about the behavior of this system for t ge 0.

Explanation / Answer

To study such a question, we consider the rate of change of the amount of salt in the tank. Let S be the amount of salt in the tank at any time t. If we can create an equation relating dS/dt to Sand t, then we will have a differential equation which we can, ideally, solve to determine the relationship between S and t. To describe dS/dt, we use the concept of concentration, the amount of salt per unit of volume of liquid in the tank. In this example, the inflow and outflow rates are the same, so the volume of liquid in the tank stays constant at 15 l. Hence, we can describe the concentration of salt in the tank by concentration of salt=s/15 gallon/litre ds/dt=-s/15

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