3. For a three tank system, pure water flows into tank 1 which - before the flow begins - initially contains 32 lb. of salt. The solution from tank 1 flows into tank 2, from tank 2 into tank 3, and from tank 3 to a holding vat. All three tanks are filled to capacity before the flow starts. But tanks 2 and 3 contain only pure water initially, before flow begins.
Assuming that the pipes connecting the tanks contribute negligible volume to the system and that each tank is continuously well-mixed, such that each has a homogeneous concentration throughout,
• Derive a system of equations for the weight of salt in each tank as a function of time.
• State A for this first-order linear system, where $x' = Ax$.
• Find the general solution for this system.
• Determine when (in minutes) the salt weight in tank 2 reaches its maximum.
• Determine what this maximum salt weight in tank 2 is.
Other conditions include:
• Flow rate once flow begins 40 gal/min into and out of each tank
• Volumes of tanks 1, 2, 3. 160, 240, 320 gal, respectively
