Consider the two tanks shown in the figure below. Each has a capacity of 300 gallons. At time t = 0, tank 1 contains 100 gallons of a brine solution and tank 2 contains 200 gallons of a brine solution. Each tank also initially contains 50 pounds of salt. Pure water flows into tank 1, then, a well-mixed solution flows out from tank 1 into tank 2. Finally a well-mixed solution drains out of tank 2. The three flow rates indicated in the figure are each 5 gal/min.
Tank 1, capacity = 300 gal.
Volume of brine = 100 gal.
x(t) = amount of salt (lbs.)
Tank 2, capacity = 300 gal.
Volume of brine = 200 gal.
y(t) = amount of salt (lbs.)
(a) Write a system of differential equations that describes the amount of salt, x(t), in tank 1 and the amount of salt, y(t), in tank 2. Use the variables x and y in writing your answers below. Do not use x(t) and y(t).
dx/dt =
dy/dt =
(b) Solve the system to find formulas for x(t) and y(t). Write your answers in terms of the variable t.
x(t) =
y(t) =
(c) Determine the maximum amount of salt in tank 2. At what time does this occur?
The maximum amount of salt in tank 2 = pounds, which occurs at time = minutes