Question

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 10 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) Explain what happens to the concentration of salt in each of the tanks. Justify your answer in a few sentences.

          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 10 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) Explain what happens to the concentration of salt in each of the tanks. Justify your answer in a few sentences.

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 10 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) Explain what happens to the concentration of salt in each of the tanks. Justify your answer in a few sentences.

Added by James M.

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