Consider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds $x_1(t)$ and $x_2(t)$ of salt in tanks A and B, respectively, at time $t$. This system is described by the system of equations $frac{dx_1}{dt} = -frac{2}{25}x_1 + frac{1}{50}x_2$ $frac{dx_2}{dt} = frac{2}{25}x_1 - frac{2}{25}x_2$ with initial conditions $x_1(0) = 25$, $x_2(0) = 0$ (see (3) and the surrounding discussion on mixtures on page 107). What is the system of differential equations if, instead of pure water, a brine solution containing 5 pounds of salt per gallon is pumped into tank A? $frac{dx_1}{dt} = $ $frac{dx_2}{dt} = $
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Since the brine solution contains 5 pounds of salt per gallon and it's being pumped in at a rate of 1 gallon per minute, the rate of salt being added to tank A is 5 pounds per minute. Second, we need to consider the rate at which salt is leaving tank A. Since Show more…
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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
Adi S.
Two very large tanks $A$ and $B$ are each partially filled with 100 gallons of brine. Initially, 100 pounds of salt is dissolved in the solution in tank $A$ and 50 pounds of salt is dissolved in the solution in $\operatorname{tank} B$. The system is closed in that the wellstirred liquid is pumped only between the tanks, as shown in FIGURE 2.9.6. (a) Use the information given in the figure to construct a mathematical model for the number of pounds of salt $x_{1}(t)$ and $x_{2}(t)$ at time $t$ in tanks $A$ and $B$, respectively. (b) Find a relationship between the variables $x_{1}(t)$ and $x_{2}(t)$ that holds at time $t$. Bxplain why this relationship makes intuitive sense. Use this relationship to help find the amount of salt in $\operatorname{tank} B$ at $t=30 \mathrm{~min}$.
First-Order Differential Equations
Modeling with Systems of First-Order DEs
Let Tank A, Tank B, and Tank C be three connected mixing tanks. Figure 1 illustrates a three-tank mixing system such that the directions of flows of liquid in and out of the tanks are indicated by the arrows. Initially, the three tanks contain 140 liters, 80 liters, and 200 liters of pure water, respectively. Suppose 0.4 kg/liter brine runs into the mixing system through Tank A at a rate of 3 liters/min. The well-stirred mixture is exchanged between the three tanks at the indicated rates. The well-stirred mixture drains out of the system from Tank C at a rate of 3 liters/min. a) Calculate the volume of liquid in each tank at time t. b) Construct a system of linear differential equations representing the mathematical model for the above-mentioned mixture problem. (Do not solve the model)
Syed Basim M.
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