Suppose two interconnected tanks partially filled with brine. Pure water flows into tank 1 at arate of 2 liters per minute. Pure water flows into tank 2 at a rate of 3 liters per minute. Brineflows out of tank 2 and into tank 1 at a rate of A liters per minute. Brine flows out of tank 1and into a pipe at a rate of B liters per minute. Then C liters of the brine in this pipe exitvia a drain in this pipe every minute. The remaining B – C liters of brine enter tank 2 everyminute. Assume uniform mixing at all times t.(a) Let A1 (t) be the amount of salt in tank 1 at timet > 0. Let A2(t) be the amount of saltin tank 2 at time t > 0. Construct a system of differential equations to model the rate ofchange of salt in both tanks at time t > 0.(b) Set up a matrix equation to solve this linear system of ODE’s. What must be true aboutA, B, and C so that the entries of the matrix are constant?(c) Choose values of A, B, and C so that the entries of the matrix in part (b) are constant.Then solve the IVP.(d) Is it possible to model the relationship between A1 and A2 using a constant-coefficient,second order ODE? If so, write down this ODE. If not, explain why and give an exampleof a constant-coefficient, second order ODE and show how you would transform it into amatrix equation.

Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Answered: Suppose two interconnected tanks…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A brine solution of salt flows at a constant rate of 5 L/min into a large tank that initially held 100 L of brine solution in which was dissolved 0.15 kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is 0.03 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.01 kg/L? Determine the mass of salt in the tank after t min. mass= kg
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Consider a large vat containing sugar water that is to be made into soft drinks (see figure below). A B Suppose: The vat contains 230 gallons of liquid, which never changes. Sugar water with a concentration of 7 tablespoons/gallon flows through pipe A into the vat at the rate of 5 gallons/minute. • Sugar water with a concentration of 3 tablespoons/gallon flows through pipe B into the vat at the rate of 25 gallons/minute. The liquid in the vat is kept well-mixed. • Sugar water leaves the vat through pipe C at the rate of 30 gallons/minute. Let S(t) represent the number of tablespoons of sugar in the vat at time t, where t is given in minutes. (A) Write the DE model for the time rate of change of sugar in the vat: dS dt (B) Solve the differential equation to find the amount of sugar in the vat as a function of time. Your function will have an arbitrary constant K in it. Assume that K > 0. S(t) = (C) Suppose that there are 22 tablespoons of sugar in the vat at t = 0. How many tablespoons…
2. A large tank initially contains 100 gallons of brine in which 10 lb of salt is dissolved. Starting at t = 0, pure water flows into the tank at the rate of 5 gal/min. The mixture is kept uniform by stirring, and the well-stirred mixture simultaneously flows out at the slower rate of 2 gal/min. a) How much salt is in that tank at the end of 15 min and what is the concentration at that time? b) If the capacity of the tank is 250 gallons, what is the concentration at the instant the tank overflows?
A brine solution of salt flows at a constant rate of 8 L/min into a large tank that initially held 100 L of brine solution in which was dissolved 0.2 kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is 0.04 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.01 kg/L? Determine the mass of salt in the tank after t min. mass 4-4e 2t 25 kg
Two large tanks, each holding 100 L of liquid, are interconnected by pipes, with the liquid flowing from tank A into tank B at a rate of 5 L/min and from B into A at a rate of 4 L/min. The liquid inside each tank is kept well stirred. A brine solution with a concentration of 0.1 kg/L of salt flows into tank A at a rate of 10 L/min. The (diluted) solution flows out of the system from tank A at 9 L/min and from tank B at 1 L/min. If initially, tank A contains pure water and tank B contains 10 kg of salt, determine the mass of salt in each tank at time t≥0. x(t) = 10 L/min- 0.1 kg/L y(t) = 9 L/min A x(t) 100 L x(0) = 0 kg 5 L/min What is the solution to the system? 18 4 L/min B y(t) 100 L y(0) = 10 kg ..... 1 L/min O ✔

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