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Suppose two interconnected tanks partially filled with brine. Pure water flows into tank 1 at a rate of 2 liters per minute. Pure water flows into tank 2 at a rate of 3 liters per minute. Brine flows out of tank 2 and into tank 1 at a rate of A liters per minute. Brine flows out of tank 1 and into a pipe at a rate of B liters per minute. Then C liters of the brine in this pipe exit via a drain in this pipe every minute. The remaining B – C liters of brine enter tank 2 every minute. 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 salt in tank 2 at time t > 0. Construct a system of differential equations to model the rate of change 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 about A, 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.

Suppose two interconnected tanks partially filled with brine. Pure water flows into tank 1 at a rate of 2 liters per minute. Pure water flows into tank 2 at a rate of 3 liters per minute. Brine flows out of tank 2 and into tank 1 at a rate of A liters per minute. Brine flows out of tank 1 and into a pipe at a rate of B liters per minute. Then C liters of the brine in this pipe exit via a drain in this pipe every minute. The remaining B – C liters of brine enter tank 2 every minute. 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 salt in tank 2 at time t > 0. Construct a system of differential equations to model the rate of change 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 about A, 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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 15EQ
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Suppose two interconnected tanks partially filled with brine. Pure water flows into tank 1 at a rate of 2 liters per minute. Pure water flows into tank 2 at a rate of 3 liters per minute. Brine flows out of tank 2 and into tank 1 at a rate of A liters per minute. Brine flows out of tank 1 and into a pipe at a rate of B liters per minute. Then C liters of the brine in this pipe exit via a drain in this pipe every minute. The remaining B – C liters of brine enter tank 2 every minute. 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 salt in tank 2 at time t > 0. Construct a system of differential equations to model the rate of change 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 about A, 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 example of a constant-coefficient, second order ODE and show how you would transform it into a matrix equation.
Transcribed Image Text:Suppose two interconnected tanks partially filled with brine. Pure water flows into tank 1 at a rate of 2 liters per minute. Pure water flows into tank 2 at a rate of 3 liters per minute. Brine flows out of tank 2 and into tank 1 at a rate of A liters per minute. Brine flows out of tank 1 and into a pipe at a rate of B liters per minute. Then C liters of the brine in this pipe exit via a drain in this pipe every minute. The remaining B – C liters of brine enter tank 2 every minute. 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 salt in tank 2 at time t > 0. Construct a system of differential equations to model the rate of change 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 about A, 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 example of a constant-coefficient, second order ODE and show how you would transform it into a matrix equation.
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