Modeling The following tank mixing problem was on our previous homework: pure water 3 gal/min mixture 1 gal/min mixture 4 gal/min mixture 3 gal/min 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. Write the IVP in matrix vector form. Use X(t) = [(0)] B Explain how we know from the physical situation, without solving the system, that both A(t) and B(t) will approach 0 as time increases. с What does the fact presented in part (a) tell us about the sign of the eigenvalues (or at least the sign of the real part of the eigenvalues) in this system of differential equations? Explain in one sentence. (You do not need to find the eigenvalues to answer this.) D. Go ahead and use wolframalpha.com to find the eigenval- ues and eigenvectors. Write the solution of the IVP.
Modeling The following tank mixing problem was on our previous homework: pure water 3 gal/min mixture 1 gal/min mixture 4 gal/min mixture 3 gal/min 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. Write the IVP in matrix vector form. Use X(t) = [(0)] B Explain how we know from the physical situation, without solving the system, that both A(t) and B(t) will approach 0 as time increases. с What does the fact presented in part (a) tell us about the sign of the eigenvalues (or at least the sign of the real part of the eigenvalues) in this system of differential equations? Explain in one sentence. (You do not need to find the eigenvalues to answer this.) D. Go ahead and use wolframalpha.com to find the eigenval- ues and eigenvectors. Write the solution of the IVP.
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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![**Modeling**
The following tank mixing problem was on our previous homework:
[Diagram Description]
- The diagram shows two tanks labeled A and B.
- Tank A receives pure water at a rate of 3 gallons per minute.
- There is a mixture flowing from tank A to tank B at 1 gallon per minute.
- Tank A also has an output mixture flowing out at 4 gallons per minute.
- Tank B has an output mixture flowing out at 3 gallons per minute.
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.
**A.** Write the IVP in matrix vector form. Use \(\vec{X}(t) = \begin{bmatrix} A(t) \\ B(t) \end{bmatrix}\).
**B.** Explain how we know from the physical situation, without solving the system, that both \(A(t)\) and \(B(t)\) will approach 0 as time increases.
**C.** What does the fact presented in part (a) tell us about the sign of the eigenvalues (or at least the sign of the real part of the eigenvalues) in this system of differential equations? Explain in one sentence. (You do not need to find the eigenvalues to answer this.)
**D.** Go ahead and use wolframalpha.com to find the eigenvalues and eigenvectors. Write the solution of the IVP.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ba71b9f-6645-4b82-be5d-cda1c9d5ec57%2Fbf663b45-a38f-4a5b-b3af-587fe14cca51%2Fkgabh6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Modeling**
The following tank mixing problem was on our previous homework:
[Diagram Description]
- The diagram shows two tanks labeled A and B.
- Tank A receives pure water at a rate of 3 gallons per minute.
- There is a mixture flowing from tank A to tank B at 1 gallon per minute.
- Tank A also has an output mixture flowing out at 4 gallons per minute.
- Tank B has an output mixture flowing out at 3 gallons per minute.
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.
**A.** Write the IVP in matrix vector form. Use \(\vec{X}(t) = \begin{bmatrix} A(t) \\ B(t) \end{bmatrix}\).
**B.** Explain how we know from the physical situation, without solving the system, that both \(A(t)\) and \(B(t)\) will approach 0 as time increases.
**C.** What does the fact presented in part (a) tell us about the sign of the eigenvalues (or at least the sign of the real part of the eigenvalues) in this system of differential equations? Explain in one sentence. (You do not need to find the eigenvalues to answer this.)
**D.** Go ahead and use wolframalpha.com to find the eigenvalues and eigenvectors. Write the solution of the IVP.
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