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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.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 61CR
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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. A(t) Write the IVP in matrix vector form. Use X(t) = [] 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 eigenval- ues and eigenvectors. Write the solution of the IVP.
Transcribed Image Text: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. A(t) Write the IVP in matrix vector form. Use X(t) = [] 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 eigenval- ues and eigenvectors. Write the solution of the IVP.
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