In a certain factory, there are two tanks used to mix brine (also known as salt water). Tai 10 gallons of brine containing 4 pounds of salt. Tank B starts with 10 gallons of fresh w salt. Five gallons per minute are pumped from Tank A into Tank B. Five gallons per mir from Tank B into Tank A. Nothing enters either tank from any other source, and nothing tank except what is pumped into the other tank. The mixtures are kept even by constant purposes of this problem, assume that both tanks have infinite capacity. Let A(t) be the number of pounds of salt in Tank A after t minutes. Let B(t) be the numb

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
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In a certain factory, there are two tanks used to mix brine (also known as salt water). Tank A starts with
10 gallons of brine containing 4 pounds of salt. Tank B starts with 10 gallons of fresh water containing no
salt. Five gallons per minute are pumped from Tank A into Tank B. Five gallons per minute is pumped
from Tank B into Tank A. Nothing enters either tank from any other source, and nothing leaves either
tank except what is pumped into the other tank. The mixtures are kept even by constant stirring. For
purposes of this problem, assume that both tanks have infinite capacity.
Let A(t) be the number of pounds of salt in Tank A after t minutes. Let B(t) be the number of pounds of
salt in Tank B after t minutes.
(a) Set up a system of differential equations for A and B to model this situation. Be sure to include
any initial conditions. Also, be sure to indicate what each of your variables represents.
(b) Solve your system of equations from (a).
(c) Find lim t-→∞ A(t) and lim t→∞ B(t).
(d) In a few brief sentences explain why your answer to (c) makes sense in the context of the
problem. (How could you have predicted this answer without solving any differential equations?)
Transcribed Image Text:In a certain factory, there are two tanks used to mix brine (also known as salt water). Tank A starts with 10 gallons of brine containing 4 pounds of salt. Tank B starts with 10 gallons of fresh water containing no salt. Five gallons per minute are pumped from Tank A into Tank B. Five gallons per minute is pumped from Tank B into Tank A. Nothing enters either tank from any other source, and nothing leaves either tank except what is pumped into the other tank. The mixtures are kept even by constant stirring. For purposes of this problem, assume that both tanks have infinite capacity. Let A(t) be the number of pounds of salt in Tank A after t minutes. Let B(t) be the number of pounds of salt in Tank B after t minutes. (a) Set up a system of differential equations for A and B to model this situation. Be sure to include any initial conditions. Also, be sure to indicate what each of your variables represents. (b) Solve your system of equations from (a). (c) Find lim t-→∞ A(t) and lim t→∞ B(t). (d) In a few brief sentences explain why your answer to (c) makes sense in the context of the problem. (How could you have predicted this answer without solving any differential equations?)
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