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A tann initrally cunsnista f 50 gal ef 1b/gal f sald Ļ5 pumped into the tan n at a, rate cf 3'gal/ mixtyre flowt out at aThe same vate A 3 gal/min. Gnd the well-5ted a) What oinitnal-value prohlem is Suhshed hy the amonnd At salt A) in the'tann ad Dime +? 5) What is he actual amount of salt in the fan n at time t? C) How much salt ir in the fan n after 20 minuter? レ d) Mow much Salt !J lafter a (ong me ? In the tan

A tann initrally cunsnista f 50 gal ef 1b/gal f sald Ļ5 pumped into the tan n at a, rate cf 3'gal/ mixtyre flowt out at aThe same vate A 3 gal/min. Gnd the well-5ted a) What oinitnal-value prohlem is Suhshed hy the amonnd At salt A) in the'tann ad Dime +? 5) What is he actual amount of salt in the fan n at time t? C) How much salt ir in the fan n after 20 minuter? レ d) Mow much Salt !J lafter a (ong me ? In the tan

Advanced Engineering Mathematics
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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differential equations 

**Title: Salt Concentration Dynamics in a Tank** **Introduction:** In this exercise, we explore the dynamics of salt concentration in a mixing tank system. We begin with initial conditions and develop a mathematical model to predict the concentration over time. **Scenario:** A tank initially contains 50 gallons of pure water. A solution containing 1 lb/gal of salt is pumped into the tank at a rate of 2 gallons per minute. Simultaneously, the well-mixed solution is pumped out of the tank at the same rate of 3 gallons per minute. **Questions:** **a) Initial-Value Problem Formulation:** - What is the initial-value problem that describes the amount of salt in the tank at any time \( t \)? **b) Actual Amount of Salt:** - What is the actual amount of salt in the tank at time \( t = 3 \) minutes? **c) Salt Amount After 20 Minutes:** - How much salt is in the tank after 20 minutes? **d) Long-Term Salt Concentration:** - How much salt is in the tank after a long time? **Conclusion:** Through solving these equations, students will understand the interplay of input and output rates in the mixing process, and predict future concentration levels in practical scenarios.
Transcribed Image Text:**Title: Salt Concentration Dynamics in a Tank** **Introduction:** In this exercise, we explore the dynamics of salt concentration in a mixing tank system. We begin with initial conditions and develop a mathematical model to predict the concentration over time. **Scenario:** A tank initially contains 50 gallons of pure water. A solution containing 1 lb/gal of salt is pumped into the tank at a rate of 2 gallons per minute. Simultaneously, the well-mixed solution is pumped out of the tank at the same rate of 3 gallons per minute. **Questions:** **a) Initial-Value Problem Formulation:** - What is the initial-value problem that describes the amount of salt in the tank at any time \( t \)? **b) Actual Amount of Salt:** - What is the actual amount of salt in the tank at time \( t = 3 \) minutes? **c) Salt Amount After 20 Minutes:** - How much salt is in the tank after 20 minutes? **d) Long-Term Salt Concentration:** - How much salt is in the tank after a long time? **Conclusion:** Through solving these equations, students will understand the interplay of input and output rates in the mixing process, and predict future concentration levels in practical scenarios.
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