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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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