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1. Consider a 100L tank containing 50L of salt water with a concentration of 1kg/L. Suppose that each second, 3L of pure water flows into the tank and 2L of the mixture flows out of the tank. The tank is kept well mixed at all times. (a) Write an IVP that governs the salt content in the tank. Make sure to explain your equation. (b) Solve the IVP from part (a).

1. Consider a 100L tank containing 50L of salt water with a concentration of 1kg/L. Suppose that each second, 3L of pure water flows into the tank and 2L of the mixture flows out of the tank. The tank is kept well mixed at all times. (a) Write an IVP that governs the salt content in the tank. Make sure to explain your equation. (b) Solve the IVP from part (a).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
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IVP: Initial-Value-Problem

1. Consider a 100L tank containing 50L of salt water with a concentration of 1kg/L. Suppose that each second, 3L of pure water flows into the tank and 2L of the mixture flows out of the tank. The tank is kept well mixed at all times. (a) Write an IVP that governs the salt content in the tank. Make sure to explain your equation. (b) Solve the IVP from part (a). 2. Now consider the same model as in question 1, but with a "salt buffer" involved. This buffer will add salt if the concentration is below 1kg/L and will remove salt if the concentration is above 1kg/L. (a) Propose an ODE that would model the salt content in the tank and explain it. If you are using any parameters for unknowns, explain what they mean. Note that there are many correct ways to set this up. (b) Without solving the ODE, answer the following: If at a certain moment there is a salt concentration of 1kg/L in the tank, and taking into account all effects, is the amount of salt increasing or decreasing at that moment?
Transcribed Image Text:1. Consider a 100L tank containing 50L of salt water with a concentration of 1kg/L. Suppose that each second, 3L of pure water flows into the tank and 2L of the mixture flows out of the tank. The tank is kept well mixed at all times. (a) Write an IVP that governs the salt content in the tank. Make sure to explain your equation. (b) Solve the IVP from part (a). 2. Now consider the same model as in question 1, but with a "salt buffer" involved. This buffer will add salt if the concentration is below 1kg/L and will remove salt if the concentration is above 1kg/L. (a) Propose an ODE that would model the salt content in the tank and explain it. If you are using any parameters for unknowns, explain what they mean. Note that there are many correct ways to set this up. (b) Without solving the ODE, answer the following: If at a certain moment there is a salt concentration of 1kg/L in the tank, and taking into account all effects, is the amount of salt increasing or decreasing at that moment?
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