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Fresh water flows into tank 1; mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3;. at the given flow rater gallons per minute. The initial amounts x₁ (0)=x₁ (lb), and x₂ (0)=0, and x3 (0) = 0 of salt in three tanks are given, as are their volumes V₁, V₂, and V3. First solve for the amounts of salt in the three tanks at time t, then determine the maximal amount of salt that tank 3 ever contains. Finally, construct a figure showing the graphs of x₁ (t), x₂(t), and x3 (t). r=80, x=60, V₁=16, V₂ = 10, V3 = 40 B X₁ (t)= X₂ (t)= X3 (t)= structor Set LCS Week 2 h Clear all Naafiri Gameplay Trailer Check answer Naafiri Abilities Rundown

Fresh water flows into tank 1; mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3;. at the given flow rater gallons per minute. The initial amounts x₁ (0)=x₁ (lb), and x₂ (0)=0, and x3 (0) = 0 of salt in three tanks are given, as are their volumes V₁, V₂, and V3. First solve for the amounts of salt in the three tanks at time t, then determine the maximal amount of salt that tank 3 ever contains. Finally, construct a figure showing the graphs of x₁ (t), x₂(t), and x3 (t). r=80, x=60, V₁=16, V₂ = 10, V3 = 40 B X₁ (t)= X₂ (t)= X3 (t)= structor Set LCS Week 2 h Clear all Naafiri Gameplay Trailer Check answer Naafiri Abilities Rundown

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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**Title: Calculating the Amount of Salt in Interconnected Tanks** **Introduction:** In this exercise, we explore the flow of fresh water and mixed brine through a series of interconnected tanks. By analyzing the initial conditions and flow rates, we will solve for the amount of salt in each tank over time. **Problem Statement:** Fresh water flows into tank 1 at a rate of \(r\) gallons per minute. Mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3 at the same rate. The initial amounts of salt in the tanks are given as \(x_1(0) = x_0\) (in pounds), \(x_2(0) = 0\), and \(x_3(0) = 0\). The volumes of the three tanks are given as \(V_1\), \(V_2\), and \(V_3\), respectively. ### Given Parameters: - \(r = 80\) (flow rate in gallons per minute) - \(x_0 = 60\) (initial amount of salt in tank 1 in pounds) - \(V_1 = 16\) (volume of tank 1 in gallons) - \(V_2 = 10\) (volume of tank 2 in gallons) - \(V_3 = 40\) (volume of tank 3 in gallons) ### Tasks: 1. Solve for the amount of salt \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\) in the three tanks at time \(t\). 2. Determine the maximal amount of salt that tank 3 ever contains. 3. Construct a graph showing \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\). **Graph Explanation:** The solution requires solving a system of differential equations that model the flow of salt through the tanks. These equations account for the input and output rates and the mixing process within each tank. The resulting system will provide us with the functions \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\). ### Example Graph Layout: - **x-axis (Time \(t\)):** Represents the time in minutes throughout the flow process. - **y-axis (Salt Amount):** Represents the amount of
Transcribed Image Text:**Title: Calculating the Amount of Salt in Interconnected Tanks** **Introduction:** In this exercise, we explore the flow of fresh water and mixed brine through a series of interconnected tanks. By analyzing the initial conditions and flow rates, we will solve for the amount of salt in each tank over time. **Problem Statement:** Fresh water flows into tank 1 at a rate of \(r\) gallons per minute. Mixed brine flows from tank 1 into tank 2, from tank 2 into tank 3, and out of tank 3 at the same rate. The initial amounts of salt in the tanks are given as \(x_1(0) = x_0\) (in pounds), \(x_2(0) = 0\), and \(x_3(0) = 0\). The volumes of the three tanks are given as \(V_1\), \(V_2\), and \(V_3\), respectively. ### Given Parameters: - \(r = 80\) (flow rate in gallons per minute) - \(x_0 = 60\) (initial amount of salt in tank 1 in pounds) - \(V_1 = 16\) (volume of tank 1 in gallons) - \(V_2 = 10\) (volume of tank 2 in gallons) - \(V_3 = 40\) (volume of tank 3 in gallons) ### Tasks: 1. Solve for the amount of salt \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\) in the three tanks at time \(t\). 2. Determine the maximal amount of salt that tank 3 ever contains. 3. Construct a graph showing \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\). **Graph Explanation:** The solution requires solving a system of differential equations that model the flow of salt through the tanks. These equations account for the input and output rates and the mixing process within each tank. The resulting system will provide us with the functions \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\). ### Example Graph Layout: - **x-axis (Time \(t\)):** Represents the time in minutes throughout the flow process. - **y-axis (Salt Amount):** Represents the amount of
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