Three tanks containing brine are shown below. Construct a model for the number of pounds of salt x₁(t), x₂(t), x²(t) at time t in tanks A, B, and C respectively. Then write it in matrix form. You do not need to solve. mixture mixture 1 gal/min 2 gal/min មា ២ A 100 gal 100 gal pure water 4 gal/min mixture 6 gal/min B 100 gal mixture 5 gal/min mixture 4 gal/min
Three tanks containing brine are shown below. Construct a model for the number of pounds of salt x₁(t), x₂(t), x²(t) at time t in tanks A, B, and C respectively. Then write it in matrix form. You do not need to solve. mixture mixture 1 gal/min 2 gal/min មា ២ A 100 gal 100 gal pure water 4 gal/min mixture 6 gal/min B 100 gal mixture 5 gal/min mixture 4 gal/min
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![### Modeling Pounds of Salt in Brine Tanks
**Problem Statement:**
Three tanks containing brine are illustrated below. We are tasked with constructing a model to represent the number of pounds of salt \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\) at time \(t\) in tanks A, B, and C respectively. The mathematical model should be formulated in matrix form. Solving the model is not required.
**Diagram Analysis:**
1. **Tank A:**
- Capacity: 100 gallons.
- Inflow:
- Pure water at 4 gallons/minute.
- Outflow:
- Mixture to Tank B at 2 gallons/minute.
- Mixture to outlet at 6 gallons/minute.
2. **Tank B:**
- Capacity: 100 gallons.
- Inflow:
- Mixture from Tank A at 2 gallons/minute.
- Outflow:
- Mixture to Tank C at 1 gallon/minute.
- Mixture to outlet at 5 gallons/minute.
3. **Tank C:**
- Capacity: 100 gallons.
- Inflow:
- Mixture from Tank B at 1 gallon/minute.
- Outflow:
- Mixture to outlet at 4 gallons/minute.
**Steps to Develop the Model:**
1. **Salt Balance Equations:**
- For each tank, establish the rate of change of salt content.
- Use inflow and outflow rates to set up differential equations.
2. **Matrix Representation:**
- Convert the system of differential equations into a matrix form for simplicity.
**Constructing the Model:**
1. **Tank A (Salt Balance):**
- Inflow of salt into Tank A is only from pure water, which contains no salt.
- Outflow of salt is to Tank B and the outlet.
- Differential equation:
\[ \frac{d x_1}{dt} = -\left(\frac{2}{100} + \frac{6}{100}\right)x_1 = -0.08 x_1 \]
2. **Tank B (Salt Balance):**
- Inflow of salt from Tank A.
- Outflow of salt to Tank C and the outlet.
- Differential equation:
\[ \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8aa10308-c186-4690-945e-554747bf90d8%2F780b4b9c-2fa1-4c77-a189-a92edc3abb18%2Fpxshil_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Modeling Pounds of Salt in Brine Tanks
**Problem Statement:**
Three tanks containing brine are illustrated below. We are tasked with constructing a model to represent the number of pounds of salt \(x_1(t)\), \(x_2(t)\), and \(x_3(t)\) at time \(t\) in tanks A, B, and C respectively. The mathematical model should be formulated in matrix form. Solving the model is not required.
**Diagram Analysis:**
1. **Tank A:**
- Capacity: 100 gallons.
- Inflow:
- Pure water at 4 gallons/minute.
- Outflow:
- Mixture to Tank B at 2 gallons/minute.
- Mixture to outlet at 6 gallons/minute.
2. **Tank B:**
- Capacity: 100 gallons.
- Inflow:
- Mixture from Tank A at 2 gallons/minute.
- Outflow:
- Mixture to Tank C at 1 gallon/minute.
- Mixture to outlet at 5 gallons/minute.
3. **Tank C:**
- Capacity: 100 gallons.
- Inflow:
- Mixture from Tank B at 1 gallon/minute.
- Outflow:
- Mixture to outlet at 4 gallons/minute.
**Steps to Develop the Model:**
1. **Salt Balance Equations:**
- For each tank, establish the rate of change of salt content.
- Use inflow and outflow rates to set up differential equations.
2. **Matrix Representation:**
- Convert the system of differential equations into a matrix form for simplicity.
**Constructing the Model:**
1. **Tank A (Salt Balance):**
- Inflow of salt into Tank A is only from pure water, which contains no salt.
- Outflow of salt is to Tank B and the outlet.
- Differential equation:
\[ \frac{d x_1}{dt} = -\left(\frac{2}{100} + \frac{6}{100}\right)x_1 = -0.08 x_1 \]
2. **Tank B (Salt Balance):**
- Inflow of salt from Tank A.
- Outflow of salt to Tank C and the outlet.
- Differential equation:
\[ \
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