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This exercise is related to heat transfer, where the objective is to determine the temperature distribution across a thin metal sheet. The boundary temperatures of the sheet are given, and we are asked to model the temperature at the internal nodes \(T_1\), \(T_2\), \(T_3\), and \(T_4\), based on the boundary conditions.

### a) Forming the system of equations
The temperature at each internal node is the average of its four neighboring points (this is based on the finite difference method for heat conduction). Let's set up the equations for each node.

1. **Node \(T_1\):**
   - Neighbors: \(T_2\), \(T_4\), \(10^\circ\), and \(20^\circ\)
   - Equation: 
   \[
   T_1 = \frac{T_2 + T_4 + 10 + 20}{4}
   \]
   Simplifying:
   \[
   4T_1 = T_2 + T_4 + 30
   \]

2. **Node \(T_2\):**
   - Neighbors: \(T_1\), \(T_3\), \(20^\circ\), and \(40^\circ\)
   - Equation:
   \[
   T_2 = \frac{T_1 + T_3 + 20 + 40}{4}
   \]
   Simplifying:
   \[
   4T_2 = T_1 + T_3 + 60
   \]

3. **Node \(T_3\):**
   - Neighbors: \(T_2\), \(T_4\), \(30^\circ\), and \(40^\circ\)
   - Equation:
   \[
   T_3 = \frac{T_2 + T_4 + 30 + 40}{4}
   \]
   Simplifying:
   \[
   4T_3 = T_2 + T_4 + 70
   \]

4. **Node \(T_4\):**
   - Neighbors: \(T_1\), \(T_3\), \(10^\circ\), and \(30^\circ\)
   - Equation:
   \[
   T_4 = \frac{T_1 + T_3 + 10 + 30}{4}
   \]
   Simplifying:
   \[
   4T_4 = T_1 + T_3 + 40
   \]

### System of Equations:
\[
\begin{aligned}
4T_1 &= T_2 + T_4 + 30 \\
4T_2 &= T_1 + T_3 + 60 \\
4T_3 &= T_2 + T_4 + 70 \\
4T_4 &= T_1 + T_3 + 40
\end{aligned}
\]

### b) Solving the system using row operations
To solve this system, we can express it in matrix form and use Gaussian elimination or row operations to find the values of \(T_1\), \(T_2\), \(T_3\), and \(T_4\).

I will solve this system now.The solution to the system of equations is:

- \( T_1 = 20^\circ \)
- \( T_2 = 27.5^\circ \)
- \( T_3 = 30^\circ \)
- \( T_4 = 22.5^\circ \)

### Reflection:
The temperatures at the internal nodes are within the expected range, given the boundary conditions. The values of \( T_1 \) and \( T_3 \) are close to the boundary temperatures they are nearest to, while \( T_2 \) and \( T_4 \) adjust based on the average of their surrounding temperatures.

Would you like a more detailed breakdown of the solution process or have any specific questions?

Here are 5 related questions you can consider:
1. How does the finite difference method work in heat transfer problems?
2. What would happen if one of the boundary temperatures changed? How would it affect the system?
3. Can this method be extended to larger grids with more internal points?
4. How do row operations (Gaussian elimination) help in solving linear systems like this one?
5. Why is it valid to assume that the temperature at a node is the average of its four neighbors?

**Tip:** In problems like these, setting up the equations carefully based on the physical model is crucial before attempting to solve them numerically.

In heat transfer, determine the temperature distribution across a thin metal sheet with known boundary conditions using the finite difference method.

Solve a heat transfer problem by determining temperature distribution on a thin metal sheet. This exercise models the internal node temperatures based on the finite difference method and solves the system using Gaussian elimination.

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