A metal rod, insulated along its sides, is 5 meters long. It is initially at a...

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A metal rod, insulated along its sides, is 5 meters long. It is initially at a uniform temperature of 20°C. At time t=0, one end of the rod is placed in a furnace at 100°C. The other end remains insulated. Find an expression for the temperature at a point P located x meters from the heated end at any time t seconds after t=0. Instructions: 1. Mathematical Formulation: ● ● 2. Boundary and Initial Conditions: ● Formulate the one-dimensional heat equation based on the given scenario. Define the terms and parameters involved. Specify the boundary conditions for the rod. Discuss the physical significance of the conditions. 3. Analytical Solution: ● Solve the one-dimensional heat equation analytically to find the expression for the temperature distribution at any point P and time t. 4. Visualization: ● Provide visualizations of the temperature distribution along the rod for different time instances. 5. Discussion: ● Discuss the behavior of the rod as it heats up over time. Analyze the impact of changing parameters (e.g., initial temperature, rod length) on the temperature distribution. A metal rod, insulated along its sides, is 5 meters long. It is initially at a uniform temperature of 20°C. At time t=0, one end of the rod is placed in a furnace at 100°C. The other end remains insulated. Find an expression for the temperature at a point P located x meters from the heated end at any time t seconds after t=0. Instructions: 1. Mathematical Formulation: ● ● 2. Boundary and Initial Conditions: ● Formulate the one-dimensional heat equation based on the given scenario. Define the terms and parameters involved. Specify the boundary conditions for the rod. Discuss the physical significance of the conditions. 3. Analytical Solution: ● Solve the one-dimensional heat equation analytically to find the expression for the temperature distribution at any point P and time t. 4. Visualization: ● Provide visualizations of the temperature distribution along the rod for different time instances. 5. Discussion: ● Discuss the behavior of the rod as it heats up over time. Analyze the impact of changing parameters (e.g., initial temperature, rod length) on the temperature distribution.

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