MAT 350 Module Five Application Project Guidelines and Rubric: Equilibrium Temperature Distribution This project gives students an introduction to one type of problem encountered in thermodynamics, that of solving for the equilibrium temperature distribution of a thin plate of metal. One way of solving this type of problem is to solve a continuous-time differential equation that can be queried as a function of (x,y) for any continuous-valued position on the plate of metal. In general, this solution could be exact given certain assumptions, but this solution is somewhat difficult to compute. A simpler way to approximately solve the problem is to discretize the plate and solve a system of linear equations. More details and background theory can be reviewed at Equilibrium Temperature Distributions. Students will solve this system of linear equations using two different methods: 1) determinant computations to solve for the matrix inverse, and 2) Cramer's rule. Make sure to show your work as you solve the problems. Problems Consider the discretized square plate in Figure 1. Figure 1: Discretized Square Plate 1. Write a system of linear equations for this system and clearly identify the matrix A in the matrix equation Ax=b. 2. Compute the determinant of A. 3. Compute the classical adjoint matrix (i.e., the adjugate matrix) of A. 4. Using your answers from problems 2 and 3, solve for A-1 and x. 5. Use Cramer's rule to check your answer by solving for just x1 and x4. Rubric Instructor Feedback: This activity uses an integrated rubric in Blackboard. Students can view instructor feedback in the Grade Center. For more information, review these instructions. Critical Elements Main Elements Exemplary (100%) Completes all of the requirements of all assigned problems All parts of the assignment are computed correctly (all answers correct) Proficient (85%) Completes most of the requirements Needs Improvement (55%) Includes some of the requirements Most parts of the assignment are computed correctly Some parts of the assignment are computed correctly Computational Analysis (Shows Work) Substantially shows work in a meaningful sequence to support answers Shows work in an ordered sequence to support answers There is some work shown, but it is neither comprehensive nor in an ordered sequence Applies Course Content All of the relevant course content is correctly applied to the assignment Effectively works with linear algebra and solves the quantitative problems in the context of thermodynamics Most of the relevant course content is correctly applied to the assignment Works with linear algebra and solves the quantitative problems in the context of thermodynamics, but with moderate effectiveness Some of the relevant course content is correctly applied to this assignment Uses linear algebra and solves the quantitative problems with minimal effectiveness Computational Result (Correct Answers) Quantitative Literacy Not Evident (0%) Includes fewer than half of the requirements; completes fewer than half of the problems There are many computational errors Value 25 Little or no work shown to support answers, or it is unclear how computations link to correct answers Does not correctly apply any of the course content 30 Is not able to connect linear algebra techniques to thermodynamics 10 Earned Total 20 15 100%