Question

2 10° 400 4 3 400 An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. The figure represents 20° 20° a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let T1, ..., T4 denote the temperatures at the four interior nodes of the figure. The temperature at a node is approximately equal to the average of the 10° four nearest nodes—to the left, above, to the right, and below. Solve the system of equations below to find the temperatures T1, 30° 30° T4. 4T1 - Tz T4 = 30 - Ty + 4T2 - T3 = 60 T +412 - TA = 70 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The unique solution of the system is ( CODD. (Type integers or decimals rounded to the nearest thousandth as needed.) B. The system has infinitely many solutions. C. The system has no solution.

Answer 1

Given system of equations es. 4TI Tz to T3 - Ta - 30 -Ti + 4Tz - Tz + OTA = 60 OTI - Tz + 4T3 T4 - 70 The augmented matrix es. 4 - 1 -1 30 -1 O 60 O -1 -1 70 Applying the row operation R2 Rit4R2 + -I - 1 30 O is 270 tt ī ī -1 70 Rz R2 +ISR3 4 - 1 o - 1 30 O is 270 o 56 -16 1320

see that Here, we there are three non-zero rows in the above matrix with a variables. Number of variables are more than the number of nonzero nonzero rows. Hence this system has infinitely many solutions. Option (B) (B) is correct.