Solve the Problem Using Matrices $\mathrm { T } _ { 1 } = \frac { 0 + 0 + 300 + \mathrm { T } _ { 2 } } { 4 }$

question 12

Multiple Choice

Solve the problem using matrices.
-In a study of heat transfer in a grid of wires, the temperature at an exterior node is maintained at a constant value (in °F) as shown in the figure. When the grid is in thermal equilibrium, the temperature at an interior
Node is the average of the temperatures at the four adjacent nodes. For instance $\mathrm { T } _ { 1 } = \frac { 0 + 0 + 300 + \mathrm { T } _ { 2 } } { 4 }$ , or $4 \mathrm {~T} _ { 1 } - \mathrm { T } _ { 2 } = 300$ . Find the temperatures $\mathrm { T } _ { 1 } , \mathrm {~T} _ { 2 }$ , and $\mathrm { T } _ { 3 }$ when the grid is in thermal equilibrium.

• A) \(\mathrm { T } _ { 1 } = \frac { 769 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 976 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 475 } { 7 } \circ \mathrm { F }\)
• B) \(\mathrm { T } _ { 1 } = 97 ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = 88 ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 745 } { 14 } ^ { \circ } \mathrm { F }\)
• C) \(\mathrm { T } _ { 1 } = 194 ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = 88 ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 745 } { 7 } { } ^ { \circ } \mathrm { F }\)
• D) \(\mathrm { T } _ { 1 } = \frac { 669 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 576 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 870 } { 7 } { } ^ { \circ } \mathrm { F }\)