Solve the Problem Using Matrices ${ } ^ { \circ } \mathrm { F }$

question 217

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 ${ } ^ { \circ } \mathrm { F }$ ) 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) \(\mathrm { T } _ { 1 } = \frac { 666 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 564 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = 105 ^ { \circ } \mathrm { F }\)
• B) \(\mathrm { T } _ { 1 } = \frac { 766 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 964 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 430 } { 7 } \circ \mathrm { F }\)
• C) \(\mathrm { T } _ { 1 } = \frac { 1352 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 604 } { 7 } \circ { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 655 } { 7 } \circ \mathrm { F }\)
• D) \(\mathrm { T } _ { 1 } = \frac { 676 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 604 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 655 } { 14 } \circ \mathrm { F }\)

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