14. [M] Study how changes in boundary temperatu plate affect the temperatures at

Question

14. [M] Study how changes in boundary temperatu plate affect the temperatures at interior points on a stee e. on the plat egin by estimating the temperatures Ti. T2. Ts. Ts at shown in each of the sets of four points on the steel plate the higure. In each case, the value of T, is approximated by e average of the temperatures at the four closest points. See Exercises 33 and 34 in Section 1.1, where the values in degrees) turn out to be (20, 27.5,30, 22.5). How is this of values related to your results for the points in set th list (a) and set (b)? b. Without making any computations, guess the interior temperatures in (a) when the boundary temperatures are all multiplied by 3. Check your guess. c. Finally, make a general conjecture about the correspon- dence from the list of eight boundary temperatures to the list of four interior temperatures. Plate A Plate B 20° 20° 0° 0° 0° 0°10° 40° 0° 0° 10 40° 20° 20° 10° 10°

Explanation / Answer

Plate A

T1 = (T2+T4 + 20 + 0)/4

T2= (T1+T3 + 20 + 0)/4

T3 = (T2+T4 + 20 + 0)/4

T4 = (T1+T3 + 20 + 0)/4

from equations we know T1 =T3 and T2 = T4

T1 = (2T2+20)/4 = 0.5T2 + 5

T2 =(2T1+20)/4 = 0.5T1 + 5 = 0.5(0.5T2+5) + 5 = 0.25T2 + 7.5

1.25T2 = 7.5

T2 = 5 =T4 and T1 = 7.5 = T3

Plate B

T1 = (T2+T4 + 10 + 0)/4 = (T2+T4 + 10)/4

T2= (T1+T3 + 40 + 0)/4 =  (T1+T3 + 40)/4

T3 = (T2+T4 + 40 +10)/4 =  (T2+T4 + 50)/4

T4 = (T1+T3 + 10 + 10)/4 = (T1+T3 + 20)/4

From above equations : T3 = T1 + 10 and T4 = T2 -5

T4 = (T1+T3 + 20)/4 = (2T1 +30)/4 and T2 = (T1+T3 + 40)/4 = (2T1+50)/4

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