Study how changes in boundary temperatures on a steel plate affect the temperatures at interior points on the plate.
a. Begin by estimating the temperatures T1, T2, T3, T4 at each of the sets of four points on the steel plate shown in the figure. In each case, the value of Tk is approximated by the 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 list of values related to your results for the points in set (a) and set (b)?
b. Without making any computations, guess the interior temperatures in (a) when the boundary temperatures are all multipled by 3. Check your guess.
c. Finally, make a general conjecture about the correspondence from the list of eight boundary temperatures to the list of four interior temperatures.