Imagine that you wish to know the temperature at all points on a plate that is insulated except for two positions on the edge where heating elements of known temperature are applied. At some point when the temperature at all positions on the plate have stabilized, what will the temperature at each point be?

To solve such a problem, we imagine the plate surface divided into equal sized "tiles" and restate the problem to find the temperature at the center of each tile. We know that the temperature of each tile depends upon its neighboring temperatures. In fact, it is a weighted average of those temperatures based on the proportion of our tile's edge that is shared with its neighbor. Here is a rectangular "plate" divided into 15 equal area tiles. Assume that each edge of each tile has identical length, so that the temperature of any one tile is a simple average of the temperatures of all neighboring tiles that share a side with the tile in question. This creates a system of 15 equations, one for each "tile" area.

TA and TB are the non-insulated positions along the edge where a heat source of known temperature will be applied. The temperatures at T1 through T15 are the unknowns that you wish to find.

a. Define a function named heatPlate that accepts two temperature values TA and TB (representing TA and TB, respectively) and returns a single matrix result that is a 53 matrix of temperatures for the unknowns T1 through T15 in the order labeled above. If you can not figure out how to return the values as a 53 matrix, return them as a column vector.

To solve this problem, identify 15 equations. Each equation represents the temperature value for one tile based upon the temperature values of each of its neighboring tiles. For example, the temperature at T8 is equal to the average of the temperatures: T5, T7, T9, and T11.

What about the corners? Well, since the gray area is insulation, the only effect is from the non-insulating tiles. Thus, T15 is equal to the average of only T12 and T14.

How do TA and TB factor into the equations? They will be in the equations, but since their values will be known when the function is called, the terms with TA and TBshould be on the right-hand side of the equations once the equations are put into general form.

DO NOT SOLVE FOR ANY OF THE TEMPERATURES OF THIS PROBLEM BY HAND. Work out the equations for each tile and then reorder the terms in general form, with the unknown terms on the left-hand side and the known terms on the right-hand side of the equation. Once you have the equations in general form, rewrite the linear system as a matrix equation and put the commands to solve this problem in your heatPlate(TA,TB) function definition.

b. Write code to call your heatPlate function to compute the values of T1 through T15 for each of the following conditions. Do not suppress the output of your function calls.

i. Find T1 through T15 when TA = 80 and TB = 80

ii. Find T1 through T15 when TA = 90 and TB = 10

iii. Find T1 through T15 when TA = 22 and TB = 67