Final Practice T1 Q4 - Looking for a detailed worked solution with theoretical explanation

This question has several parts, Please scroll down and try to answer all suthuestions until you get to "END OF THIS QUESTION", A small house in Antarctica has one living area at ground level and a stoiage area upstairs Ground The temperature outside the house is constant at tJD C The oor of the house is perfectly insulated, so that no heat is exchanged between the living area and the ground Other interiaces are partially insulated, so that the rate of change ot temperature across any surface is propomonal to the dierence in the temperatures on each side It yl is the temperature in the storage area and 92 is the temperature in the living area [both in degrees C) at time t hours, we have the differential equations 4'51 4392 f :ak1ll1+k2(y2'!l11i j :k2(y1*y2)'k3312 where k1,k2,k3 are the constants of proportionality tor the roof above the storage area, and the ceiling between the living area and the storage area, and the walls of the living area, respectively (a) lfthis pair of diflerential equations is written in matrix/'vectorlonn as (322%) (ii) , enter the matrix A here, using the variable names k1, k2, k3: Give your answer to pan (a) as a matrix by clicking the button above Select 2 rows and 2 columns, then click on each entry to enter your answer Use Maple syntax for your answers below For example, use it for multiplication, exp (73*t) for (5'3, 5 scrz:(51/(3)for g and for (g). You can either enter exact answers or decimal approximations that are correct to at least 6 decimal places, such as 19. 123456 For the rest ofthisouestion, suppose that k1 : 0.12 and k2 : 0.02 and k3 : 0.04 (h) Substitute these values into A, and use Maple to nd the eigenvalues and eigenvectors of A Enter the set of eigenvalues here. ED 9 one eigenvector for each eigenvalue here' @D The syntax tor the set ot numbers {1, 2} is (1,2) The syntax forthe set of vectors { (g) , (2)} is ((1)2), 3,42} You may use the preview button to check your syntax is correct [2) Hence write down a formula for the solution ( ) of the system of diflerential equations in terms of the variable 312 13 Call the arbitrary constants in your answer Ct and ()2: You may use the p'EV/ew button to check that yoursynlax is correct = @D {d} The small house in Antarctica is heated to a temperature of 110 C in the storage area, and 18 C in the living area At midnight there is a power failure, and from this point onwards, there is no heating in the house Find the temperature in the living area at 6 40 a m and enter your answer here: oc Give your answer rounded to six decimal places, such as 18.123456, l END OF THIS QUES'HON