Question 1 Box-shaped design are commonly seen for small satellites. The box-shaped design is simple and easy to manufacture. However, the box-shaped design requires careful calculation of the dimensions, so that the expandable solar panels can be mounted on the satellite when folded. Figure 1 shows the unfolding process of a box-shaped satellite with the solar panels mounted on the side and bottom faces. For this satellite, to ensure the power supply, nine solar panels are mounted on the side and bottom faces, where eight of them on the side faces are foldable. The ninth on the bottom face is fixed on the bottom face and is not seen in Figure 1. Figure 1: Unfolding process of solar panels on a box-shaped satellite. The production cost of the solar panels depends on their total area A, which is further determined by the designable dimensions of the satellite. Let us assume that top and bottom faces of the satellite are both square with edge length a, and the height between the two faces is b, then the total area of the solar panels is given by A = a+8ab The capability of this satellite for carrying payload is determined by the volume of the satellite, denoted by V, which also depends on the designable dimensions a and b, namely V = ab. In this question, we assume that the V = 0.125 m, and the thickness of the solar panels is not included in the payload volume. Therefore, to minimise the total cost of the satellite, we need to find the optimal designable dimensions a and b. (a) Write a MATLAB function named SolarPanelArea.m to calculate the total arca A of the solar panels. In case of need, use the file name Solar Panel AreaDeriv.m for the derivative, and file name Solar Panel AreaDeriv2.m for the second derivative. The headers of the functions must be function area Solar PanelArea(a) function areaDeriv = SolarPanelAreaDeriv(a) function areaDeriv2 = Solar PanelAreaDeriv2(a) Page 2 of 4 [5 MARKS] (b) Write a MATLAB script to find the optimal designable dimensions a and b that minimise the total area A of the solar panels. The script must be named optimal Dimensions.m. The last line of the script must be fprintf('The optimal dimensions are a = %f m and b = %f m. ', a, b); In your report, detail the process you have followed to find the optimal dimensions, generate plots when necessary. The precision of your results must be of 10-9 or higher in terms of absolute tolerance, please indicate the stopping criteria you have chosen. [10 MARKS] function xk = NewtonIteration (func, dfunc, x0, aTol) % Example implementation of the Newton's method for single variable root % finding. %% Input: % func: function handle for the function f of equation f(x) = 0 % dfunc: function handle for the derivative function f' % x0: the initial guess that is close to the desired root % aTol: the user-specified absolute tolerance %% Output: % x: the root within the given tolerance %% Implementation % Check input to ensure the validity of the tolerance if (aTol < eps) end error('Invalid input: exiting') % Set an initial error that is bigger than the tolerance erraTol + 1; % Set the iterative index k = 0; % Set the iterative root xk as the initial guess xk = x0; % Print to command window the header fprintf('Iteration \t\t\t x \t\t\t\t [f(x)| ') % Iterative calculation while err > aTol % Evaluate the function and its derivative at xk fk = func(xk); dfkdfunc(xk); % If |fk| is smaller than the machine precision, stop iteration if abs(fk)