An object is composed of a semicircle sitting atop a rectangle, as in the figure below. The rectangle has height h and width 1. (i) Use the Lagrange multiplier technique to find the height-to-length ratio that maximises area A, with perimeter P held constant (say, P = Po). Ensure the behaviour is tested in the neighbour- hood of the critical point. When determining a strategy for the neighbourhood test, you might want to consider your answer to (iii) first. You should note that each ray of constant height-to-length ratio intersects the constraint. At the point of intersection there will also be an intersection with a contour belonging to the optimising function. Your task is to determine whether that contour value is higher/lower than the contour value at the critical point. With h = ml, try writing A in terms of Po and m, then adjust m accordingly. (ii) Use the Lagrange multiplier technique to find the height-to-length ratio that minimises perimeter with area held constant (say, A = A0). Ensure the behaviour is tested in the neighbourhood of the critical point. (iii) The figures below show two scenarios. In both cases the contours belong to the function being optimised and the white curve depicts the constraint equation for an arbitrary value of P or A. For reference, overlaid on both plots are dashed lines corresponding to constant height-to-length ratios: h = ml where the slope, m = tan(kπ/16) with k = 1, 2,...,7. • Connect the plots (left and right) to the questions (i) and (ii). • Describe where your height-to-length ratio findings in (i) and (ii) sit with respect to the dashed lines. • Are your results consistent with the plots? h 12 10 12 150 135 10 120 6 105 90 2 0 2 4 1 60 h 612 60 2 48 45 0 18 8 10 0 2 4 6 8 10