Problem 2. The curved roof of a new building is constructed by a thin shell structure....

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Problem 2. The curved roof of a new building is constructed by a thin shell structure. To avoid curved walls and ceilings in the interior of the building, a room (with vertical walls and a horizontal ceiling) should be constructed so that it has the maximum possible volume that fits beneath the roof. One possible example (in which the volume of the room is not the maximum possible one) is depicted in Figure 1a). As can be seen, the curved roof does not change in the z-direction. As a consequence, the 3D problem reduces to the 2D problem of finding the maximum possible rectangular area Arec under the curve given by the function a) which describes the roof in the x-y plane, see Figure 1b). y f(x) X 1 4 = COS X - -x² + 3, 2 10 A rec -3 -1 3 2 y 1 1 0 b) y=f(x) A rec 1 2 Figure 1: a) Curved roof in 3D with room beneath it, b) 2D description of the problem. →X 3 a) Use the bisection method with tolerance & = 0.3 to find a reasonable approximation ã of the positive root a of f(x) in the initial interval [2, 3]. Based on this result, what is the maximum possible length of the rectangle /? b) Turn the problem of finding the maximum possible rectangular area Arec under f(x) into a new rootfinding problem of the form g(x) = 0. Use Newton's method with initial guess xo = ã, where a is the approximated root found in Part a), and two iterations to find an approximation of the positive root ß of g(x) in the interval [0, ã]. Based on this result, what is the maximum possible area of the rectangle Arec? Problem 2. The curved roof of a new building is constructed by a thin shell structure. To avoid curved walls and ceilings in the interior of the building, a room (with vertical walls and a horizontal ceiling) should be constructed so that it has the maximum possible volume that fits beneath the roof. One possible example (in which the volume of the room is not the maximum possible one) is depicted in Figure 1a). As can be seen, the curved roof does not change in the z-direction. As a consequence, the 3D problem reduces to the 2D problem of finding the maximum possible rectangular area Arec under the curve given by the function a) which describes the roof in the x-y plane, see Figure 1b). y f(x) X 1 4 = COS X - -x² + 3, 2 10 A rec -3 -1 3 2 y 1 1 0 b) y=f(x) A rec 1 2 Figure 1: a) Curved roof in 3D with room beneath it, b) 2D description of the problem. →X 3 a) Use the bisection method with tolerance & = 0.3 to find a reasonable approximation ã of the positive root a of f(x) in the initial interval [2, 3]. Based on this result, what is the maximum possible length of the rectangle /? b) Turn the problem of finding the maximum possible rectangular area Arec under f(x) into a new rootfinding problem of the form g(x) = 0. Use Newton's method with initial guess xo = ã, where a is the approximated root found in Part a), and two iterations to find an approximation of the positive root ß of g(x) in the interval [0, ã]. Based on this result, what is the maximum possible area of the rectangle Arec?