### The City Of St Louis In The United States Has A

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# The city of St Louis in the United States has a monument known as the "Gateway Arch" which is pictured in Figure 1 below. You have been asked to help design a small-scale replica of this arch…

The city of St Louis in the United States has a monument known as the "Gateway Arch" which is pictured in Figure 1 below. You have been asked to help design a small-scale replica of this arch for Brisbane Figure 1 – Gateway Arch in St Louis, Missouri

To simplify the design problem, you can assume that the arch follows a parabolic equation (and that the thickness of the arch is negligible). This equation that the arch follows will be of the form y ax2+bx+ where a, b and care constants which will need to be solved for. Figure 2 shows a schematic diagram for the replica arch referenced against the x-y axes shown. The arch is to be 4 metres wide at its base (and thus pass through P2 and P3 as shown in the diagram). The arch height is limited by a nearby overhanging tree. As a result, it has been decided that the arch needs to pass through the point P, in order to prevent the structure from interfering with the tree. X(m) P (13) P3 (2,0) P2(-2,0) Figure 2-Replica arch schematic Write out the system of simultaneous linear equations that need to be satisfied for the a) parabola to pass through the required points. b) Write the simultaneous linear equations from part (a) as a matrix equation AX-B. c What two conditions must be satisfied for matrix A to have an inverse? Show that these conditions are satisfied in this problem. d) Find A1 using Gaussian elimination. e) Solve for a, b and c using the matrix equation from part (b) and A- from part (d). Showthe matrix multiplication step in full. What is the equation of the parabola?