Question

please show all your work

2 Optimization and root finding The pilot of a stunt plane wishes to travel along the trajectory y 5 cos(z + 2) 150, where y is the height of the plane (in metres) and az is the horizontal displacement (in meters), as indicated in Figure 2 As Figure 2 illustrates, there is a transmission tower located at 0 of height 144 m. Safety standards require a minimum distance of 5 m between the trajectory of the plane and the top of the transmission tower (which is located at position (0, 144 m)). 160 140 120 proposed 100 light to trajectory 80 wer 60 40 20 Figure 2: The propoued trajectory of the stunt plane proposed trajectory s0 60 40 20 Figure 2: The proposed trajectory of the stunt plane. (a) Write down the equation for the dist ance (in metres) between the plane and the top of the transmission tower. Your answer should be a function of only the horizontal displacement z. (b) Determine the minimum distance between the plane and the top of the transmission tower. You will find it easier to minimize the square of the distance (which is equivalent to minimizing the distance itself). Performing the minimization task will require you to determine the root of a complicated function. Use Newton's Method with the initial guess 0.5 m to determine the root correct to five decimal places. Once you have found the root to the requested accuracy, don't forget to calculate the corresponding minimum distance. (c) Does the trajectory of the plane pass the required safety standards? If so, by how many metres does the plane clear the safety standards? If not, by how many metres does the plane fall short of the safety standards?

Answer 1

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