The objective is to find the path from point A to point B that minimizes undesirables. You won’t find the numerical values of the path coefficients in this exercise; you’ll just set up the...

The objective is to find the path from point A to point B that minimizes undesirables. You won’t find the numerical values of the path coefficients in this exercise; you’ll just set up the optimization statement. The cubic relation for the y–x path projected on the flat surface of the Earth is of this relation: y = y_A
+ a x−x_A
+ b x−x_A
x−x_B
+ c x−x_A
x−x_B
x−x_C
in which (x_A, y_A) are the longitude and latitude coordinates of point A in Cartesian (flat, planar) space and points B and C have similar notation. This function is termed a Newton interpolating polynomial, here truncated to the cubic functionality. Point C is an intermediate point on the path, but its coordinates are yet to be determined. It’s a clever equation: at x = x_A, all the x-terms are zero, leaving the intercept as y_A. At x_B, y must be y_B, because the path must end at point B. This requires the value of coefficient a to be a = y_B
−y_A
x_B
−x_A. The path goes through mountains where elevation, z, depends on (x,y) location: z = f x,y . So the path distance, S, is the distance in three dimensions. Along the path in (x,y,z) space, there are bothersome flying bugs. The bug density (number per m3 of air, essentially the number encountered in each meter of distance) is ρ = g x,y z . We wish to have our optimizer find the path coefficient values to minimize path distance and to minimize the number of bugs encountered along the path. “I’d walk an extra 50 m just to avoid a bug,” one friend says. The answers to this exercise will be equations in terms of the coordinates of points A and B, path coefficients, penalty weighting values, and functions f and g. (i) Show how to derive the aforementioned relation for the value of coefficient a. (ii) Derive a similar relation for the b coefficient value in terms of the coordinates for points A, B, and C. (iii) Clearly indicate the DVs for the optimization. (iv) Use calculus to represent how to obtain the OF value. This will use the path integral techniques. (v) State the optimization application in standard form. (vi) Weight the two OF terms to match the friend’s statement about bugs. (Feel free to implement your solution.)