A Monte Carlo analysis can be used for optimization. For example, the trajectory of a ball...

A Monte Carlo analysis can be used for optimization. For example, the trajectory of a ball can be computed with

 (P14.35)

where y = the height (m), u0 = the initial angle (radians), v0 = the initial velocity (m/s), g = the gravitational constant = 9.81 m/s2, and y0 = the initial height (m). Given y0 = 1 m, v0 = 25 m/s, and 00 = 50o, determine the maximum height and the corresponding x distance (a) analytically with calculus and (b) numerically with Monte Carlo simulation. For the latter, develop a script that generates a vector of 10,000 uniformly-distributed values of x between 0 and 60 m. Use this vector and Eq. P14.35 to generate a vector of heights. Then, employ the max function to determine the maximum height and the associated x distance.