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

where y = the height (m), θ₀ = 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 θ₀ = 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.

y = (tano).x - 8 2v cos00 x + yo