Question

1. (10 points A Monte Carlo analysis can be used for optimization. For example, the trajectory of a ball can be computed with 2vo cos2 Go where y is the achieved height in meters (m), 6% is the initial angle (radians), is the horizontal distance (m), g is the gravitational acceleration (-9.81 m/s2), vo is the initial velocity (m/s), and yo is the initial height. Given yo 1 m, vo 25 m/s, and 500 (or T/180 x 50 rad), determine the maximum height and the corresponding z distance numerically with Monte Carlo simulation. Develop a function calc trajectory that generates a vector of 10,000 uniformly distributed values ofr between 0 and 60 m using the rand function. Use this vector and the above equation to generate a vector of heights. Determine the maximum height and the associated z distance. Your function template is as follows: 1 function [hmax xmax] calc trajectory(s) 3 rng(s); 8 Initialize the random number generator with the integer s 5 8 Rest of your code goes here Your code must return the maximum height 6 8 achieved in the variable hmax and the corresponding x distance in xmax. 8 end Note that we control the random number generator using the rng function. It seeds the random number generator using the non-negative integer s supplied to your function so that rand produces a predictable sequence of numbers each time the function is called.

Answer 1

Matlab Code:

format short
gg=9.81;
y0=1;
v0=25;
theta0=50*pi/180;
s=10000;
xmin=0;
xmax=60;
r=rand(s,1);
xrand=xmin+(xmax-xmin)*r;
yrand=tan(theta0)*xrand-gg/(2*v0^2*cos(theta0)^2)*xrand.^2+y0;
[ymax,i]=max(yrand);
xmax=xrand(i);
y=@(x)-tan(theta0)*x-gg/((2*v0^2*cos(theta0)^2)*x.^2+y0);
[xmax,hmax]=calc_trajectory(y,0,60)

Output:

xmax =
   59.9999
hmax =
-71.5052

Answer 2

#include <iostream>
#include <cstdlib>
#include <cmath>
#include <string>
#include <ctime>

using namespace std;

int main ()
{
    srand (time(0));
    string operation;
    cout << "Using the letters 'o', or 'q', please indicate if you would like to simulate once, or quit the program: " << endl;
    cin >> operation;
    string reservoir_name; // Creating variables for reservoir
    double reservoir_capacity;
    double outflow;
    double inflow_min;
    double inflow_max;

    if (operation == "q")
    {
        cout << "Exiting program." << endl;
        system ("pause");
        return 0;
    }

    while (operation == "o") // Choose one or multiple simulations.
        {

                string reservoir_name; // Creating variables for reservoir function
                double reservoir_capacity;

                double inflow_min = 0;
                double inflow_max = 0;
                double inflow_range = inflow_min + inflow_max;
                double inflow_difference = inflow_max - inflow_min;
                double inflow_threshold = .9 * inflow_range/2; // Math for acceptable flow threshold.

                cout << "What is the name of the reservoir?" << endl;
                cin.ignore ();
                getline (cin,reservoir_name); // Grab whole string for reservoir name.
                cout << "What is the capacity of the reservoir in MAF (Millions of Acre Feet)?" << endl;
                cin >> reservoir_capacity;
                cout << "What is the minimum inflow?" << endl;
                cin >> inflow_min;
                cout << "What is the maximum inflow?" << endl;
                cin >> inflow_max;
                cout << "What is the required outflow?" << endl;
                cin >> outflow;
                inflow_range = inflow_min + inflow_max;
                inflow_threshold = .9 * inflow_range/2;
                cin.ignore ();

                if (outflow > inflow_threshold) // Check for unacceptable outflow levels.
                {
                    cout << "Warning! The outflow is over 90% of the average inflow. Simulation aborted. Returning to main menu." << endl;
                }
                else
                {
                    const int number_simulations = 10;
                    double fill_level = 0;
                    int years = 1;
                    cout << "Running simulation." << endl;
                    for (int i = 1; i < number_simulations; i++) // Each year
                    {

                        for (years; fill_level < reservoir_capacity; years++ )
                        {
                            double r = rand() * 1.0 / RAND_MAX;
                            double x = inflow_min + inflow_range * r;// SHOULD be between minimum inflow and maximum inflow.
                            if (fill_level < 0)
                            {
                            fill_level = 0;
                            }
                        }   // Simulate the change of water level.
                    cout << years << endl;
                    }

                }
                    cout << "What would you like to do now?" << endl; // Saving for later. The menu re-prompt message and code.
                    cout << "Using the letters 'o', or 'q', please indicate if you would like to simulate once, or quit the program: " << endl;
                    cin >> operation;
    }

    system ("pause");
    return 0;
}