Exercise 2: Suppose that 100 spheres of radius 1 are dropped into a cylinder of radius 11 from random locations above the cylinder. For each of the cases k = 2.0 and k = 4.0 do the following: run the program given in the text (for n = 100) and create a three-dimensional scatter plot of the final positions of the centers of the spheres. What is noticeably different about the two scatter plots?
Exercise 3: The value of
/4 can be approximated by finding the fraction of times that a point (x, y) placed randomly in the square [-1, 1]×[-1, 1] (the region -1 
x 1 and -1 y 1) also lies in the unit circle x2 + y2 1. Approximate the value of /4 = 0.78539816... in this manner using 100 points placed randomly in this square. With the random placement of 100 points, the best approximation that can be obtained in this manner is 79/100 = .79. With the random placement of 100 or fewer points, the best approximation that can be obtained is 11/14 = .785714..., which can occur only when n is a positive multiple of 14 (and so, in this case, more points does not necessarily produce a better approximation). How close is your estimate to this value?Exercise 4: Repeat the work for Exercise 3, but use 1000 points placed randomly in [-1, 1]×[-1, 1]. Since we are utilizing more points, and a number that is an integer multiple of the number of points from Exercise 3, we would expect a better approximation. Does this yield a better approximation of
/4? If so, then how much better? With the random placement of 1000 points, the best approximation that can be obtained in this manner is 785/1000 = .785. With the random placement of 1000 or fewer points, the best approximation that can be obtained is 355/452 = .78539823..., which can occur only when n is a positive multiple of 452. How close is your estimate to this value?Exercise 5: Consider the function f(x) = e-cos(x). For 0
x 1, this function satisfies 0 f(x) 1. Thus the area of the region below the graph of y = f(x) and above the x-axis is the same as the fraction that this area represents in the square [0, 1]×[0, 1]. This area can then be approximated by randomly generating points (x, y) in [0, 1]×[0, 1], and finding the fraction of these points that satisfy y e-cos(x). Approximate this area in this manner using 100 randomly generated points.