Problem 2 In n-dimensional space, the solid sphere of radius r at point p is the set of points x such that Dist (x, P) Er. Write a function VolumeSphereIntersect (P, Q, R, S, T, W) which uses a Monte Carlo method with T trials to compute the volume of the intersection of a sphere of radius R centered at P with a sphere of radius S centered at Q. The function should return a confidence interval with confidence W. Here P and Q are n-dimensional vectors; R, S are positive floating numbers, T is a positive integer: and W is an floating point number between 0 and 1. 30 (It is possible to compute the exact answer, with some geometry and some calculus, but that is not the assignment.) To carry out the Monte Carlo method, use a uniform distribution within the box [B1, Ci] x [B2, C2] x [Bn, Ca] where BR = max(P[K] - R., Q[K] - S) and Ck = min P[K] + R, Q[K] + S). Note that if the distance from P to Q is greater than R + S, then the two spheres do not intersect, and so the function should simply return [0, 0]. Thus for example with P= [1,0, -1] Q=[2, 1,0] R=8 S=3 T=10000 W=0.95 you should generate 10,000 points within the box [-1, 5] x [-2, 4] x [-3, 3). In this case, the sphere around Q is inside the sphere about P, so it is particularly easy to calculate the exact answer: namely, the volume of the sphere around Q, 47 53/3 = 367 = 113.097. The volume of the box is 63 = 216, so the fraction of the points in the box is f = 0.5236. The standard deviation of the fraction o therefore should be about vf(1 - f)/10000 = 0.005. The 95% confidence interval should therefore be about [113 - 1.95 . 0 . 216, 113 + 1.95 . 0 . 216] ~ [111, 115], with, of course, some random variation depending on the particular sample. Thus VolumeSphereIntersect (P, Q, R, S, T, W) should return approximately [111, 115]