A company manufactures chains by connecting the links produced by a certain machine. The length of each link produced by this machine is uniformly distributed over [60,80] millimeters. The seams on each link are also distributed uniformly between [0,3] millimeters. Assume that the lengths of dierent links and seams are all independent from each other, and if X1,X2,...,Xn are the lengths of n links, and if Y1,Y2,...,Yn1 are the lengths of n1 seams, then the net length of a chain obtained by connecting these links is (X1 Y1) + (X2 Y2) + ... + (Xn1 Yn1) + Xn.

(a) Compute the expected value of the net length of a chain with n links.

(b) Compute the variance of the net length of a chain with n links.

(c) The company would like to produce chains with a 2-meter (that is, 2,000-millimeter) net length, but since the length of each link and seam is random, it is impossible to produce chains that measure exactly 2 meters. Needless to say, measuring the length of each chain and seam is very tedious. The company decides to construct such chains by connecting n links, so that the lengths of the chains exceed 2 meters 99.9% of the time. Compute approximately what n should be. Hint: Use central limit theory.