Question: (a) It is observed that three machines A, B, and C produce, respectively, 60, 80, and 40 units of a product during one workday lasting 8 hours (or 480 minutes). The average output is then 60 units per day. We see that A, B, and C use, respectively, 8, 6, and 12 minutes to make one unit. If all machines were equally efficient and jointly produced 60 + 80 + 40 = 180 units during a day, then how much time would be required to produce each unit? (Note that the answer is not (8 + 6 + 12)/3.)
(b) Suppose that n machines A1, A2, ... , An produce the same product simultaneously during a time interval of length T . Given that the production times per unit are respectively t1, t2, ... , tn, find the total output Q. Show that if all the machines were equally efficient and together had produced exactly the same total amount Q in the time span T , then the time needed for each machine to produce one unit would be precisely the harmonic mean t ¯ H of t1, t2, ... , tn.