9 Each machine on Widgetco’s assembly line gets out of whack an average of once a minute. Laborers are assigned to reset a machine that gets out of whack. The company pays each laborer cs dollars per hour and estimates that each hour of idle machine time costs the company cm dollars in lost production. Data indicate that the time between successive breakdowns of a machine and the time to reset a machine are exponential. Widgetco plans to assign each worker a certain number of machines to watch over and repair. Let M total number of Widgetco machines, w

number of laborers hired by Widgetco, and R = M/w = machines assigned to each laborer.

a Express Widgetco’s hourly cost in terms of R and M.

b Show that the optimal value of R does not depend on the value of M.

c Use calculus to show that costs are minimized by choosing

d Suppose cm 78¢ and cs $2.75. Widgetco has 200 machines, and a laborer can reset a machine in an average of 7.8 seconds. How can Widgetco minimize costs?
e In parts (a)–(d), we have tacitly assumed that at any point in time, the rate at which the machines assigned to a worker break down does not depend on the number of his or her assigned machines that are currently working properly. Does this assumption seem reasonable?

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60 R =