Let be the performance level of someone learning a skill as a function of the tr

Question

Let be the performance level of someone learning a skill as a function of the training time . The derivative represents the rate at which performance improves. If is the maximum level of performance of which the learner is capable, then a model for learning is given by the differential equation dP/dt = k(M-P(t)) where is a positive constant. Two new workers, Andy and Bob, were hired for an assembly line. Andy could process units per minute after one hour and units per minute after two hours. Bob could process units per minute after one hour and units per minute after two hours. Using the above model and assuming that , estimate the maximum number of units per minute that each worker is capable of processing. Andy: Bob:

Explanation / Answer

Bill learns at a rate of k= 1/24 to a maximum of 13, approximately. Bob: k = 1/70, M = 17, approx. P(t) = M (1 - exp (-k t)) where t is in minutes. More precisely, with t in hours: For Bill, (e^k - 1) ( e^k -12) = 0 Since k is not 0, k = ln ( 12) = 2.4849 Then solve for M = 13.0909 Bob: 4 e^2k - 14 e^k + 10 = 0 which factors to: (2e^k - 5)(e^k - 1) = 0 so k = ln (5/2) = 0.91629 Solving for M = 16.6667

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