Let P(t) be the performance level of someone learning a skill as a function of t

Question

Let P(t) be the performance level of someone
learning a skill as a function of the training time t. The derivative
(dP/dt) represents the rate at which performance improves.
If M 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 k is a positive constant.
Two new workers, Andy and Peter, were hired for an assembly
line. Andy could process 11 units per minute after one hour
and 13 units per minute after two hours. Peter could process 10
units per minute after one hour and 15 units per minute after two
hours. Using the above model and assuming that P(0) = 0, estimate
the maximum number of units per minute that each worker
is capable of processing.
Andy: ?
Peter: ?

Explanation / Answer

suppose Andy= bill Peter= bob 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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