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, David and Peter, were hired for an assembly line. David could process 11 units per minute after one hour and 14 units per minute after two hours.

Answer 1

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.

Answer 2

Peter:
David:

Answer 3

have you solved the diff equ for P(t)

Answer 4

\[\int\limits_{}^{}\frac{dP}{M-P} = \int\limits_{}^{}k dt\]
\[-\ln (M-P) = kt +C\]
\[M-P = Ce^{-kt}\]
\[P = M-Ce^{-kt}\]
given, P(0)=0 --> C=M
\[P(t) = M(1-e^{-kt})\]
plug in the 2 given points to solve for M

Answer 5

Thank you so much!!

Answer 6

sorry did you figure it out

Answer 7

David:
\[\frac{11}{1-e^{-k}} = M\]
\[\frac{14}{1-e^{-2k}} = M\]
Let u = e^-k, --> k = ln(1/u)
set equations equal
\[\frac{11}{1-u}=\frac{14}{1-u^{2}}\]
\[\rightarrow 11u^{2}-14u+3 = 0\]
\[(11u-3)(u-1) = 0\]
u = 3/11 and 1
k can't be 0 so u cannot be 1
substituting back in to get M
\[M = \frac{11}{1-u} = \frac{11}{1-3/11} = \frac{121}{8}\]

repeat same process for peter

Answer 8

Thank you!!