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## Homework Statement

Problem:

(1 pt) Let P(t) be the performance level of someone learning a skill as a function of the training time t. The derivative \displaystyle \frac{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

\frac{dP}{dt} = k(M-P(t))

where k is a positive constant.

Two new workers, Mark and Peter, were hired for an assembly line. Mark could process 12 units per minute after one hour and 14 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.

## Homework Equations

Method of solving separable differential equations, basic integration...

## The Attempt at a Solution

I have solved up to the equation for P(t) = M + A*e^-kt (where A represents +/- e^c), and found that the limit of P(t) as t approaches infinity is M. After that, I don't know how to find the maximum number of units per minute for each of Mark and Peter. Any help would be appreciated. Thanks!