The floor manager of a processing plant has to train a new employee on a machine which creates a critical part. From past experience, the company has derived a model for the number of parts a new employee can produce per hour of

 p'(t) = 0.018(500 - P) 

Where P represents the number of parts produced per hour and t is the number of hours since training began. The employee can be released to operate the machine without supervision, when he or she can produce 350 parts per hour. Find the function of the number of parts produced as a function of the number of hours of training the employee has had. 

a) Use an appropriate limit to determine the maximum number of parts this machine can produce. 

b) What is the average rate of increase of parts over the first week? 

c) At the 40th hour of training, how quickly is the employee increasing his or her output? 

d) When will the new employee be able to work unsupervised?