SOLUTION: The management of a factory found out that the model N=30(1-e^-0.09t) describes the number of units "N" produced after a new employee has worked for "t" days. A) Find the numb

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Question 1116665: The management of a factory found out that the model
N=30(1-e^-0.09t)
describes the number of units "N" produced after a new employee has worked for "t" days.
A) Find the number of units produced by a new employee after 5 days.
B) What is the rate of change after the first week? Interpret the meaning of the result.
C) Is there a limit as to the number of units per day an employee can produce no matter how long that person has been working for?

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
the formula is n = 30 * (1 - e^(-.09T)

e^(-.09T) is the same as 1/e^(.09T)

the formula becomes n = 30 * (1 - 1/e^(.09T)

when T = 5, n = 30 * (1 - 1/e^(.09*5).

this makes n = 10.87115545 units per day.

the maximum number of units the worker can produce per day appears to be 30.

this is because, as T gets larger and larger, 1/e^(.09T) gets smaller and smaller and approaches 0, making the formula approach n = 30 * 1 = 30.

i'm not sure how to answer the rate of changes after the first week, but i think it would start getting smaller and smaller.

a look at the graph of the equation might shed some light on this.

in the first 7 days the average rate of change is 2.003... units per day.

in the next 7 days, the average rate of change is 1.066... units per day.

in the next 7 days, the average rate of change is .568... units per day.

in the next 7 days, the average rate of change is .302... units per day.

the curve is definitely flattening out as the number of weeks progress.

each succeeding week has the increase in the average number of units per day for that week being cut very roughly in half.

here's what the graph looks like.