# Optimization Problems

#### Tristan

##### New member
Need some help...
Working on optimization and I don't know how to start a certain problem about worker efficiency.

An efficiency study of the first shift (from 8 A.M. to 5 P.M.) at a certain factory indicates that an average worker who arrives on the job at 8 A.M. will have assembled
Q(t)= -t^3 + 6t^2 +15t transistor radios t hours later. At what time during the shift is the worker performing most efficiently?

#### tkhunny

##### Moderator
Staff member
Need some help...
Working on optimization and I don't know how to start a certain problem about worker efficiency.

An efficiency study of the first shift (from 8 A.M. to 5 P.M.) at a certain factory indicates that an average worker who arrives on the job at 8 A.M. will have assembled
Q(t)= -t^3 + 6t^2 +15t transistor radios t hours later. At what time during the shift is the worker performing most efficiently?

You'll have to define "efficiently". Does that mean "greatest rate"? Could there be a first derivative in your future?

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#### Harry_the_cat

##### Senior Member
I'm going to take efficiency to mean making the greatest number of radios per unit time. So you're looking for the maximum of $$\displaystyle Q(t)$$

The usual method is to solve $$\displaystyle \dfrac{dQ}{dt} = 0$$ for $$\displaystyle t$$ and ensuring this is a maximum by checking that $$\displaystyle \dfrac{d^2Q}{dt^2} < 0$$

Can you do that?
I think you need the maximum of Q ' (t).