. A project management consultant estimated that if a particular project was completed, that t years after its date of completion, P thousand persons would be direct beneficiaries of the project, where

Question

.  A project management consultant estimated that if a particular project was completed, that t years after its  date of completion, P thousand persons would be direct beneficiaries of the project, where  
P (t) = t3/3 - 5t2 +20t, 		0<t<12

For what value of t will the maximum number of people receive direct benefits?

Steve · Answer

to find where P(t) is a maximum, find where its derivative is zero:

P'(t) = t^2 - 10t + 20
t = 5 ± √5

That is where P(t) has a min or a max. To determine which, check

P''(t) = 2t - 10

At t = 5 - √5, P'' < 0, so P(t) has a max.

So, at t = 5 - √5 = 2.764, P will be at a maximum.