. 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?
dear
dear teacher school back hint not she
To find the value of t at which the maximum number of people will receive direct benefits, we need to find the maximum value of the function P(t) = t^3/3 - 5t^2 + 20t within the given range of 0 < t < 12.
To find the maximum value, we can take the derivative of P(t) with respect to t and find where it equals zero.
P'(t) = (d/dt) (t^3/3 - 5t^2 + 20t)
= t^2 - 10t + 20
Setting P'(t) equal to zero and solving for t:
t^2 - 10t + 20 = 0
We can solve this equation by factoring or using the quadratic formula:
(t - 2)(t - 10) = 0
This gives us two possible solutions: t = 2 and t = 10.
Since the given range is 0 < t < 12, we can discard t = 10 as it is outside the range.
Therefore, the value of t at which the maximum number of people will receive direct benefits is t = 2.