. 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?
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.
To find the value of t when the maximum number of people receive direct benefits, we need to find the maximum point of the given function P(t) = t3/3 - 5t2 + 20t within the given interval 0 < t < 12.
To do that, we can use calculus and find the critical points of the function by taking the derivative and setting it equal to zero.
Step 1: Take the derivative of P(t):
P'(t) = d/dt (t3/3 - 5t2 + 20t)
= t2 - 10t + 20
Step 2: Set the derivative equal to zero and solve for t:
t2 - 10t + 20 = 0
This is a quadratic equation, which we can solve using the quadratic formula or by factoring. In this case, it's easier to factor the equation:
(t - 2)(t - 10) = 0
Setting each factor equal to zero gives us two possible values for t:
t - 2 = 0 -> t = 2
t - 10 = 0 -> t = 10
Step 3: Check if the critical points are within the given interval:
We have found two critical points: t = 2 and t = 10. Since t has to be within the interval 0 < t < 12, we can see that both critical points are within this range.
Step 4: Evaluate the function at the critical points:
P(2) = (2)3/3 - 5(2)2 + 20(2)
= (8/3) - 20 + 40
= 40/3
P(10) = (10)3/3 - 5(10)2 + 20(10)
= 1000/3 - 500 + 200
= 700/3
Step 5: Compare the two values obtained:
We now compare the two values P(2) = 40/3 and P(10) = 700/3. We see that P(10) = 700/3 is the larger value, indicating the maximum number of people will receive direct benefits at t = 10 years.
Therefore, the maximum number of people will receive direct benefits when t = 10 years.