. 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 the value of t at which the maximum number of people receive direct benefits, we can differentiate the function P(t) with respect to t and set it equal to zero.
P'(t) = (t^3/3 - 5t^2 + 20t)' = 3t^2/3 - 10t + 20
Setting P'(t) = 0:
3t^2/3 - 10t + 20 = 0
To solve this equation, we can multiply through by 3:
t^2 - 10t + 60 = 0
Factoring the quadratic equation:
(t - 6)(t - 10) = 0
Setting each factor equal to zero, we get:
t - 6 = 0 or t - 10 = 0
Solving for t:
t = 6 or t = 10
So, the maximum number of people will receive direct benefits at t = 6 or t = 10.
To find the value of t that will result in the maximum number of people receiving direct benefits, we need to find the value of t at which the derivative of P(t) is equal to zero.
The derivative of P(t) can be found by applying the power rule of differentiation. Let's find the derivative of t³/3, -5t², and 20t separately.
The derivative of t³/3 is (1/3)*3t², which simplifies to t².
The derivative of -5t² is -5*2t, which simplifies to -10t.
The derivative of 20t is 20.
Now we can find the derivative of P(t) by adding the derivatives of each term: P'(t) = t² - 10t + 20.
To find the value of t where the derivative is equal to zero, we set P'(t) equal to zero and solve for t:
t² - 10t + 20 = 0.
This is a quadratic equation that we can solve using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula.
The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b² - 4ac)) / 2a.
In our case, a = 1, b = -10, and c = 20. Substituting these values into the quadratic formula, we get:
t = (-(-10) ± √((-10)² - 4*1*20)) / (2*1).
Simplifying further, we have:
t = (10 ± √(100 - 80)) / 2.
t = (10 ± √20) / 2.
t = 5 ± √5.
So, the solutions for t are t = 5 + √5 and t = 5 - √5.
However, since the range of t is specified as 0 < t < 12, the only valid solution is t = 5 - √5, which falls within the specified range.
Therefore, for the maximum number of people to receive direct benefits, t should be approximately 5 - √5.