Find P′′(5) . Using correct units, interpret the meaning of P′′(5) in the context of the problem.

Question

"For time 0 ≤ t ≤ 8, people arrive at a venue for an outdoor concert at a rate modeled by the function A defined by A(t) = 0.3sin(1.9t) + 0.3cos(0.6t) + 1.3. For time 0 ≤ t ≤ 1, no one leaves the venue, and for time 1 ≤ t ≤ 8, people leave the venue at a rate modeled by the function L defined by L(t) = 0.2cos(1.9t) + 0.2sin (t) + 0.8. Both A(t) and L(t) are measured in hundreds of people per hour, and t is measured in hours. The number of people at the venue, in hundreds, at time t hours is given by P(t)."

Answer 1

Hi Sophie P.,

The answer is pretty much stock:

P''(5) is the rate of change/hr of the rate of change/hr of people in the venue at the exactly 5 hours time point. The units are in people * hr^(-2) . To find the value, just differentiate the net change function (entry minus exit) vs. t and evaluate at t=5.

Since these are fairly simple functions, it would not be difficult to integrate the entry function from t=0 to 1 and the net change function (entry minus exit) from t = 1 to 8 to see how many persons remain (sentient or not) at the end of the concert.

-- Cheers, -- Mr. d.