Is there a time t , for 1<t<8 , at which the rate of change of the number of people at the venue changes from negative to positive? Give a reason for your answer.

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

The rate of change of the number of people at the venue refers to the net rate, which means the rate at which people are arriving minus the rate at which they are leaving, i.e. A(t) - L(t). This net rate is negative whenever L(t) > A(t) and it is positive whenever A(t) > L(t). This makes good intuitive sense: If at a specific time, t₀, A(t₀) = 1.2 and L(t₀) = .8 it means people are arriving at a rate of 120 ppl / hr and leaving at a rate of only 80 ppl / hr. Thus the net rate is positive, .4, and this means that the number of people at the venue is increasing at that time.

The question asks if there is any time at which the rate changes from negative to positive, in other words, is there a time when the arrival rate is smaller than the leaving rate and becomes greater.

To decide, we can simply graph both A(t) and L(t) on a calculator (since this is a calculator-active AP-style question), on the window 1 < t < 8 and see if the graph of A(t) crosses from below L(t) to above it.