Question

17. For time t on the interval [0, 8], people arrive at a venue for an outdoor concert at a rate
modeled by A(t) = 0.3sin(1.9t) + 0.3cos(0.6t) + 1.3. For time t on the interval [0, 1], no one
leaves the venue, but for t on the interval [1, 8], people leave the venue at a rate modeled by
L(t) = 0.2cos(1.9t) + 0.2sin t + 0.8. Both functions 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 the function P(t).
a. At time t = 2, there are 6 hundred people at the venue. Write an equation for the locally linear
approximation of P at time t = 2, and use it to approximate the number of people at the venue at
time t = 2.5 hours. (Locally linear means we can pretend that the graph of function P is a line.)

b. What is the instantaneous rate of change of P at time t = 4 hours? Indicate units of measure.

c. At time t = 2, there are 6 hundred people at the venue. Find how many people are at the
venue at time t = 5 hours.

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

e. Is there a time t on the interval (1, 8), at which the rate of change of the number of people at
the venue changes from negative to positive? Justify your answer.

f. A rectangular “standing-only” section at the venue changes size at t increases in order to
manage the flow of people. Let x represent the length, in feet, of the section, and let y represent
the width, in feet, of the section. The length of the section is increasing at a rate of 6 feet per
hour and the width of the section is decreasing at a rate of 3 feet per hour. What is the rate of
change of the area of the section with respect to time when x = 16 and y = 10?
Indicate units of measure in your answer.

Answer 1

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