A rectangular “standing-only” section at the venue changes size as t increases in order to manage the flow of people.

Question

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.

DESCRIPTION: "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

length = x = 16 ft

width = y = 10 ft

rate of change of length = dx = 6 ft/hr

rate of change of width = dy = -3 ft/hr

area = A = x * y

rate of change of area = dA = (x)(dy) + (dx)(y)

dA = (16 ft) (-3 ft/hr) + (6 ft/hr) (10 ft) = 12 ft²/hr