Final answer:
The rate of change of the area of a circle with a decreasing radius of 192 ft at a rate of 4 ft/min is -4,835.2π square feet per minute.
Step-by-step explanation:
The student's question involves a circle with a decreasing radius and requires us to find the rate of change of area of that circle. The formula for the area of a circle is A = πr², where A is the area and r is the radius. To find how the area changes with time, we take the derivative of the area with respect to time (dA/dt).
Using the chain rule, we have dA/dt = 2πr (dr/dt). We know the radius of the circle is decreasing at 4 ft/min, so dr/dt is -4 ft/min. Substituting r with 192 ft (since you want the rate of change of area at the instant the radius is 192 ft) and dr/dt with -4 ft/min into the derived formula, we get dA/dt = 2π(192) × (-4), which calculates the rate of change in the area in square feet per minute.
Therefore, the rate of change of the area of the circle at the moment when the radius is 192 ft is -4,835.2π square feet per minute, meaning the area of the circle is decreasing at this rate.