Final answer:
To answer the student's question, we must calculate the rate of change of a sphere's surface area using differentiation and related rates, given the constant rate of volume increase.
Step-by-step explanation:
The question is asking about the rate at which the surface area of a sphere changes given that the volume of the sphere is increasing at a constant rate. To find this rate, we need to use calculus, specifically the concept of related rates, and the formulae for the volume and surface area of a sphere. The formula for the volume V of a sphere is given by V = (4/3)πr^3 and the formula for the surface area S is S = 4πr^2. Since the volume is changing at a rate of 5409 cubic meters per second, we denote this by dV/dt = 5409 m^3/sec. To find the rate of change of the surface area (dS/dt), we need to find the relationship between the surface area and the volume. First, express the radius r in terms of the volume, r = (3V/4π)^(1/3), and then differentiate the surface area with respect to time using the chain rule.