Final answer:
The rate of change of the area of a circle with respect to time, dA/dt, is equal to 2πr(dr/dt), which combines the circle's radius and the rate at which the radius changes with time.
Step-by-step explanation:
The student has asked how to find the rate of change of the area of a circle with respect to time, given the rate of change of the circle's radius with respect to time. To solve this, we start with the formula for the area of a circle, A = πr², and differentiate both sides with respect to time t.
The differentiation process, using the chain rule, would be as follows:
Firstly, taking the derivative of the area with respect to time gives us dA/dt. Secondly, for the right side of the equation, we have d(πr²)/dt which simplifies to 2πr(dr/dt). Therefore, the rate of change of the area of the circle with respect to time is dA/dt = 2πr(dr/dt).
This represents the relationship between the rate of change of the area and the rate of change of the radius over time. The rate at which the area of the circle expands depends directly on the radius of the circle and the rate at which the radius itself is changing.