1. **State the problem:** We want to find the rate of change of the area $A$ of a circle with respect to time $t$, denoted $\frac{dA}{dt}$, in terms of the rate of change of the radius $r$ with respect to time, $\frac{dr}{dt}$. 2. **Formula for the area of a circle:** The area $A$ of a circle with radius $r$ is given by $$A = \pi r^2$$ 3. **Differentiate both sides with respect to time $t$:** Using the chain rule, $$\frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = \pi \cdot 2r \cdot \frac{dr}{dt} = 2\pi r \frac{dr}{dt}$$ 4. **Interpretation:** This means the rate of change of the area depends on the current radius $r$ and how fast the radius is changing $\frac{dr}{dt}$. **Final answer:** $$\boxed{\frac{dA}{dt} = 2\pi r \frac{dr}{dt}}$$