The radius of a circle is decreasing at a constant rate of $0.2$ centimeters per second. In terms of the circumference $C$, what is the rate of the change of the area of the circle, in square centimeters per second?

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We are given the following quantities:

$$\frac{dr}{dt}=-0.1$$

We are asked to find $\dfrac{dA}{dt}$, which we can derive using the area formula.

$$A=\pi r^2$$ $$\frac{dA}{dt}=2\pi r \frac{dr}{dt}$$

Note that circumference is equal to $2\pi r$, so we can make the following substitution:

$$A = C \frac{dr}{dt}$$ $$A = \boxed{-(0.1)C}$$